%!PS-Adobe-2.0 %%Creator: dvips(k) 5.98 Copyright 2009 Radical Eye Software %%Title: arXiv:1006.3714v2 [gr-qc] 31 Aug 2010 %%CreationDate: Tue Apr 9 04:45:06 2019 %%Pages: 29 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -K1 -z HTTFourierMain.dvi -o %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 2019.04.09:0445 %%BeginProcSet: texc.pro 0 0 /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round 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%matrix defaultmatrix setmatrix 90 rotate /stampsize 20 def /Times-Roman findfont stampsize scalefont setfont currentfont /FontBBox get aload pop /pdf@top exch 1000 div stampsize mul def pop /pdf@bottom exch 1000 div stampsize mul def pop 256 -32 moveto currentpoint /pdf@lly exch pdf@bottom add def /pdf@llx exch 2 sub def 0.5 setgray (arXiv:1006.3714v2 [gr-qc] 31 Aug 2010) show currentpoint /pdf@ury exch pdf@top add def /pdf@urx exch 2 add def /pdfmark where{pop}{userdict /pdfmark /cleartomark load put}ifelse [ /H /I /Border [0 0 1] /BS <</S/D/D[2 6]/W 1>> /Color [0 1 1] /Action << /Subtype /URI /URI (http://arxiv.org/abs/1006.3714v2)>> /Subtype /Link /Rect[pdf@llx pdf@lly pdf@urx pdf@ury] /ANN pdfmark grestore TeXDict begin HPSdict begin 1 0 bop 0 0 a SDict begin [/Producer (dvips + Distiller)/Title ()/Subject ()/Creator (LaTeX with hyperref package)/Author ()/Keywords () /DOCINFO pdfmark end 0 0 a 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.1) cvn /DEST pdfmark end -158 -515 a Black Black -150 -166 a SDict begin [/Count 0/Dest (section*.1) cvn/Title (I Orbital Motion of compact binaries) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.2) cvn/Title (II The 1pN-accurate transverse and traceless far-zone radiation field for eccentric compact binaries) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.3) cvn/Title (III Ingredients for Fourier decomposition) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.4) cvn/Title (IV Decomposition of the elements appearing in the multipoles) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.5) cvn/Title (V Decomposition of the multipole moments) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 7/Dest (section*.6) cvn/Title (VI The effect of radiation reaction and the stationary phase approximation of the GW field) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.7) cvn/Title (A Radiative dynamics) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.8) cvn/Title (B SPA of the GW signal) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.9) cvn/Title (C Solution to the SPA condition equation, j=0) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.10) cvn/Title (D Solution to the SPA condition equation, j=0 and m=0. 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Fourier transformation of a slow-in-time signal) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.12) cvn/Title (F The limit et 0: the quasi-circular case) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.13) cvn/Title (G Some concluding remarks for the eccentric inspiral templates) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.14) cvn/Title (VII Conclusions and outlook) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.15) cvn/Title (VIII Acknowledgments) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.16) cvn/Title (A Proof of some summation formulas) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.17) cvn/Title (B Fourier representation of the sinmu and cosmu terms) /OUT pdfmark end -150 -166 a -150 -166 a SDict begin [/Count 0/Dest (section*.18) cvn/Title (C Fourier representation of products of two sin 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b(et)h(al.)36 b(pro)n(vided)24 b(a)g(formal)h(frequency-domain)-150 2922 y(decomp)r(osition)30 b(in)h(m)n(ultiples)g(of)f(t)n(w)n(o)g (irreducible)g(frequencies:)42 b(the)31 b Fw(r)l(adial)i FB(frequency)d Fv(f)2797 2934 y Fu(r)2833 2922 y FB(,)i(describing)d (the)i(elapsed)f(time)i(form)-150 3018 y(one)d(p)r(eriastron)f(passage) f(to)j(the)f(next,)h(and)f(the)h(frequency)f(asso)r(ciated)f(to)h(the)h (p)r(eriastron)e(adv)-5 b(ance)28 b(parameter,)h Fv(f)3672 3030 y Fu(\036)3715 3018 y FB(,)h(but)g(used)-150 3113 y(this)h(decomp)r(osition)f(only)g(for)g(the)h(computation)f(of)g(the)h (far-zone)f(energy)f(\035ux,)i(where)f(some)g(mathematical)g(asp)r (ects)g(made)h(it)-150 3209 y(unnecessary)26 b(to)h(compute)h(the)g(F) -7 b(ourier)26 b(co)r(e\036cien)n(ts)i(directly)-7 b(.)-67 3304 y(Ref.)40 b([)142 3304 y SDict begin H.S end 142 3304 a FB(7)184 3304 y SDict begin 11.5 H.L end 184 3304 a 184 3304 a SDict begin [/Subtype /Link/Dest (cite.Tessmer:Gopakumar:2006) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 184 3304 a FB(])28 b(pro)n(vided)g(n)n (umerical)f(insigh)n(ts)h(to)g(the)h(case)f(of)g(no)g(radiation)f (reaction)h(\(RR\))h(with)g(the)g(use)f(of)h(essen)n(tially)e(the)i (same)-150 3400 y(frequencies)e(that)h(app)r(eared)e(in)i([)932 3400 y SDict begin H.S end 932 3400 a FB(5)974 3400 y SDict begin 11.5 H.L end 974 3400 a 974 3400 a SDict begin [/Subtype /Link/Dest (cite.Galtsov:Matiukhin:Petukhov:1980) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 974 3400 a FB(].)-67 3495 y(Y)-7 b(unes)29 b(et)g(al.)41 b([)429 3495 y SDict begin H.S end 429 3495 a FB(6)471 3495 y SDict begin 11.5 H.L end 471 3495 a 471 3495 a SDict begin [/Subtype /Link/Dest (cite.Yunes:Arun:Berti:Will:2009) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 471 3495 a -1 w FB(])30 b(established)e(analytic)g(F)-7 b(ourier-domain)27 b(inspiral)h(templates)h(for)g(eccen)n(tric)f (binary)g(emission)g(for)g(the)i(exem-)-150 3591 y(plary)f(case)f(of)i (Newtonian)f(equations)g(of)g(motion)g(\(EOM\))i(and)e(the)h (leading-order)d(GW)j(amplitude.)43 b(Those)29 b(authors)f(used)i(the) -150 3686 y(expansion)21 b(of)h(the)g(Kepler)f(equation)g(and)h (related)f(trigonometric)f(function)i(com)n(binations)f(of)h(the)g (eccen)n(tric)f(anomaly)g(in)h(terms)f(of)-150 3782 y(Bessel)26 b(functions)h(and)g(applied)g(this)g(to)g(a)f(series)g(expansion)g(of)g (the)i(leading-order)c(GW)j(amplitude)h(in)f(the)g(orbital)f(eccen)n (tricit)n(y)-7 b(.)-67 3877 y(What)35 b(w)n(e)f(lik)n(e)g(to)g(do)g(in) h(this)g(article)f(is)g(the)h(generalisation)d(of)j(Y)-7 b(unes)34 b(et)h(al.)57 b(to)35 b(1pN)f(order)f(of)i(the)f(conserv)-5 b(ativ)n(e)33 b(orbital)-150 3973 y(dynamics.)64 b(The)37 b(w)n(ork)f(will)h(include,)j(in)d(addition)f(to)h(Y)-7 b(unes)37 b(et)h(al.,)h(the)e(e\033ect)g(of)g(p)r(eriastron)f(adv)-5 b(ance)36 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666 2284 a Black FB(9)p Black 707 2231 a SDict begin H.R end 707 2231 a 707 2284 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.1) cvn H.B /ANN pdfmark end 707 2284 a FB(\),)h Fv(\017)f FB(coun)n(ts)f(the)h(in)n(v)n(erse)e (order)g(of)i(the)g(sp)r(eed)g(of)f(ligh)n(t)h Fv(c)p FB(,)g Fv(\017)2575 2254 y Fo(2)2635 2284 y Fr(=)f Fv(c)2759 2254 y FA(\000)p Fo(2)2848 2284 y FB(,)h Fv(r)j FB(is)d(the)g(scaled)e (relativ)n(e)h(separation)-150 2380 y(of)e(the)h(t)n(w)n(o)e(b)r(ound)i (ob)5 b(jects,)22 b Fv(\036)g FB(is)f(the)h(orbital)e(phase,)i Fv(v)j FB(is)c(the)h(true)f(anomaly)-7 b(,)21 b Fp(k)h FB(is)f(the)h(p)r(eriastron)e(adv)-5 b(ance)20 b(parameter,)h Fv(e)3803 2392 y Fu(r)3861 2380 y FB(and)g Fv(e)4055 2392 y Fu(t)-150 2475 y FB(are)e(some)g(\020radial\021)25 b(and)19 b(\020time\021)26 b(eccen)n(tricit)n(y)-7 b(.)34 b(The)20 b(latter)f(app)r(ears)f(in)i(the)g(Kepler)f(equation)g (\(KE\),)i(Eq.)f(\()3230 2475 y SDict begin H.S end 3230 2475 a Black FB(2)p Black 3272 2422 a SDict begin H.R end 3272 2422 a 3272 2475 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.1) cvn H.B /ANN pdfmark end 3272 2475 a FB(\),)h(connecting)e(the)h(mean)-150 2571 y(anomaly)28 b Fp(M)h FB(\(whic)n(h)g(is)g(directly)g(prop)r(ortional)f(to)h(the)g (elapsed)g(time)g Fv(t)h FB(with)f(the)h(prop)r(ortionalit)n(y)d (factor)h Fv(n)e Fr(=)f(2)p Fv(\031)s(=P)3739 2583 y Fu(r)3802 2571 y Fr(=)g(2)p Fv(\031)s(f)4025 2583 y Fu(r)4061 2571 y FB(,)-150 2666 y(where)g Fv(P)141 2678 y Fu(r)205 2666 y FB(is)g(the)i(time)f(b)r(et)n(w)n(een)g(t)n(w)n(o)f(consecutiv)n (e)g(p)r(eriastron)g(passages\))f(to)i(the)g(eccen)n(tric)f(anomaly)g Fv(u)p FB(,)h(see)g([)3453 2666 y SDict begin H.S end 3453 2666 a FB(8)3495 2666 y SDict begin 11.5 H.L end 3495 2666 a 3495 2666 a SDict begin [/Subtype /Link/Dest (cite.Memmesheimer:Gopakumar:Schafer:2004) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 3495 2666 a -1 w FB(])g(for)g (geometrical)-150 2762 y(insigh)n(t.)45 b(The)30 b(shorthands)f Fv(\021)s FB(,)i Fv(\026)p FB(,)h(and)e Fq(j)p Fv(E)5 b Fq(j)30 b FB(denote)g(the)h(symmetric)f(mass)g(ration)f Fv(\021)i Fr(=)c(\()p Fv(m)2817 2774 y Fo(1)2854 2762 y Fv(m)2927 2774 y Fo(2)2964 2762 y Fr(\))p Fv(=)p Fr(\()p Fv(m)3143 2774 y Fo(1)3201 2762 y Fr(+)20 b Fv(m)3359 2774 y Fo(2)3396 2762 y Fr(\))3428 2732 y Fo(2)3496 2762 y FB(\()p Fv(m)3601 2774 y Fo(1)3669 2762 y FB(and)30 b Fv(m)3906 2774 y Fo(2)3973 2762 y FB(are)-150 2857 y(the)e(individual)g(masses)e(with)i Fv(m)918 2869 y Fu(t)970 2857 y Fr(=)23 b Fv(m)1131 2869 y Fo(1)1187 2857 y Fr(+)18 b Fv(m)1343 2869 y Fo(2)1380 2857 y FB(\),)28 b(the)g(reduced)f(mass,)g(and)g(the)h(absolute)f(v)-5 b(alue)27 b(of)h(the)g(orbital)e(binding)i(energy)-7 b(.)-67 2953 y(This)22 b(parameterization)e(is)i(essen)n(tial)f(for)g (the)i(computation)e(of)h(the)g(higher-order)e(time)i(deriv)-5 b(ativ)n(es)21 b(of)h(the)g(radiativ)n(e)f(m)n(ultip)r(ole)-150 3048 y(momen)n(ts.)35 b(W)-7 b(e)22 b(can)f(easily)g(obtain)g(them)h (using)f(the)h(KE.)g(Ho)n(w)f(the)h(orbital)f(parameterization)f (e\033ects)i(the)g(far-zone)e(gra)n(vitational)-150 3144 y(\034eld)28 b(will)g(b)r(e)g(depicted)g(in)f(the)h(next)g(sections.) -75 3420 y SDict begin H.S end -75 3420 a -75 3420 a SDict begin 10.5 H.A end -75 3420 a -75 3420 a SDict begin [/View [/XYZ H.V]/Dest (section*.2) cvn /DEST pdfmark end -75 3420 a Fs(I)r(I.)89 b(THE)30 b(1PN-A)n(CCURA)-7 b(TE)29 b(TRANSVERSE)i(AND)d(TRA)n(CELESS)i(F)-10 b(AR-ZONE)30 b(RADIA)-7 b(TION)30 b(FIELD)f(F)n(OR)1221 3507 y(ECCENTRIC)h(COMP)-7 b(A)n(CT)29 b(BINARIES)-67 3720 y FB(W)-7 b(e)24 b(start)f(collecting)g (the)h(expressions)e(of)h(the)h(transv)n(erse-traceless)c(far-zone)i (\034eld)i Fv(h)2688 3690 y Fo(TT)2810 3720 y FB(in)g(Junk)n(er)e(and)i (Sc)n(h\344fer)f([)3640 3720 y SDict begin H.S end 3640 3720 a FB(9)3682 3720 y SDict begin 11.5 H.L end 3682 3720 a 3682 3720 a SDict begin [/Subtype /Link/Dest (cite.Junker:Schafer:1992) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 3682 3720 a -1 w FB(].)36 b(They)23 b(are)-150 3836 y(giv)n(en)c(in)h(tensor-spherical)d (harmonics)h(and)i(it)g(is)f(remark)-5 b(able)18 b(to)i(note)f(their)h (structure)f(as)g(they)g(app)r(ear)g(as)g Fv(I)3333 3793 y Fo(\()p Fu(l)p Fo(\))3326 3861 y Fu(lm)3434 3836 y Fr(=)k Fv(e)3561 3806 y FA(\000)p Fu(im\036)3752 3836 y Fv(f)3793 3848 y Fu(lm)3877 3836 y Fr(\()p Fv(u)p Fr(\()p Fv(t)p Fr(\)\))-150 3956 y FB(and)k Fv(S)67 3913 y Fo(\()p Fu(l)p Fo(\))62 3981 y Fu(lm)170 3956 y Fr(=)22 b Fv(e)296 3926 y FA(\000)p Fu(im\036)488 3956 y Fv(g)528 3968 y Fu(lm)612 3956 y Fr(\()p Fv(u)p Fr(\()p Fv(t)p Fr(\)\))p FB(.)38 b(T)-7 b(aking)27 b(as)f(basis)536 4222 y SDict begin H.S end 536 4222 a 536 4222 a SDict begin 11.5 H.A end 536 4222 a 536 4222 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.10) cvn /DEST pdfmark end 536 4222 a Fv(h)584 4188 y Fo(TT)584 4243 y Fu(ij)732 4222 y Fr(=)50 b Fv(\017)881 4188 y Fo(4)928 4166 y Fv(G)p 928 4203 66 4 v 929 4279 a(R)1003 4080 y Fn(\()1151 4118 y Fo(2)1107 4143 y Fn(X)1070 4319 y Fu(m)p Fo(=)p FA(\000)p Fo(2)1282 4149 y(\(2\))1300 4230 y Fv(I)1364 4234 y Fo(2)p Fu(m)1474 4222 y Fv(T)1535 4182 y Fu(E)s Fo(2)p Fu(;)p Fo(2)p Fu(m)1523 4245 y(ij)1753 4222 y Fr(+)18 b Fv(\017)1870 4080 y Fn(")1999 4118 y Fo(2)1956 4143 y Fn(X)1918 4319 y Fu(m)p Fo(=)p FA(\000)p Fo(2)2129 4149 y(\(2\))2141 4230 y Fv(S)2212 4234 y Fo(2)p Fu(m)2322 4222 y Fv(T)2383 4182 y Fu(B)s Fo(2)p Fu(;)p Fo(2)p Fu(m)2371 4245 y(ij)2602 4222 y Fr(+)2766 4118 y Fo(3)2723 4143 y Fn(X)2685 4319 y Fu(m)p Fo(=)p FA(\000)p Fo(3)2897 4149 y(\(3\))2915 4230 y Fv(I)2979 4234 y Fo(3)p Fu(m)3089 4222 y Fv(T)3150 4182 y Fu(E)s Fo(2)p Fu(;)p Fo(3)p Fu(m)3138 4245 y(ij)3350 4080 y Fn(#)847 4517 y Fr(+)p Fv(\017)946 4483 y Fo(2)982 4375 y Fn(")1112 4413 y Fo(3)1068 4438 y Fn(X)1031 4614 y Fu(m)p Fo(=)p FA(\000)p Fo(3)1242 4444 y(\(3\))1254 4525 y Fv(S)1324 4529 y Fo(3)p Fu(m)1434 4517 y Fv(T)1495 4477 y Fu(B)s Fo(2)p Fu(;)p Fo(3)p Fu(m)1483 4540 y(ij)1715 4517 y Fr(+)1879 4413 y Fo(4)1835 4438 y Fn(X)1798 4614 y Fu(m)p Fo(=)p FA(\000)p Fo(4)2010 4444 y(\(4\))2028 4525 y Fv(I)2092 4529 y Fo(4)p Fu(m)2202 4517 y Fv(T)2263 4477 y Fu(E)s Fo(2)p Fu(;)p Fo(4)p Fu(m)2251 4540 y(ij)2463 4375 y Fn(#\))2592 4517 y Fv(;)p Black 1322 w FB(\(10\))p Black -150 4801 a(where)28 b Fv(R)h FB(is)f(the)h(distance)f(from)h(the)f (observ)n(er)f(to)h(the)h(source,)e Fv(T)1991 4761 y Fu(E)s Fo(2)p Fu(;lm)1979 4826 y(j)s(k)2208 4801 y FB(and)h Fv(T)2431 4761 y Fu(B)s Fo(2)p Fu(;lm)2419 4826 y(j)s(k)2649 4801 y FB(are)g(the)g(\020electric\021)35 b(and)28 b(\020magnetic\021) 34 b(tensor)-150 4916 y(spherical)29 b(harmonics)f([)621 4916 y SDict begin H.S end 621 4916 a FB(10)705 4916 y SDict begin 11.5 H.L end 705 4916 a 705 4916 a SDict begin [/Subtype /Link/Dest (cite.Thorne:1980) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 705 4916 a -1 w FB(],)i(the)g(sup)r(erscript)g Fr(\()p Fv(i)p Fr(\))f FB(allo)n(wing)g Fv(i)d Fr(=)g(2)p Fv(;)14 b Fr(3)p Fv(;)g Fr(4)28 b FB(denotes)h(the)h Fv(i)2656 4886 y Fo(th)2752 4916 y FB(time)h(di\033eren)n(tiation.)42 b(They)30 b(read)f(\025)g(up)-150 5011 y(to)e(the)h(required)f(o)n(v)n (erall)e(order)i(of)g Fv(\017)1031 4981 y Fo(6)1091 5011 y Fr(=)c Fv(c)1215 4981 y FA(\000)p Fo(6)1331 5011 y FB(in)28 b(Eq.)37 b(\()1620 5011 y SDict begin H.S end 1620 5011 a Black FB(10)p Black 1704 4958 a SDict begin H.R end 1704 4958 a 1704 5011 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 1704 5011 a FB(\))28 b(\025)1833 5011 y SDict begin H.S end 1833 5011 a 1833 5011 a SDict begin 11.5 H.A end 1833 5011 a 1833 5011 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11) cvn /DEST pdfmark end 1833 5011 a -60 5196 a Fo(\(2\))-42 5277 y Fv(I)22 5281 y Fo(22)92 5269 y Fr(=8)199 5138 y Fn(r)p 281 5138 112 4 v 291 5213 a Fr(2)p Fv(\031)p 291 5250 92 4 v 316 5326 a Fr(5)393 5269 y Fv(E)5 b(\026e)548 5235 y FA(\000)p Fo(2)p Fu(i\036)700 5127 y Fn(\()777 5213 y Fr(2)18 b Fq(\000)g Fr(2)p Fv(e)1001 5183 y Fo(2)1001 5233 y Fu(t)p 777 5250 261 4 v 858 5326 a Fv(A)920 5302 y Fo(2)1066 5269 y Fq(\000)1169 5213 y Fr(1)p 1159 5250 63 4 v 1159 5326 a Fv(A)1250 5269 y Fr(+)g(1)g(+)g Fq(j)p Fv(E)5 b Fq(j)p Fv(\017)1622 5235 y Fo(2)1659 5127 y Fn(")1717 5207 y Fr(2\(15)p Fv(\021)21 b Fq(\000)d Fr(82\))2150 5140 y Fn(\000)2187 5207 y Fv(e)2226 5177 y Fo(2)2226 5228 y Fu(t)2281 5207 y Fq(\000)g Fr(1)2406 5140 y Fn(\001)p 1717 5250 727 4 v 1989 5326 a Fr(21)p Fv(A)2135 5302 y Fo(3)2472 5269 y Fr(+)2575 5175 y Fo(3)p 2575 5189 34 4 v 2575 5236 a(7)2618 5208 y Fr(\(3)p Fv(\021)k Fq(\000)c Fr(1\))p Fv(e)2951 5178 y Fo(2)2951 5228 y Fu(t)3006 5208 y Fr(+)g(7)p Fv(\021)j Fq(\000)d Fr(15)p 2565 5250 794 4 v 2912 5326 a Fv(A)2974 5302 y Fo(2)p Black Black eop end end %%Page: 3 3 TeXDict begin HPSdict begin 3 2 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.3) cvn /DEST pdfmark end -158 -515 a Black 4043 -299 a FB(3)p Black 175 -21 a Fr(+)268 -77 y(3\(3)p Fv(\021)21 b Fq(\000)d Fr(1\))p 268 -40 335 4 v 363 36 a(14)p Fv(A)631 -21 y Fr(+)745 -77 y(3)p 724 -40 84 4 v 724 36 a(14)817 -21 y(\(1)h Fq(\000)f Fr(3)p Fv(\021)s Fr(\))1111 -163 y Fn(#\))157 262 y Fr(+)p Fv(i)265 120 y Fn(\()341 205 y Fr(2)383 132 y Fn(p)p 465 132 219 4 v 465 205 a Fr(1)g Fq(\000)g Fv(e)647 177 y Fo(2)647 226 y Fu(t)684 205 y Fv(e)723 217 y Fu(t)766 205 y Fr(sin\()p Fv(u)p Fr(\))p 341 242 640 4 v 610 318 a Fv(A)672 294 y Fo(2)1008 262 y Fr(+)g Fq(j)p Fv(E)5 b Fq(j)p Fv(\017)1237 227 y Fo(2)1288 120 y Fn(")1347 205 y Fr(2)1389 132 y Fn(p)p 1471 132 219 4 v 1471 205 a Fr(1)18 b Fq(\000)g Fv(e)1653 177 y Fo(2)1653 226 y Fu(t)1690 205 y Fr(\(79)g Fq(\000)g Fr(6)p Fv(\021)s Fr(\))p Fv(e)2064 217 y Fu(t)2106 205 y Fr(sin\()p Fv(u)p Fr(\))p 1347 242 974 4 v 1742 318 a(21)p Fv(A)1888 294 y Fo(3)2349 262 y Fr(+)2442 200 y Fv(e)2481 212 y Fu(t)2524 200 y Fr(sin\()p Fv(u)p Fr(\))2752 133 y Fn(\000)2789 200 y Fr(\(9)p Fv(\021)k Fq(\000)c Fr(3\))p Fv(e)3122 170 y Fo(2)3122 221 y Fu(t)3177 200 y Fr(+)g(19)p Fv(\021)j Fq(\000)d Fr(25)3573 133 y Fn(\001)p 2442 242 1169 4 v 2804 332 a Fr(7)p Fv(A)2908 308 y Fo(2)2945 259 y Fn(p)p 3028 259 219 4 v 73 x Fr(1)g Fq(\000)g Fv(e)3210 303 y Fo(2)3210 353 y Fu(t)3620 120 y Fn(#\))3749 262 y Fv(;)3895 262 y SDict begin H.S end 3895 262 a 3895 262 a SDict begin 14.5 H.A end 3895 262 a 3895 262 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11a) cvn /DEST pdfmark end 3895 262 a FB(\(11a\))p -150 369 4235 4 v -60 421 a Fo(\(2\))-42 501 y Fv(I)22 505 y Fo(21)92 493 y Fr(=0)c Fv(;)3890 493 y SDict begin H.S end 3890 493 a 3890 493 a SDict begin 14.5 H.A end 3890 493 a 3890 493 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11b) cvn /DEST pdfmark end 3890 493 a FB(\(11b\))p -150 525 4235 4 v -60 598 a Fo(\(2\))-42 679 y Fv(I)22 683 y Fo(20)92 671 y Fr(=)k Fq(\000)g Fr(16)342 549 y Fn(r)p 424 549 103 4 v 451 614 a Fv(\031)p 434 652 84 4 v 434 728 a Fr(15)527 671 y Fv(E)5 b(\026)643 529 y Fn(\()710 671 y Fr(1)18 b Fq(\000)929 614 y Fr(1)p 863 652 175 4 v 863 728 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1066 671 y(+)g Fq(j)p Fv(E)5 b Fq(j)p Fv(\017)1295 636 y Fo(2)1346 529 y Fn(")1404 609 y Fr(2\()p Fv(\021)22 b Fq(\000)c Fr(26\))1754 542 y Fn(\000)1791 609 y Fv(e)1830 579 y Fo(2)1830 630 y Fu(t)1885 609 y Fq(\000)g Fr(1)2010 542 y Fn(\001)p 1404 652 644 4 v 1599 728 a Fr(7)p Fv(A)p Fr(\()p Fv(u)p Fr(\))1815 704 y Fo(3)2076 671 y Fr(+)2169 614 y(4)g Fq(\000)g Fr(19)p Fv(\021)p 2169 652 271 4 v 2178 728 a Fr(7)p Fv(A)p Fr(\()p Fv(u)p Fr(\))2394 704 y Fo(2)2468 671 y Fr(+)2561 614 y(51)p Fv(\021)i Fq(\000)f Fr(115)p 2561 652 354 4 v 2609 728 a(14)p Fv(A)p Fr(\()p Fv(u)p Fr(\))2942 671 y Fq(\000)3056 614 y Fr(3)p 3035 652 84 4 v 3035 728 a(14)3128 671 y(\(3)p Fv(\021)j Fq(\000)c Fr(1\))3422 529 y Fn(#\))3551 671 y Fv(;)3900 671 y SDict begin H.S end 3900 671 a 3900 671 a SDict begin 14.5 H.A end 3900 671 a 3900 671 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11c) cvn /DEST pdfmark end 3900 671 a FB(\(11c\))p -150 778 4235 4 v -61 830 a Fo(\(2\))-49 911 y Fv(S)22 915 y Fo(22)92 903 y Fr(=0)c Fv(;)3890 903 y SDict begin H.S end 3890 903 a 3890 903 a SDict begin 14.5 H.A end 3890 903 a 3890 903 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11d) cvn /DEST pdfmark end 3890 903 a FB(\(11d\))p -150 934 4235 4 v -61 1011 a Fo(\(2\))-49 1092 y Fv(S)22 1096 y Fo(21)92 1084 y Fr(=)167 1020 y(32)251 955 y Fn(p)p 333 955 61 4 v 343 987 a Fu(\031)p 343 1001 41 4 v 347 1048 a Fo(5)407 1020 y Fr(\()q Fv(m)513 1032 y Fo(1)568 1020 y Fq(\000)k Fv(m)724 1032 y Fo(2)761 1020 y Fr(\))d(\()p Fq(\000)p Fv(E)5 b Fr(\))1003 978 y Fo(3)p Fu(=)p Fo(2)1107 1020 y Fv(\021)s(e)1190 990 y FA(\000)p Fu(i\036)1309 947 y Fn(p)p 1392 947 219 4 v 73 x Fr(1)18 b Fq(\000)g Fv(e)1574 991 y Fo(2)1574 1040 y Fu(t)p 167 1065 1445 4 v 762 1141 a Fr(3)p Fv(A)p Fr(\()p Fv(u)p Fr(\))978 1117 y Fo(2)1635 1084 y Fv(;)3900 1084 y SDict begin H.S end 3900 1084 a 3900 1084 a SDict begin 14.5 H.A end 3900 1084 a 3900 1084 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11e) cvn /DEST pdfmark end 3900 1084 a FB(\(11e\))p -150 1165 4235 4 v -61 1217 a Fo(\(2\))-49 1297 y Fv(S)22 1301 y Fo(20)92 1289 y Fr(=0)c Fv(;)3905 1289 y SDict begin H.S end 3905 1289 a 3905 1289 a SDict begin 14.5 H.A end 3905 1289 a 3905 1289 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11f) cvn /DEST pdfmark end 3905 1289 a FB(\(11f)6 b(\))p -150 1321 4235 4 v -60 1394 a Fo(\(3\))-42 1475 y Fv(I)22 1479 y Fo(33)92 1467 y Fr(=8)199 1336 y Fn(r)p 281 1336 112 4 v 291 1411 a Fr(2)p Fv(\031)p 291 1448 92 4 v 295 1524 a Fr(21)393 1467 y(\()p Fq(\000)p Fv(E)f Fr(\))588 1432 y Fo(3)p Fu(=)p Fo(2)706 1467 y Fr(\()q Fv(m)812 1479 y Fo(1)867 1467 y Fq(\000)18 b Fv(m)1023 1479 y Fo(2)1060 1467 y Fr(\))d Fv(\017\021)s(e)1224 1432 y FA(\000)p Fo(3)p Fu(i\036)1376 1325 y Fn(\()1442 1467 y Fv(e)1481 1479 y Fu(t)1524 1467 y Fr(sin\()p Fv(u)p Fr(\))1752 1325 y Fn( )1828 1405 y Fr(4)1884 1338 y Fn(\000)1921 1405 y Fv(e)1960 1375 y Fo(2)1960 1426 y Fu(t)2016 1405 y Fq(\000)j Fr(1)2141 1338 y Fn(\001)p 1828 1448 351 4 v 1897 1524 a Fv(A)p Fr(\()p Fv(u)p Fr(\))2071 1500 y Fo(3)2207 1467 y Fq(\000)2366 1411 y Fr(1)p 2300 1448 175 4 v 2300 1524 a Fv(A)p Fr(\()p Fv(u)p Fr(\))2484 1325 y Fn(!)175 1749 y Fr(+)g Fv(i)301 1647 y Fn(q)p 384 1647 219 4 v 102 x Fr(1)g Fq(\000)g Fv(e)566 1720 y Fo(2)566 1769 y Fu(t)616 1632 y Fn(\022)687 1693 y Fr(4)g Fq(\000)g Fr(4)p Fv(e)911 1663 y Fo(2)911 1713 y Fu(t)p 687 1730 261 4 v 712 1806 a Fv(A)p Fr(\()p Fv(u)p Fr(\))886 1782 y Fo(3)976 1749 y Fr(+)1136 1693 y(3)p 1069 1730 175 4 v 1069 1806 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1272 1749 y Fq(\000)1471 1693 y Fr(5)p 1365 1730 254 4 v 1365 1806 a(2)p Fv(A)p Fr(\()p Fv(u)p Fr(\))1581 1782 y Fo(2)1628 1632 y Fn(\023)1689 1607 y(\))1770 1749 y Fv(;)3894 1749 y SDict begin H.S end 3894 1749 a 3894 1749 a SDict begin 14.5 H.A end 3894 1749 a 3894 1749 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11g) cvn /DEST pdfmark end 3894 1749 a FB(\(11g)q(\))p -150 1856 4235 4 v -60 1908 a Fo(\(3\))-42 1989 y Fv(I)22 1993 y Fo(32)92 1981 y Fr(=0)c Fv(;)3890 1981 y SDict begin H.S end 3890 1981 a 3890 1981 a SDict begin 14.5 H.A end 3890 1981 a 3890 1981 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11h) cvn /DEST pdfmark end 3890 1981 a FB(\(11h\))p -150 2013 4235 4 v -60 2122 a Fo(\(3\))-42 2203 y Fv(I)22 2207 y Fo(31)92 2195 y Fr(=8)199 2064 y Fn(r)p 281 2064 112 4 v 291 2139 a Fr(2)p Fv(\031)p 291 2176 92 4 v 295 2252 a Fr(35)393 2195 y(\()p Fq(\000)p Fv(E)5 b Fr(\))588 2160 y Fo(3)p Fu(=)p Fo(2)706 2195 y Fr(\()q Fv(m)812 2207 y Fo(1)867 2195 y Fq(\000)18 b Fv(m)1023 2207 y Fo(2)1060 2195 y Fr(\))d Fv(\021)s(e)1190 2160 y FA(\000)p Fu(i\036)1323 2024 y Fn(8)1323 2099 y(<)1323 2249 y(:)1406 2139 y Fv(e)1445 2151 y Fu(t)1488 2139 y Fr(sin\()p Fv(u)p Fr(\))p 1406 2176 296 4 v 1467 2252 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1731 2195 y Fq(\000)1824 2108 y Fv(i)1853 2042 y Fq(p)p 1921 2042 249 4 v 1921 2108 a Fr(1)j Fq(\000)g Fv(e)2103 2120 y Fu(t)2132 2084 y Fo(2)2183 2016 y Fn(\020)2233 2108 y Fr(1)g Fq(\000)2456 2076 y Fo(5)p 2386 2090 175 4 v 2386 2137 a(6)p Fu(A)p Fo(\()p Fu(u)p Fo(\))2570 2016 y Fn(\021)p 1824 2176 796 4 v 2134 2252 a Fv(A)p Fr(\()p Fv(u)p Fr(\))2630 2024 y Fn(9)2630 2099 y(=)2630 2249 y(;)2731 2195 y Fv(;)3914 2195 y SDict begin H.S end 3914 2195 a 3914 2195 a SDict begin 14.5 H.A end 3914 2195 a 3914 2195 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11i) cvn /DEST pdfmark end 3914 2195 a FB(\(11i\))p -150 2327 4235 4 v -60 2379 a Fo(\(3\))-42 2459 y Fv(I)22 2463 y Fo(30)92 2451 y Fr(=0)c Fv(;)3911 2451 y SDict begin H.S end 3911 2451 a 3911 2451 a SDict begin 14.5 H.A end 3911 2451 a 3911 2451 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11j) cvn /DEST pdfmark end 3911 2451 a FB(\(11j\))p -150 2483 4235 4 v -61 2535 a Fo(\(3\))-49 2616 y Fv(S)22 2620 y Fo(33)92 2608 y Fr(=0)g Fv(;)3893 2608 y SDict begin H.S end 3893 2608 a 3893 2608 a SDict begin 14.5 H.A end 3893 2608 a 3893 2608 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11k) cvn /DEST pdfmark end 3893 2608 a FB(\(11k\))p -150 2640 4235 4 v -61 2704 a Fo(\(3\))-49 2785 y Fv(S)22 2789 y Fo(32)92 2777 y Fr(=)p Fv(e)196 2743 y FA(\000)p Fu(i)d Fo(2)g Fu(\036)394 2721 y Fr(8)p 394 2758 42 4 v 394 2834 a(3)446 2647 y Fn(r)p 529 2647 112 4 v 539 2721 a Fr(2)p Fv(\031)p 539 2758 92 4 v 564 2834 a Fr(7)641 2777 y Fv(E)707 2743 y Fo(2)744 2675 y Fn(q)p 827 2675 219 4 v 102 x Fr(1)18 b Fq(\000)g Fv(e)1009 2749 y Fo(2)1009 2798 y Fu(t)1046 2777 y Fv(\026)p Fr(\(1)g Fq(\000)g Fr(3)p Fv(\021)s Fr(\))1399 2649 y Fn(\000)1437 2716 y Fv(e)1476 2728 y Fu(t)1519 2716 y Fr(sin\()p Fv(u)p Fr(\))g Fq(\000)g Fr(4)p Fv(i)1905 2649 y Fq(p)p 1974 2649 249 4 v 67 x Fr(1)g Fq(\000)g Fv(e)2156 2728 y Fu(t)2185 2692 y Fo(2)2222 2649 y Fn(\001)p 1399 2758 861 4 v 1724 2834 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1898 2810 y Fo(3)2284 2777 y Fv(;)3914 2777 y SDict begin H.S end 3914 2777 a 3914 2777 a SDict begin 14.5 H.A end 3914 2777 a 3914 2777 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11l) cvn /DEST pdfmark end 3914 2777 a FB(\(11l\))p -150 2858 4235 4 v -61 2910 a Fo(\(3\))-49 2991 y Fv(S)22 2995 y Fo(31)92 2983 y Fr(=0)c Fv(;)3867 2983 y SDict begin H.S end 3867 2983 a 3867 2983 a SDict begin 14.5 H.A end 3867 2983 a 3867 2983 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11m) cvn /DEST pdfmark end 3867 2983 a FB(\(11m\))p -150 3015 4235 4 v -61 3070 a Fo(\(3\))-49 3151 y Fv(S)22 3155 y Fo(30)92 3143 y Fr(=)k Fq(\000)g Fr(16)342 3022 y Fn(r)p 424 3022 145 4 v 471 3087 a Fv(\031)p 434 3124 125 4 v 434 3200 a Fr(105)569 3143 y Fv(E)635 3109 y Fo(2)672 3143 y Fv(e)711 3155 y Fu(t)740 3064 y Fn(p)p 823 3064 249 4 v 79 x Fr(1)g Fq(\000)g Fv(e)1005 3155 y Fu(t)1034 3119 y Fo(2)1071 3143 y Fv(\026)p Fr(\(1)g Fq(\000)g Fr(3)p Fv(\021)s Fr(\))1424 3087 y(sin\()p Fv(u)p Fr(\))p 1424 3124 215 4 v 1425 3200 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1599 3176 y Fo(3)3890 3143 y SDict begin H.S end 3890 3143 a 3890 3143 a SDict begin 14.5 H.A end 3890 3143 a 3890 3143 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11n) cvn /DEST pdfmark end 3890 3143 a FB(\(11n\))p -150 3224 4235 4 v -60 3300 a Fo(\(4\))-42 3381 y Fv(I)22 3385 y Fo(44)92 3373 y Fr(=)167 3317 y(4)p 167 3354 42 4 v 167 3430 a(9)218 3243 y Fn(r)p 301 3243 112 4 v 311 3317 a Fr(2)p Fv(\031)p 311 3354 92 4 v 336 3430 a Fr(7)413 3373 y Fv(E)479 3339 y Fo(2)516 3373 y Fv(e)555 3339 y FA(\000)p Fo(4)p Fu(i\036)707 3373 y Fv(\026)p Fr(\(1)h Fq(\000)f Fr(3)p Fv(\021)s Fr(\))1051 3231 y Fn(\()1128 3312 y Fr(48)1226 3244 y Fn(\000)1262 3312 y Fr(1)g Fq(\000)g Fv(e)1444 3324 y Fu(t)1473 3282 y Fo(2)1511 3244 y Fn(\001)1549 3262 y Fo(2)p 1128 3354 459 4 v 1251 3430 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1425 3406 y Fo(4)1614 3373 y Fq(\000)1707 3312 y Fr(27)1805 3244 y Fn(\000)1842 3312 y Fr(1)g Fq(\000)g Fv(e)2024 3324 y Fu(t)2053 3282 y Fo(2)2090 3244 y Fn(\001)p 1707 3354 422 4 v 1812 3430 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1986 3406 y Fo(3)2157 3373 y Fr(+)2250 3317 y(43)f Fq(\000)h Fr(48)p Fv(e)2557 3329 y Fu(t)2585 3287 y Fo(2)p 2250 3354 373 4 v 2330 3430 a Fv(A)p Fr(\()p Fv(u)p Fr(\))2504 3406 y Fo(2)2651 3373 y Fq(\000)2810 3317 y Fr(6)p 2744 3354 175 4 v 2744 3430 a Fv(A)p Fr(\()p Fv(u)p Fr(\))2947 3373 y(+)g(6)175 3742 y(+)268 3631 y(6)p Fv(ie)378 3643 y Fu(t)406 3564 y Fq(p)p 475 3564 249 4 v 67 x Fr(1)g Fq(\000)g Fv(e)657 3643 y Fu(t)686 3607 y Fo(2)737 3631 y Fr(sin\()p Fv(u)p Fr(\))965 3513 y Fn(\022)1036 3578 y Fo(8)1069 3584 y Fr(\()1102 3578 y Fo(1)p FA(\000)p Fu(e)1218 3586 y Fm(t)1246 3553 y Fl(2)1278 3584 y Fr(\))p 1036 3611 275 4 v 1087 3659 a Fu(A)p Fo(\()p Fu(u)p Fo(\))1228 3643 y Fl(2)1339 3631 y Fr(+)1486 3598 y Fo(1)p 1432 3612 142 4 v 1432 3659 a Fu(A)p Fo(\()p Fu(u)p Fo(\))1601 3631 y Fr(+)g(4)1726 3513 y Fn(\023)p 268 3723 1519 4 v 922 3799 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1096 3775 y Fo(2)1797 3600 y Fn(\))1877 3742 y Fv(;)3895 3742 y SDict begin H.S end 3895 3742 a 3895 3742 a SDict begin 14.5 H.A end 3895 3742 a 3895 3742 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11o) cvn /DEST pdfmark end 3895 3742 a FB(\(11o\))p -150 3849 4235 4 v -60 3901 a Fo(\(4\))-42 3982 y Fv(I)22 3986 y Fo(43)92 3974 y Fr(=0)c Fv(;)3890 3974 y SDict begin H.S end 3890 3974 a 3890 3974 a SDict begin 14.5 H.A end 3890 3974 a 3890 3974 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11p) cvn /DEST pdfmark end 3890 3974 a FB(\(11p\))p -150 4006 4235 4 v -60 4115 a Fo(\(4\))-42 4195 y Fv(I)22 4199 y Fo(42)92 4187 y Fr(=)187 4131 y(8)p 167 4168 84 4 v 167 4244 a(63)260 4115 y Fq(p)p 329 4115 92 4 v 72 x Fr(2)p Fv(\031)s(E)487 4153 y Fo(2)524 4187 y Fv(\026)p Fr(\(1)k Fq(\000)g Fr(3)p Fv(\021)s Fr(\))p Fv(e)906 4153 y FA(\000)p Fo(2)p Fu(i\036)1058 4045 y Fn(\()1125 4187 y Fq(\000)1200 4131 y Fr(7)g Fq(\000)g Fr(12)p Fv(e)1466 4143 y Fu(t)1494 4101 y Fo(2)p 1200 4168 332 4 v 1260 4244 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1434 4220 y Fo(2)1559 4187 y Fr(+)1652 4126 y(3)1708 4059 y Fn(\000)1746 4126 y Fr(1)g Fq(\000)g Fv(e)1928 4138 y Fu(t)1957 4096 y Fo(2)1994 4059 y Fn(\001)p 1652 4168 380 4 v 1736 4244 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1910 4220 y Fo(3)2060 4187 y Fr(+)2220 4131 y(6)p 2153 4168 175 4 v 2153 4244 a Fv(A)p Fr(\()p Fv(u)p Fr(\))2356 4187 y Fq(\000)g Fr(6)g Fq(\000)2592 4101 y Fr(3)p Fv(ie)2702 4113 y Fu(t)2730 4035 y Fq(p)p 2799 4035 249 4 v 66 x Fr(1)g Fq(\000)g Fv(e)2981 4113 y Fu(t)3010 4077 y Fo(2)3061 4009 y Fn(\020)3175 4068 y Fo(1)p 3121 4082 142 4 v 3121 4130 a Fu(A)p Fo(\()p Fu(u)p Fo(\))3290 4101 y Fr(+)g(4)3415 4009 y Fn(\021)3478 4101 y Fr(sin\()p Fv(u)p Fr(\))p 2592 4168 1101 4 v 3036 4244 a Fv(A)p Fr(\()p Fv(u)p Fr(\))3210 4220 y Fo(2)3702 4045 y Fn(\))3783 4187 y Fv(;)3893 4187 y SDict begin H.S end 3893 4187 a 3893 4187 a SDict begin 14.5 H.A end 3893 4187 a 3893 4187 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11q) cvn /DEST pdfmark end 3893 4187 a FB(\(11q\))p -150 4295 4235 4 v -60 4346 a Fo(\(4\))-42 4427 y Fv(I)22 4431 y Fo(41)92 4419 y Fr(=0)c Fv(;)3904 4419 y SDict begin H.S end 3904 4419 a 3904 4419 a SDict begin 14.5 H.A end 3904 4419 a 3904 4419 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11r) cvn /DEST pdfmark end 3904 4419 a FB(\(11r\))p -150 4451 4235 4 v -60 4524 a Fo(\(4\))-42 4605 y Fv(I)22 4609 y Fo(40)92 4597 y Fr(=)187 4540 y(8)p 167 4577 84 4 v 167 4653 a(21)260 4475 y Fn(r)p 343 4475 71 4 v 353 4540 a Fv(\031)p 353 4577 51 4 v 357 4653 a Fr(5)413 4597 y Fv(E)479 4562 y Fo(2)516 4597 y Fv(\026)p Fr(\(1)k Fq(\000)g Fr(3)p Fv(\021)s Fr(\))873 4455 y Fn(\()950 4535 y Fr(5)1006 4468 y Fn(\000)1044 4535 y Fr(1)g Fq(\000)g Fv(e)1226 4547 y Fu(t)1254 4505 y Fo(2)1292 4468 y Fn(\001)p 950 4577 380 4 v 1034 4653 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1208 4630 y Fo(3)1358 4597 y Fq(\000)1518 4540 y Fr(6)p 1451 4577 175 4 v 1451 4653 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1654 4597 y Fq(\000)1832 4540 y Fr(5)p 1747 4577 212 4 v 1747 4653 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1921 4630 y Fo(2)1987 4597 y Fr(+)g(6)2112 4455 y Fn(\))2206 4597 y Fv(;)3904 4597 y SDict begin H.S end 3904 4597 a 3904 4597 a SDict begin 14.5 H.A end 3904 4597 a 3904 4597 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11s) cvn /DEST pdfmark end 3904 4597 a FB(\(11s\))-150 4843 y(where)34 b Fv(\026)h FB(is)g(the)h(reduced)e(mass)g(and)h Fv(A)p Fr(\()p Fv(u)p Fr(\))h(:=)f(1)23 b Fq(\000)g Fv(e)1645 4855 y Fu(t)1688 4843 y Fr(cos)13 b Fv(u)p FB(.)58 b(The)35 b(ab)r(o)n(v)n(e)f(m)n (ultip)r(ole)h(momen)n(ts)g(con)n(tain)f(terms)g(with)i(certain)-150 4938 y(equal)28 b(structure.)39 b(Ev)-5 b(aluating)28 b(the)h(exp)r(onen)n(tial)e(functions)i Fv(e)1867 4908 y FA(\000)p Fu(im\036)2074 4938 y FB(with)f Fv(m)d Fr(=)f(1)p Fv(;)14 b Fr(2)p Fv(;)g Fr(3)p Fv(;)g Fr(4)26 b FB(to)i Fr(1)p FB(pN)h(accuracy)-7 b(,)27 b(one)h(obtains)f(factors)-150 5034 y(of)g(the)h(form)f Fv(A)p Fr(\()p Fv(u)p Fr(\))457 5004 y FA(\000)p Fu(n)581 5034 y FB(and)g Fr(sin\()p Fv(u)p Fr(\))p Fv(=)-5 b(A)p Fr(\()p Fv(u)p Fr(\))1167 5004 y Fu(n)1213 5034 y FB(.)1273 5004 y Fk(1)1347 5034 y FB(Those)26 b(terms)h(can)g(b)r(e)g(expanded)g(\034rst)g(in)g Fr(sin)h FB(and)f Fr(cos)f FB(series)g(of)h Fv(u)g FB(and)g(afterw)n (ards)p Black -150 5203 300 4 v -150 5413 a SDict begin H.S end -150 5413 a -150 5413 a SDict begin 9.5 H.A end -150 5413 a -150 5413 a SDict begin [/View [/XYZ H.V]/Dest (footnote@1) cvn /DEST pdfmark end -150 5413 a -138 5390 a Fj(1)-56 5413 y Fi(sin)11 b Fh(u)24 b Fx(and)g Fi(cos)12 b Fh(u)24 b Fx(of)f(higher)i(p)r(o)n(w)n(er)f(than)h(one)f (in)g(the)g(denominator)h(can)f(alw)n(a)n(ys)g(b)r(e)g(expressed)h (again)f(in)g Fh(A)p Fi(\()p Fh(u)p Fi(\))g Fx(and)g Fi(sin)12 b Fh(u)p Fx(.)p Black Black Black eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.4) cvn /DEST pdfmark end -158 -515 a Black 4043 -299 a FB(4)p Black -150 -83 a(in)30 b(terms)g(of)f Fp(M)h FB(with)g(the)h(help)f(of)f (Eqs.)44 b(\()1247 -83 y SDict begin H.S end 1247 -83 a Black FB(16)p Black 1330 -136 a SDict begin H.R end 1330 -136 a 1330 -83 a SDict 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1205 a 1441 1148 a Fr(1)p 1223 1186 478 4 v 1223 1262 a(\(1)18 b Fq(\000)g Fv(e)c Fr(cos)e Fv(u)p Fr(\))1655 1238 y Fu(n)1772 1205 y Fr(=)60 b(1)18 b(+)g Fv(b)2076 1162 y Fo(\()p Fu(n)p Fo(\))2076 1227 y(0)2192 1205 y Fr(+)2302 1101 y FA(1)2275 1126 y Fn(X)2277 1303 y Fu(j)s Fo(=1)2408 1205 y Fv(b)2444 1162 y Fo(\()p Fu(n)p Fo(\))2444 1228 y Fu(j)2555 1205 y Fr(cos)13 b Fv(j)5 b(u)14 b(;)p Black 1133 w FB(\(12\))p Black 1577 1492 a Fv(b)1613 1449 y Fo(\()p Fu(n)p Fo(\))1613 1515 y(0)1772 1492 y Fr(=)1924 1389 y FA(1)1897 1414 y Fn(X)1904 1590 y Fu(i)p Fo(=1)2031 1492 y Fv(\014)2082 1449 y Fo(\()p Fu(n)p Fo(\))2078 1516 y(2)p Fu(i;i)2196 1492 y Fv(;)p Black 1718 w FB(\(13\))p Black 1555 1769 a Fv(b)1591 1726 y Fo(\()p Fu(n)p Fo(\))1591 1792 y Fu(j)s(>)p Fo(0)1772 1769 y Fr(=)1924 1665 y FA(1)1897 1690 y Fn(X)1904 1867 y Fu(i)p Fo(=0)2031 1769 y Fv(\014)2082 1726 y Fo(\()p Fu(n)p Fo(\))2078 1792 y Fu(j)s Fo(+2)p Fu(i;i)2282 1769 y Fr(+)k Fv(\014)2416 1726 y Fo(\()p Fu(n)p Fo(\))2412 1792 y Fu(j)s Fo(+2)p Fu(i;j)s Fo(+)p Fu(i)2694 1769 y Fv(;)p Black 1220 w FB(\(14\))p Black 1544 2029 a Fv(\014)1595 1986 y Fo(\()p Fu(n)p Fo(\))1591 2054 y Fu(m;k)1760 2029 y Fr(:=)1907 1973 y(\()p Fv(n)h Fr(+)f Fv(m)g Fq(\000)g Fr(1\)!)p 1907 2010 455 4 v 1995 2086 a(\()p Fv(n)g Fq(\000)g Fr(1\)!)2423 1973 y(1)p 2396 2010 96 4 v 2396 2086 a Fv(m)p Fr(!)2527 1973 y Fv(e)2566 1943 y Fu(m)p 2526 2010 105 4 v 2526 2086 a Fr(2)2568 2062 y Fu(m)2640 1912 y Fn(\022)2701 1973 y Fv(m)2715 2086 y(k)2774 1912 y Fn(\023)1772 2291 y Fr(=)60 b Fv(e)1936 2256 y Fu(m)2055 2234 y Fr(1)p 2023 2271 V 2023 2347 a(2)2065 2323 y Fu(m)2151 2173 y Fn(\022)2212 2234 y Fv(m)2226 2347 y(k)2285 2173 y Fn(\023)2398 2234 y Fr(1)p 2370 2271 96 4 v 2370 2347 a Fv(m)p Fr(!)2504 2187 y Fu(m)p FA(\000)p Fo(1)2523 2212 y Fn(Y)2522 2388 y Fu(i)p Fo(=0)2648 2291 y Fr(\()p Fv(n)18 b Fr(+)g Fv(i)p Fr(\))c Fv(:)p Black 1008 w FB(\(15\))p Black 58 2547 a(Equation)28 b(\()449 2547 y SDict begin H.S end 449 2547 a Black FB(12)p 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b(functions)j(of)g(m)n(ultiples)f(of)h(the)g(eccen)n(tric)f(anomaly)f Fv(u)h FB(in)h(terms)f(of)h(Bessel)f(functions)g(\(see)h([)3615 2803 y SDict begin H.S end 3615 2803 a FB(11)3699 2803 y SDict begin 11.5 H.L end 3699 2803 a 3699 2803 a SDict begin [/Subtype /Link/Dest (cite.Watson:1980) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 3699 2803 a -1 w FB(],)g(p.)37 b(555\),)380 3049 y SDict begin H.S end 380 3049 a 380 3049 a SDict begin 11.5 H.A end 380 3049 a 380 3049 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.16) cvn /DEST pdfmark end 380 3049 a 14 w Fr(sin)14 b Fv(mu)61 b Fr(=)f Fv(m)934 2945 y FA(1)907 2970 y Fn(X)904 3146 y Fu(n)p Fo(=1)1057 2993 y Fr(1)p 1053 3030 50 4 v 1053 3106 a Fv(n)1127 3049 y Fr(\()p Fv(J)1205 3061 y Fu(n)p FA(\000)p Fu(m)1361 3049 y Fr(\()p Fv(ne)1482 3061 y Fu(t)1511 3049 y Fr(\))19 b(+)f Fv(J)1691 3061 y Fu(n)p Fo(+)p Fu(m)1846 3049 y Fr(\()p Fv(ne)1967 3061 y Fu(t)1996 3049 y Fr(\)\))c(sin\()p Fv(n)p 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2063 y FB(in)30 b(the)h(phase)f(exp)r(onen)n(tials) f(of)h(Eqs.)45 b(\()1139 2063 y SDict begin H.S end 1139 2063 a Black FB(11a)p Black 1264 2010 a SDict begin H.R end 1264 2010 a 1264 2063 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.11a) cvn H.B /ANN pdfmark end 1264 2063 a FB(\))31 b(-)f(\()1417 2063 y SDict begin H.S end 1417 2063 a Black FB(11s)p Black 1533 2010 a SDict begin H.R end 1533 2010 a 1533 2063 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.11s) cvn H.B /ANN pdfmark end 1533 2063 a FB(\))g(\))h(whic)n(h)f(w)n(e)g(shall)f(call)h Fv(m)h FB(in)f(the)h(m)n(ultip)r(oles)f(dictates)g(terms)g(of)g(the)h(form) -150 2158 y Fv(e)-111 2128 y FA(\000)p Fu(im\036)67 2158 y FB(,)d(whic)n(h)f(ha)n(v)n(e)g(to)g(b)r(e)h(expanded)f(in)h Fv(\017)p FB(.)276 2397 y SDict begin H.S end 276 2397 a 276 2397 a SDict begin 11.5 H.A end 276 2397 a 276 2397 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.26) cvn /DEST pdfmark end 276 2397 a Fv(v)316 2409 y Fu(e)347 2417 y Fm(t)429 2397 y Fr(:=)50 b(2)27 b(arctan)881 2280 y Fn(\024)924 2272 y(r)p 1007 2272 231 4 v 1017 2340 a Fr(1)18 b(+)g Fv(e)1199 2352 y Fu(t)p 1017 2378 211 4 v 1017 2454 a Fr(1)g Fq(\000)g Fv(e)1199 2466 y Fu(t)1266 2397 y Fr(tan)1410 2340 y Fv(u)p 1410 2378 48 4 v 1413 2454 a Fr(2)1467 2280 y Fn(\025)1539 2397 y Fv(;)p Black 2375 w FB(\(26\))p Black 264 2659 a Fv(v)304 2671 y Fu(e)335 2680 y Fm(\036)429 2659 y Fr(:=)50 b(2)27 b(arctan)881 2517 y Fn(")929 2515 y(s)p 1012 2515 246 4 v 1022 2603 a Fr(1)18 b(+)g Fv(e)1204 2615 y Fu(\036)p 1022 2640 226 4 v 1022 2716 a Fr(1)g Fq(\000)g Fv(e)1204 2728 y Fu(\036)1285 2659 y Fr(tan)1429 2603 y Fv(u)p 1429 2640 48 4 v 1432 2716 a Fr(2)1487 2517 y Fn(#)1563 2659 y Fv(;)p Black 2351 w FB(\(27\))p Black 140 2879 a Fv(e)179 2845 y FA(\000)p Fu(i)11 b(m)g(\036)441 2879 y Fr(=)61 b Fv(e)606 2845 y FA(\000)p Fu(im\036)780 2853 y Fl(0)830 2879 y Fv(e)869 2839 y FA(\000)p Fu(im)p Fo(\(1+)p Fg(k)p Fo(\))p Fu(v)1212 2847 y Fm(e)1240 2862 y(\036)1310 2879 y Fr(=)23 b Fv(e)1437 2845 y FA(\000)p Fu(im\036)1611 2853 y Fl(0)1661 2879 y Fv(e)1700 2839 y FA(\000)p Fu(imv)1867 2847 y Fm(e)1895 2862 y(\036)1957 2879 y Fv(e)1996 2845 y FA(\000)p Fu(im)p Fg(k)p Fu(v)2203 2853 y Fm(e)2231 2864 y(t)2280 2879 y Fv(;)441 3053 y Fr(=)61 b Fv(e)606 3019 y FA(\000)p Fu(im\036)780 3027 y Fl(0)844 2936 y Fn(\022)943 2997 y Fr(cos)13 b Fv(u)18 b Fq(\000)g Fv(e)1256 3009 y Fu(\036)p 915 3034 413 4 v 915 3110 a Fr(1)g Fq(\000)g Fv(e)1097 3122 y Fu(\036)1154 3110 y Fr(cos)13 b Fv(u)1355 3053 y Fq(\000)18 b Fv(i)1481 2959 y Fn(q)p 1564 2959 226 4 v 94 x Fr(1)g Fq(\000)g Fv(e)1746 3025 y Fo(2)1746 3079 y Fu(\036)1926 2997 y Fr(sin)c Fv(u)p 1813 3034 389 4 v 1813 3110 a Fr(1)k Fq(\000)g Fv(e)1995 3122 y Fu(\036)2039 3110 y Fv(cosu)2212 2936 y Fn(\023)2273 2954 y Fu(m)2363 3053 y Fv(e)2402 3019 y FA(\000)p Fu(im)p Fg(k)p Fu(v)2609 3027 y Fm(e)2637 3038 y(t)441 3338 y Fr(=)61 b Fv(e)606 3303 y FA(\000)p Fu(im\036)780 3311 y Fl(0)830 3221 y Fn(\022)929 3281 y Fr(cos)13 b Fv(u)18 b Fq(\000)g Fv(e)1242 3293 y Fu(\036)p 901 3319 413 4 v 901 3395 a Fr(1)g Fq(\000)g Fv(e)1083 3407 y Fu(\036)1141 3395 y Fr(cos)13 b Fv(u)1342 3338 y Fq(\000)18 b Fv(i)1468 3244 y Fn(q)p 1550 3244 226 4 v 1550 3338 a Fr(1)g Fq(\000)g Fv(e)1732 3309 y Fo(2)1732 3363 y Fu(\036)1912 3281 y Fr(sin)c Fv(u)p 1800 3319 389 4 v 1800 3395 a Fr(1)j Fq(\000)h Fv(e)1981 3407 y Fu(\036)2025 3395 y Fv(cosu)2198 3221 y Fn(\023)2259 3238 y Fu(m)2336 3338 y Fr(exp)2476 3171 y Fn(2)2476 3320 y(4)2532 3338 y Fq(\000)p Fv(im)p Fp(k)2763 3171 y Fn(0)2763 3320 y(@)2835 3338 y Fp(M)g Fr(+)3054 3234 y FA(1)3027 3259 y Fn(X)3029 3436 y Fu(j)s Fo(=1)3161 3338 y Fv(G)3226 3350 y Fu(j)3261 3338 y Fr(\()p Fv(e)3332 3350 y Fu(t)3361 3338 y Fr(\))c(sin)g Fv(j)5 b Fp(M)3653 3171 y Fn(1)3653 3320 y(A)3725 3171 y(3)3725 3320 y(5)441 3624 y Fr(=)61 b Fv(e)606 3590 y FA(\000)p Fu(im\036)780 3598 y Fl(0)830 3507 y Fn(\022)929 3568 y Fr(cos)13 b Fv(u)18 b Fq(\000)g Fv(e)1242 3580 y Fu(\036)p 901 3605 413 4 v 901 3681 a Fr(1)g Fq(\000)g Fv(e)1083 3693 y Fu(\036)1141 3681 y Fr(cos)13 b Fv(u)1342 3624 y Fq(\000)18 b Fv(i)1468 3530 y Fn(q)p 1550 3530 226 4 v 1550 3624 a Fr(1)g Fq(\000)g Fv(e)1732 3596 y Fo(2)1732 3649 y Fu(\036)1924 3568 y Fr(sin)c Fv(u)p 1800 3605 413 4 v 1800 3681 a Fr(1)j Fq(\000)h Fv(e)1981 3693 y Fu(\036)2039 3681 y Fr(cos)13 b Fv(u)2222 3507 y Fn(\023)2283 3525 y Fu(m)2364 3624 y Fq(\002)567 3908 y Fr(exp)g([)p Fq(\000)p Fv(im)p Fp(kM)p Fr(])1088 3742 y Fn(0)1088 3891 y(@)1161 3908 y Fr(1)18 b Fq(\000)g Fv(im)p Fp(k)1510 3805 y FA(1)1483 3830 y Fn(X)1486 4006 y Fu(j)s Fo(=1)1617 3908 y Fv(G)1682 3920 y Fu(j)1717 3908 y Fr(\()p Fv(e)1788 3920 y Fu(t)1818 3908 y Fr(\))c(sin)g Fv(j)5 b Fp(M)2110 3742 y Fn(1)2110 3891 y(A)2209 3908 y Fv(:)p Black 1705 w FB(\(28\))p Black -150 4191 a(F)-7 b(or)26 b(the)i(\034rst)e(line)h (w)n(e)g(to)r(ok)f(the)i(de\034nition)f(of)g Fv(v)j FB(as)d(a)f (function)i(of)e Fv(u)h FB(and)g Fv(e)2315 4203 y Fu(t)2344 4191 y FB(,)g(Eq.)37 b(\()2586 4191 y SDict begin H.S end 2586 4191 a Black FB(4)p Black 2628 4137 a SDict begin H.R end 2628 4137 a 2628 4191 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.1) cvn H.B /ANN pdfmark end 2628 4191 a FB(\),)27 b(and)g(Eq.)37 b(\()3063 4191 y SDict begin H.S end 3063 4191 a Black FB(3)p Black 3105 4137 a SDict begin H.R end 3105 4137 a 3105 4191 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.1) cvn H.B /ANN pdfmark end 3105 4191 a FB(\).)g(In)27 b(the)g(last)g(line)g(ab)r(o)n(v)n(e) f(w)n(e)-150 4286 y(truncated)g(the)h(exp)r(onen)n(tial)f(of)g(the)h (sin)g(series)e(after)h(the)h(linear)f(order)f(in)i Fp(k)p FB(,)g(b)r(ecause)f Fp(k)h FB(is)f(already)f(of)i(1pN)f(order)f(and)h (used)h(the)-150 4382 y(F)-7 b(ourier-domain)26 b(expansion)i(Eq.)40 b(\()1015 4382 y SDict begin H.S end 1015 4382 a Black FB(22)p Black 1098 4328 a SDict begin H.R end 1098 4328 a 1098 4382 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.22) cvn H.B /ANN pdfmark end 1098 4382 a FB(\))29 b(for)e Fv(v)s FB(.)40 b(W)-7 b(e)29 b(are)e(allo)n(w)n(ed)g(to)i(do)f(this)g (b)r(ecause)g(there)h(is)f(no)g(secular)f(con)n(tribution)h(coming)-150 4477 y(up)g(in)g(the)g(exp)r(onen)n(t)f(due)h(to)f(its)h(p)r(erio)r (dicit)n(y)f(of)h Fr(2)p Fv(\031)i FB(in)e Fp(M)f FB(and)h(its)g(a)n(v) n(erage)c(to)k(zero)e(in)i(the)g(in)n(terv)-5 b(al)27 b Fr([0)p Fv(;)14 b Fr(2)p Fv(\031)s Fr(])p FB(.)-67 4573 y(In)28 b(the)h(m)n(ultip)r(ole)g(momen)n(ts)e(the)i(n)n(um)n(b)r (ers)f Fv(m)c Fr(=)f(\(0)p Fv(;)14 b Fr(1)p Fv(;)g Fr(2)p Fv(;)g Fr(3)p Fv(;)g Fr(4\))26 b FB(app)r(ear,)i(so)f(w)n(e)h(preserv)n (e)f(all)h(con)n(tributions)f(with)h(these)h Fv(m)f FB(up)-150 4668 y(to)f(the)h(order)f(of)g Fv(\017)g FB(required.)-106 4921 y SDict begin H.S end -106 4921 a -106 4921 a SDict begin 11.5 H.A end -106 4921 a -106 4921 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.29) cvn /DEST pdfmark end -106 4921 a Fv(e)-67 4881 y FA(\000)p Fu(iv)41 4889 y Fm(e)69 4904 y(\036)167 4921 y Fr(=)305 4865 y(1)p 291 4902 68 4 v 291 4978 a Fv(e)330 4990 y Fu(t)369 4779 y Fn(\()446 4865 y Fv(i)475 4792 y Fn(p)p 558 4792 219 4 v 73 x Fr(1)18 b Fq(\000)g Fv(e)740 4836 y Fo(2)740 4885 y Fu(t)777 4865 y Fv(e)816 4877 y Fu(t)858 4865 y Fr(sin)c Fv(u)k Fq(\000)g Fv(e)1162 4835 y Fo(2)1162 4885 y Fu(t)1217 4865 y Fr(+)g(1)p 446 4902 896 4 v 807 4978 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1370 4921 y Fq(\000)g Fr(1)1495 4779 y Fn(\))1580 4921 y Fr(+)g Fq(O)r Fr(\()p Fv(\017)1797 4887 y Fo(2)1835 4921 y Fr(\))c Fv(;)p Black 2033 w FB(\(29\))p Black -139 5216 a Fv(e)-100 5176 y FA(\000)p Fu(i)p Fo(2)p Fu(v)41 5184 y Fm(e)69 5199 y(\036)167 5216 y Fr(=)309 5160 y(1)p 291 5197 76 4 v 291 5273 a Fv(e)330 5244 y Fo(2)330 5293 y Fu(t)377 5074 y Fn(\()444 5216 y Fr(2)k Fq(\000)g Fv(e)626 5182 y Fo(2)626 5236 y Fu(t)681 5216 y Fr(+)774 5154 y(4)830 5087 y Fn(\000)868 5154 y Fv(e)907 5124 y Fo(2)907 5175 y Fu(t)962 5154 y Fq(\000)g Fr(1)1087 5087 y Fn(\001)1143 5154 y Fr(+)g(2)p Fv(ie)1336 5166 y Fu(t)1364 5081 y Fn(p)p 1447 5081 219 4 v 73 x Fr(1)g Fq(\000)g Fv(e)1629 5126 y Fo(2)1629 5175 y Fu(t)1680 5154 y Fr(sin)c Fv(u)p 774 5197 1069 4 v 1222 5273 a(A)p Fr(\()p Fv(u)p Fr(\))1872 5216 y(+)1965 5154 y(2)2021 5087 y Fn(\000)2058 5154 y Fv(e)2097 5124 y Fo(2)2097 5175 y Fu(t)2152 5154 y Fq(\000)k Fr(1)2277 5087 y Fn(\001)2329 5124 y Fo(2)2385 5154 y Fq(\000)g Fr(2)p Fv(ie)2578 5166 y Fu(t)2606 5081 y Fn(p)p 2689 5081 219 4 v 73 x Fr(1)g Fq(\000)g Fv(e)2871 5126 y Fo(2)2871 5175 y Fu(t)2907 5095 y Fo(3)2959 5154 y Fr(sin)13 b Fv(u)p 1965 5197 1158 4 v 2437 5273 a(A)p Fr(\()p Fv(u)p Fr(\))2611 5249 y Fo(2)p Black Black eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark end -158 -515 a Black 4043 -299 a FB(6)p Black 281 46 a Fr(+)p Fv(\017)380 11 y Fo(2)417 46 y Fq(j)p Fv(E)5 b Fq(j)529 -96 y Fn(")578 46 y Fr(8\()p Fv(\021)21 b Fq(\000)d Fr(4\))g(+)982 -57 y(8\()p Fv(\021)k Fq(\000)c Fr(4\))1290 -125 y Fn(\000)1327 -57 y Fv(e)1366 -87 y Fo(2)1366 -37 y Fu(t)1422 -57 y Fq(\000)g Fr(3)1547 -125 y Fn(\001)1603 -57 y Fq(\000)1696 -110 y Fo(4)p Fu(i)p Fo(\()p Fu(\021)r FA(\000)p Fo(4\))p Fu(e)1956 -102 y Fm(t)1985 -104 y Fr(\()2017 -110 y Fu(e)2048 -135 y Fl(2)2048 -93 y Fm(t)2081 -110 y FA(\000)p Fo(2)2166 -104 y Fr(\))2209 -110 y Fo(sin\()p Fu(u)p Fo(\))p 1696 -76 687 4 v 1930 -66 a Fq(p)p 1999 -66 149 4 v 56 x Fo(1)p FA(\000)p Fu(e)2115 -30 y Fl(2)2115 7 y Fm(t)p 982 27 1410 4 v 1600 103 a Fv(A)p Fr(\()p Fv(u)p Fr(\))2421 46 y(+)2514 -57 y Fq(\000)p Fr(24\()p Fv(\021)i Fq(\000)e Fr(4\))2927 -125 y Fn(\000)2965 -57 y Fv(e)3004 -87 y Fo(2)3004 -37 y Fu(t)3059 -57 y Fq(\000)g Fr(1)3184 -125 y Fn(\001)3240 -57 y Fr(+)3333 -110 y Fo(16)p Fu(i)p Fo(\()p Fu(\021)r FA(\000)p Fo(4\))p Fu(e)3626 -102 y Fm(t)3655 -104 y Fr(\()3687 -110 y Fu(e)3718 -135 y Fl(2)3718 -93 y Fm(t)3751 -110 y FA(\000)p Fo(1)3836 -104 y Fr(\))3880 -110 y Fo(sin\()p Fu(u)p Fo(\))p 3333 -76 720 4 v 3584 -66 a Fq(p)p 3653 -66 149 4 v 56 x Fo(1)p FA(\000)p Fu(e)3769 -30 y Fl(2)3769 7 y Fm(t)p 2514 27 1550 4 v 3182 103 a Fv(A)p Fr(\()p Fv(u)p Fr(\))3356 79 y Fo(2)281 328 y Fr(+)356 267 y Fq(\000)p Fr(8\()p Fv(\021)j Fq(\000)d Fr(4\))728 199 y Fn(\000)766 267 y Fv(e)805 236 y Fo(2)805 287 y Fu(t)860 267 y Fq(\000)g Fr(1)985 199 y Fn(\001)1036 236 y Fo(2)1092 267 y Fq(\000)g Fr(8)p Fv(i)1246 193 y Fn(p)p 1328 193 219 4 v 1328 267 a Fr(1)g Fq(\000)g Fv(e)1510 238 y Fo(2)1510 287 y Fu(t)1547 267 y Fr(\()p Fv(\021)k Fq(\000)c Fr(4\))p Fv(e)1838 279 y Fu(t)1881 199 y Fn(\000)1919 267 y Fv(e)1958 236 y Fo(2)1958 287 y Fu(t)2013 267 y Fq(\000)g Fr(1)2138 199 y Fn(\001)2189 267 y Fr(sin\()p Fv(u)p Fr(\))p 356 309 2048 4 v 1274 385 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1448 361 y Fo(3)2413 186 y Fn(#)2480 328 y Fr(+)g Fq(O)r Fr(\()p Fv(\017)2697 294 y Fo(4)2735 328 y Fr(\))c Fv(;)p Black 1133 w FB(\(30\))p Black -139 677 a Fv(e)-100 637 y FA(\000)p Fu(i)p Fo(3)p Fu(v)41 645 y Fm(e)69 660 y(\036)167 677 y Fr(=)309 620 y(1)p 291 658 76 4 v 291 734 a Fv(e)330 705 y Fo(3)330 754 y Fu(t)377 535 y Fn(\()444 677 y Fr(3)p Fv(e)525 642 y Fo(2)525 697 y Fu(t)580 677 y Fq(\000)k Fr(4)g(+)816 574 y(3)872 506 y Fn(\000)909 574 y Fv(e)948 544 y Fo(4)948 594 y Fu(t)1004 574 y Fq(\000)g Fr(5)p Fv(e)1168 544 y Fo(2)1168 594 y Fu(t)1223 574 y Fr(+)g(4)1348 506 y Fn(\001)1404 574 y Fq(\000)1497 521 y Fu(ie)1551 529 y Fm(t)1579 527 y Fr(\()1611 521 y Fu(e)1642 496 y Fl(4)1642 538 y Fm(t)1675 521 y FA(\000)p Fo(5)p Fu(e)1791 496 y Fl(2)1791 538 y Fm(t)1824 521 y Fo(+4)1908 527 y Fr(\))1952 521 y Fo(sin)11 b Fu(u)p 1497 555 588 4 v 1681 565 a Fq(p)p 1751 565 149 4 v 1751 621 a Fo(1)p FA(\000)p Fu(e)1867 601 y Fl(2)1867 638 y Fm(t)p 816 658 1279 4 v 1368 734 a Fv(A)p Fr(\()p Fv(u)p Fr(\))2123 677 y(+)2216 615 y Fq(\000)p Fr(12)2379 548 y Fn(\000)2415 615 y Fv(e)2454 585 y Fo(2)2454 636 y Fu(t)2509 615 y Fq(\000)18 b Fr(1)2634 548 y Fn(\001)2686 585 y Fo(2)2742 615 y Fr(+)g(8)p Fv(ie)2935 627 y Fu(t)2963 542 y Fn(p)p 3046 542 219 4 v 73 x Fr(1)g Fq(\000)g Fv(e)3228 587 y Fo(2)3228 636 y Fu(t)3264 556 y Fo(3)3316 615 y Fr(sin)13 b Fv(u)p 2216 658 1264 4 v 2741 734 a(A)p Fr(\()p Fv(u)p Fr(\))2915 710 y Fo(2)281 972 y Fr(+)356 910 y Fq(\000)p Fr(4)477 843 y Fn(\000)514 910 y Fv(e)553 880 y Fo(2)553 931 y Fu(t)608 910 y Fq(\000)18 b Fr(1)733 843 y Fn(\001)785 880 y Fo(3)840 910 y Fq(\000)g Fr(4)p Fv(ie)1033 922 y Fu(t)1061 837 y Fn(p)p 1144 837 219 4 v 73 x Fr(1)g Fq(\000)g Fv(e)1326 882 y Fo(2)1326 931 y Fu(t)1363 851 y Fo(5)1414 910 y Fr(sin)c Fv(u)p 356 953 1222 4 v 861 1029 a(A)p Fr(\()p Fv(u)p Fr(\))1035 1005 y Fo(3)1588 830 y Fn(\))1673 972 y Fr(+)k Fq(O)r Fr(\()p Fv(\017)1890 937 y Fo(2)1928 972 y Fr(\))c Fv(;)p Black 1940 w FB(\(31\))p Black -139 1254 a Fv(e)-100 1214 y FA(\000)p Fu(i)p Fo(4)p Fu(v)41 1222 y Fm(e)69 1237 y(\036)167 1254 y Fr(=)309 1198 y(1)p 291 1235 76 4 v 291 1311 a Fv(e)330 1282 y Fo(4)330 1331 y Fu(t)377 1112 y Fn(\()444 1254 y Fv(e)483 1220 y Fo(4)483 1274 y Fu(t)538 1254 y Fq(\000)k Fr(8)p Fv(e)702 1220 y Fo(2)702 1274 y Fu(t)757 1254 y Fr(+)g(8)g(+)993 1193 y Fq(\000)p Fr(16)1156 1125 y Fn(\000)1193 1193 y Fv(e)1232 1162 y Fo(4)1232 1213 y Fu(t)1287 1193 y Fq(\000)g Fr(3)p Fv(e)1451 1162 y Fo(2)1451 1213 y Fu(t)1506 1193 y Fr(+)g(2)1631 1125 y Fn(\001)1687 1193 y Fq(\000)g Fr(4)p Fv(ie)1880 1205 y Fu(t)1908 1119 y Fn(p)p 1991 1119 219 4 v 74 x Fr(1)g Fq(\000)g Fv(e)2173 1164 y Fo(2)2173 1213 y Fu(t)2224 1125 y Fn(\000)2262 1193 y Fv(e)2301 1162 y Fo(2)2301 1213 y Fu(t)2356 1193 y Fq(\000)g Fr(2)2481 1125 y Fn(\001)2533 1193 y Fr(sin)13 b Fv(u)p 993 1235 1703 4 v 1757 1311 a(A)p Fr(\()p Fv(u)p Fr(\))281 1532 y(+)356 1471 y Fq(\000)p Fr(8)477 1403 y Fn(\000)514 1471 y Fv(e)553 1440 y Fo(2)553 1491 y Fu(t)608 1471 y Fq(\000)18 b Fr(6)733 1403 y Fn(\001)c(\000)823 1471 y Fv(e)862 1440 y Fo(2)862 1491 y Fu(t)917 1471 y Fq(\000)k Fr(1)1042 1403 y Fn(\001)1094 1440 y Fo(2)1149 1471 y Fq(\000)g Fr(4)p Fv(ie)1342 1483 y Fu(t)1370 1398 y Fn(p)p 1453 1398 219 4 v 73 x Fr(1)g Fq(\000)g Fv(e)1635 1442 y Fo(2)1635 1491 y Fu(t)1686 1403 y Fn(\000)1724 1471 y Fv(e)1763 1440 y Fo(4)1763 1491 y Fu(t)1818 1471 y Fq(\000)g Fr(7)p Fv(e)1982 1440 y Fo(2)1982 1491 y Fu(t)2037 1471 y Fr(+)g(6)2162 1403 y Fn(\001)2214 1471 y Fr(sin)13 b Fv(u)p 356 1513 2022 4 v 1261 1589 a(A)p Fr(\()p Fv(u)p Fr(\))1435 1565 y Fo(2)281 1778 y Fr(+)356 1717 y(32)454 1649 y Fn(\000)491 1717 y Fv(e)530 1687 y Fo(2)530 1737 y Fu(t)585 1717 y Fq(\000)18 b Fr(1)710 1649 y Fn(\001)762 1687 y Fo(3)817 1717 y Fr(+)g(24)p Fv(ie)1052 1729 y Fu(t)1094 1649 y Fn(\000)1132 1717 y Fr(1)g Fq(\000)g Fv(e)1314 1687 y Fo(2)1314 1737 y Fu(t)1351 1649 y Fn(\001)1403 1687 y Fo(5)p Fu(=)p Fo(2)1521 1717 y Fr(sin)13 b Fv(u)p 356 1759 1329 4 v 914 1835 a(A)p Fr(\()p Fv(u)p Fr(\))1088 1811 y Fo(3)281 2034 y Fr(+)356 1973 y(8)412 1906 y Fn(\000)449 1973 y Fv(e)488 1943 y Fo(2)488 1993 y Fu(t)544 1973 y Fq(\000)18 b Fr(1)669 1906 y Fn(\001)720 1943 y Fo(4)776 1973 y Fq(\000)g Fr(8)p Fv(ie)969 1985 y Fu(t)1011 1906 y Fn(\000)1049 1973 y Fr(1)g Fq(\000)g Fv(e)1231 1943 y Fo(2)1231 1993 y Fu(t)1268 1906 y Fn(\001)1319 1943 y Fo(7)p Fu(=)p Fo(2)1438 1973 y Fr(sin\()p Fv(u)p Fr(\))p 356 2015 1296 4 v 898 2091 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1072 2067 y Fo(4)1662 1892 y Fn(\))1747 2034 y Fr(+)g Fq(O)r Fr(\()p Fv(\017)1964 2000 y Fo(2)2002 2034 y Fr(\))c Fv(:)p Black 1866 w FB(\(32\))p Black -150 2292 a(Those)26 b(terms)h(will)h(mix)f(with)g Fv(f)874 2304 y Fu(lm)986 2292 y FB(and)g Fv(g)1187 2304 y Fu(lm)1271 2292 y FB(.)37 b(Let)27 b(us)g(get)g(more)f(precise)g(no)n(w)h(and)g(start)f(decomp)r (osing)h(the)g Fv(A)p Fr(\()p Fv(u)p Fr(\))3542 2262 y FA(\000)p Fu(n)3667 2292 y FB(term.)36 b(With)-150 2388 y(the)28 b(help)g(of)f(Eq.)38 b(\()460 2388 y SDict begin H.S end 460 2388 a Black FB(12)p Black 543 2335 a SDict begin H.R end 543 2335 a 543 2388 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.12) cvn H.B /ANN pdfmark end 543 2388 a FB(\))28 b(w)n(e)f(get)1042 2634 y SDict begin H.S end 1042 2634 a 1042 2634 a SDict begin 11.5 H.A end 1042 2634 a 1042 2634 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.33) cvn /DEST pdfmark end 1042 2634 a 1291 2578 a Fr(1)p 1052 2615 521 4 v 1052 2691 a(\(1)18 b Fq(\000)g Fv(e)1266 2703 y Fu(t)1323 2691 y Fr(cos)12 b Fv(u)p Fr(\))1527 2667 y Fu(n)1632 2634 y Fr(=)50 b(1)18 b(+)1928 2530 y FA(1)1901 2555 y Fn(X)1890 2731 y Fu(m)p Fo(=1)2077 2530 y Fu(m)2047 2555 y Fn(X)2046 2734 y Fu(k)q Fo(=0)2181 2634 y Fv(\014)2232 2591 y Fo(\()p Fu(n)p Fo(\))2228 2659 y Fu(m;k)2361 2634 y Fr(cos)o([\()p Fv(m)h Fq(\000)f Fr(2)p Fv(k)s Fr(\))p Fv(u)p Fr(])1632 2912 y(=)50 b(1)18 b(+)g Fv(b)1926 2869 y Fo(\()p Fu(n)p Fo(\))1926 2935 y(0)2041 2912 y Fr(+)2151 2809 y FA(1)2124 2834 y Fn(X)2124 3012 y Fu(k)q Fo(=1)2258 2912 y Fv(b)2294 2869 y Fo(\()p Fu(n)p Fo(\))2294 2938 y Fu(k)2419 2912 y Fr(cos)13 b Fv(k)s(u)h(;)p Black 1262 w FB(\(33\))p Black -150 3166 a(where)27 b(w)n(e)g(ha)n(v)n(e)f(collected)g(those)h (terms)g(with)h(the)g(same)e Fw(p)l(ositive)j FB(frequency)-7 b(.)36 b(App)r(endix)2808 3166 y SDict begin H.S end 2808 3166 a Black FB(A)p Black 2871 3109 a SDict begin H.R end 2871 3109 a 2871 3166 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section*.16) cvn H.B /ANN pdfmark end 2871 3166 a 27 w FB(will)27 b(giv)n(e)g(deep)r(er)g(explanation)f(ho)n(w)-150 3262 y(to)h(select)g(the)h(frequencies)e(with)i(the)g(same)e(absolute)h (v)-5 b(alue.)37 b(The)27 b(same)f(con)n(tribution)h(\025)g(b)r(eing)g (m)n(ultiplied)h(with)g Fr(sin)14 b Fv(u)27 b FB(\025)f(will)i(b)r(e) -150 3357 y(decomp)r(osed)f(analogously)-7 b(,)25 b(but)j(ha)n(ving)f Fr(sin)13 b Fv(mu)28 b FB(this)f(time)h(as)f(an)h(o)r(dd)f(function)h (of)g Fv(u)p FB(.)584 3632 y SDict begin H.S end 584 3632 a 584 3632 a SDict begin 11.5 H.A end 584 3632 a 584 3632 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.34) cvn /DEST pdfmark end 584 3632 a 772 3575 a Fr(sin)14 b Fv(u)p 594 3613 521 4 v 594 3689 a Fr(\(1)k Fq(\000)g Fv(e)808 3701 y Fu(t)865 3689 y Fr(cos)13 b Fv(u)p Fr(\))1070 3665 y Fu(n)1186 3632 y Fr(=)1312 3465 y Fn(0)1312 3614 y(@)1384 3632 y Fr(1)18 b(+)g Fv(b)1563 3588 y Fo(\()p Fu(n)p Fo(\))1563 3654 y(0)1679 3632 y Fr(+)1788 3528 y FA(1)1762 3553 y Fn(X)1764 3730 y Fu(j)s Fo(=1)1895 3632 y Fv(b)1931 3588 y Fo(\()p Fu(n)p Fo(\))1931 3655 y Fu(j)2056 3632 y Fr(cos)13 b Fv(j)5 b(u)2268 3465 y Fn(1)2268 3614 y(A)2367 3632 y Fr(sin)14 b Fv(u)1186 3939 y Fr(=)1312 3797 y Fn( )1377 3939 y Fr(1)k(+)1548 3835 y FA(1)1521 3860 y Fn(X)1520 4039 y Fu(k)q Fo(=1)1655 3939 y Fv(\014)1706 3896 y Fo(\()p Fu(n)p Fo(\))1702 3964 y(2)p Fu(k)q(;k)1832 3797 y Fn(!)1912 3939 y Fr(sin)c Fv(u)k Fr(+)2187 3883 y(1)p 2187 3920 42 4 v 2187 3996 a(2)2252 3797 y Fn( )2356 3835 y FA(1)2329 3860 y Fn(X)2318 4036 y Fu(m)p Fo(=2)2474 3847 y Fn(h)2514 3939 y Fv(b)2550 3896 y Fo(\()p Fu(n)p Fo(\))2550 3961 y Fu(m)p FA(\000)p Fo(1)2716 3939 y Fq(\000)g Fv(b)2835 3896 y Fo(\()p Fu(n)p Fo(\))2835 3961 y Fu(m)p Fo(+1)2982 3847 y Fn(i)3035 3939 y Fr(sin)13 b Fv(mu)3271 3797 y Fn(!)1186 4217 y Fr(=)1339 4113 y FA(1)1312 4138 y Fn(X)1314 4315 y Fu(j)s Fo(=1)1445 4217 y Fv(S)1501 4174 y Fo(\()p Fu(n)p Fo(\))1496 4240 y Fu(j)1626 4217 y Fr(sin)h Fv(j)5 b(u)13 b(;)p Black 2072 w FB(\(34\))p Black 972 4505 a Fv(S)1028 4462 y Fo(\()p Fu(n)p Fo(\))1023 4527 y(1)1174 4505 y Fr(:=)50 b(1)18 b(+)1482 4401 y FA(1)1455 4426 y Fn(X)1455 4605 y Fu(k)q Fo(=1)1589 4505 y Fv(\014)1640 4462 y Fo(\()p Fu(n)p Fo(\))1636 4530 y(2)p Fu(k)q(;k)1785 4505 y Fq(\000)1878 4449 y Fr(1)p 1878 4486 V 1878 4562 a(2)1930 4505 y Fv(b)1966 4462 y Fo(\()p Fu(n)p Fo(\))1966 4527 y(2)2076 4505 y Fv(;)p Black 1838 w FB(\(35\))p Black 954 4756 a Fv(S)1010 4713 y Fo(\()p Fu(n)p Fo(\))1005 4779 y Fu(j)s(>)p Fo(1)1174 4756 y Fr(:=)1322 4700 y(1)p 1322 4737 V 1322 4813 a(2)1387 4664 y Fn(\020)1437 4756 y Fv(b)1473 4713 y Fo(\()p Fu(n)p Fo(\))1473 4779 y Fu(j)s FA(\000)p Fo(1)1611 4756 y Fq(\000)g Fv(b)1730 4713 y Fo(\()p Fu(n)p Fo(\))1730 4779 y Fu(j)s Fo(+1)1848 4664 y Fn(\021)1926 4756 y Fv(:)p Black 1988 w FB(\(36\))p Black -67 4964 a(Remem)n(b)r(ering)27 b(Eqs.)37 b(\()685 4964 y SDict begin H.S end 685 4964 a Black FB(16)p Black 768 4910 a SDict begin H.R end 768 4910 a 768 4964 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.16) cvn H.B /ANN pdfmark end 768 4964 a FB(\))28 b(and)g(\()1022 4964 y SDict begin H.S end 1022 4964 a Black FB(18)p Black 1105 4910 a SDict begin H.R end 1105 4910 a 1105 4964 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.16) cvn H.B /ANN pdfmark end 1105 4964 a FB(\),)g(w)n(e)f(get)323 5318 y SDict begin H.S end 323 5318 a 323 5318 a SDict begin 11.5 H.A end 323 5318 a 323 5318 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.37) cvn /DEST pdfmark end 323 5318 a 565 5261 a Fr(1)p 333 5298 507 4 v 333 5375 a(\(1)18 b Fq(\000)g Fv(e)547 5387 y Fu(t)590 5375 y Fr(cos)13 b Fv(u)p Fr(\))795 5351 y Fu(n)911 5318 y Fr(=)61 b(1)18 b(+)g Fv(b)1216 5274 y Fo(\()p Fu(n)p Fo(\))1216 5340 y(0)1331 5318 y Fr(+)1441 5214 y FA(1)1414 5239 y Fn(X)1414 5417 y Fu(k)q Fo(=1)1549 5318 y Fv(b)1585 5274 y Fo(\()p Fu(n)p Fo(\))1585 5343 y Fu(k)1695 5318 y Fr(cos)13 b Fv(k)s(u)23 b Fr(=)f(1)c(+)g Fv(b)2203 5274 y Fo(\()p Fu(n)p Fo(\))2203 5340 y(0)2318 5318 y Fr(+)2429 5214 y FA(1)2402 5239 y Fn(X)2401 5417 y Fu(k)q Fo(=1)2536 5318 y Fv(b)2572 5274 y Fo(\()p Fu(n)p Fo(\))2572 5343 y Fu(k)2682 5151 y Fn(0)2682 5300 y(@)2782 5214 y FA(1)2755 5239 y Fn(X)2758 5416 y Fu(j)s Fo(=1)2892 5318 y Fr(\026)-45 b Fv(\015)2937 5283 y Fu(k)2932 5338 y(j)2991 5318 y Fr(cos)13 b Fv(j)5 b Fp(M)3246 5151 y Fn(1)3246 5300 y(A)p Black Black eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark end -158 -515 a Black 4043 -299 a FB(7)p Black 911 -21 a Fr(=)61 b(1)18 b(+)g Fv(b)1216 -64 y Fo(\()p Fu(n)p Fo(\))1216 1 y(0)1331 -21 y Fr(+)1441 -125 y FA(1)1414 -100 y Fn(X)1416 77 y Fu(j)s Fo(=1)1548 -163 y Fn( )1641 -125 y FA(1)1614 -100 y Fn(X)1613 79 y Fu(k)q Fo(=1)1751 -21 y Fr(\026)-45 b Fv(\015)1796 -55 y Fu(k)1791 0 y(j)1836 -21 y Fv(b)1872 -64 y Fo(\()p Fu(n)p Fo(\))1872 4 y Fu(k)1969 -163 y Fn(!)2049 -21 y Fr(cos)13 b Fv(j)5 b Fp(M)911 267 y Fr(=)61 b(1)18 b(+)g Fv(b)1216 224 y Fo(\()p Fu(n)p Fo(\))1216 289 y(0)1331 267 y Fr(+)1443 163 y FA(1)1417 188 y Fn(X)1414 364 y Fu(n)p Fo(=1)1553 267 y Fq(A)1619 224 y Fo(\()p Fu(n)p Fo(\))1619 290 y Fu(j)1730 267 y Fr(cos)13 b Fv(j)5 b Fp(M)14 b Fv(;)p Black 1915 w FB(\(37\))p Black 504 520 a Fr(sin)g Fv(u)p 333 557 507 4 v 333 633 a Fr(\(1)k Fq(\000)g Fv(e)547 645 y Fu(t)590 633 y Fr(cos)13 b Fv(u)p Fr(\))795 609 y Fu(n)911 576 y Fr(=)1064 472 y FA(1)1037 497 y Fn(X)1037 676 y Fu(k)q Fo(=1)1171 576 y Fv(S)1227 533 y Fo(\()p Fu(n)p Fo(\))1222 601 y Fu(k)1338 576 y Fr(sin)h Fv(k)s(u)22 b Fr(=)1685 472 y FA(1)1658 497 y Fn(X)1658 676 y Fu(k)q Fo(=1)1792 576 y Fv(S)1848 533 y Fo(\()p Fu(n)p Fo(\))1843 601 y Fu(k)1959 409 y Fn(0)1959 558 y(@)2059 472 y FA(1)2032 497 y Fn(X)2034 674 y Fu(j)s Fo(=1)2170 576 y Fr(\026)-47 b Fv(\033)2215 541 y Fu(k)2212 596 y(j)2271 576 y Fr(sin)13 b Fv(j)5 b Fp(M)2516 409 y Fn(1)2516 558 y(A)2612 576 y Fr(=)2726 472 y FA(1)2699 497 y Fn(X)2702 674 y Fu(j)s Fo(=1)2833 434 y Fn( )2926 472 y FA(1)2899 497 y Fn(X)2899 676 y Fu(k)q Fo(=1)3038 576 y Fr(\026)-47 b Fv(\033)3083 541 y Fu(k)3080 596 y(j)3125 576 y Fv(S)3181 533 y Fo(\()p Fu(n)p Fo(\))3176 601 y Fu(k)3277 434 y Fn(!)3357 576 y Fr(cos)13 b Fv(j)5 b Fp(M)911 875 y Fr(=)1064 771 y FA(1)1037 796 y Fn(X)1039 973 y Fu(j)s Fo(=1)1171 875 y Fq(S)1227 832 y Fo(\()p Fu(n)p Fo(\))1221 898 y Fu(j)1338 875 y Fr(sin)14 b Fv(j)5 b Fp(M)14 b Fv(;)p Black 2316 w FB(\(38\))p Black 686 1112 a Fq(A)752 1069 y Fo(\()p Fu(n)p Fo(\))752 1134 y(0)899 1112 y Fr(:=)50 b(1)18 b(+)g Fv(b)1216 1069 y Fo(\()p Fu(n)p Fo(\))1216 1134 y(0)1326 1112 y Fv(;)p Black 2588 w FB(\(39\))p Black 663 1313 a Fq(A)729 1270 y Fo(\()p Fu(n)p Fo(\))729 1336 y Fu(j)s(>)p Fo(0)899 1313 y Fr(:=)1037 1171 y Fn( )1141 1209 y FA(1)1114 1234 y Fn(X)1103 1410 y Fu(m)p Fo(=1)1262 1313 y Fr(\026)-45 b Fv(\015)1307 1279 y Fu(m)1302 1333 y(j)1370 1313 y Fv(b)1406 1279 y Fo(\()p Fu(n)p Fo(\))1406 1333 y Fu(m)1503 1171 y Fn(!)1596 1313 y Fv(;)p Black 2318 w FB(\(40\))p Black 679 1597 a Fq(S)735 1554 y Fo(\()p Fu(n)p Fo(\))729 1620 y Fu(j)s(>)p Fo(0)899 1597 y Fr(:=)1037 1455 y Fn( )1141 1493 y FA(1)1114 1518 y Fn(X)1103 1694 y Fu(m)p Fo(=1)1264 1597 y Fr(\026)e Fv(\033)1309 1562 y Fu(m)1306 1617 y(j)1373 1597 y Fv(S)1429 1562 y Fo(\()p Fu(n)p Fo(\))1424 1617 y Fu(m)1526 1455 y Fn(!)1619 1597 y Fv(:)p Black 2295 w FB(\(41\))p Black -150 1842 a(The)20 b(abbreviations)e Fq(A)586 1799 y Fo(\()p Fu(n)p Fo(\))586 1865 y Fu(j)703 1842 y FB(and)h Fq(S)912 1799 y Fo(\()p Fu(n)p Fo(\))906 1865 y Fu(j)1030 1842 y FB(stand)g(for)g(the)h(j)1527 1812 y Fo(th)1614 1842 y FB(con)n(tribution)f(of)h Fv(A)p Fr(\()p Fv(u)p Fr(\))2340 1812 y FA(\000)p Fu(n)2437 1842 y FB(and)g Fv(A)p Fr(\()p Fv(u)p Fr(\))2765 1812 y FA(\000)p Fu(n)2890 1842 y Fr(sin)13 b Fv(u)p FB(.)34 b(Of)20 b(course,)g(b)r(oth)g(expressions)-150 1955 y(will)33 b(not)f(tak)n(e)g(their)g(place)g(alone,)h(but)f(will)h(b)r(e)g(m)n (ultiplied)g(with)g(the)f(expansion)g(of)g(the)h Fv(e)2868 1925 y Fu(in)p Fg(k)p Fu(v)3005 1933 y Fm(e)3033 1944 y(t)3101 1955 y FB(term)g(due)f(to)g(Eq.)52 b(\()3779 1955 y SDict begin H.S end 3779 1955 a Black FB(26)p Black 3862 1902 a SDict begin H.R end 3862 1902 a 3862 1955 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.26) cvn H.B /ANN pdfmark end 3862 1955 a FB(\).)g(W)-7 b(e)-150 2051 y(F)g(ourier)26 b(decomp)r(ose)g(the)i(follo)n(wing)d(pro)r(ducts) i(of)g(series)f(as)g(they)h(app)r(ear)f(in)i(the)f(\034eld)g(m)n(ultip) r(ole)h(momen)n(ts.)36 b(The)27 b(\034rst)g(quan)n(tit)n(y)-150 2146 y(consists)h(of)i(t)n(w)n(o)e(pure)h(sin)h(series.)41 b(F)-7 b(actor)28 b(1)h(is)g(the)h(expansion)e(of)h Fr(sin\()p Fv(u)p Fr(\))14 b Fv(A)p Fr(\()p Fv(u)p Fr(\))2442 2116 y FA(\000)p Fu(n)2569 2146 y FB(in)30 b Fp(M)f FB(and)g(factor)f(2)h (is)h(the)f(non-secular)f(part)-150 2242 y(of)g(the)f(exp)r(onen)n (tial)g(of)h(the)g Fv(v)s Fr(\()p Fv(l)r Fr(\))p FB(-term.)291 2461 y SDict begin H.S end 291 2461 a 291 2461 a SDict begin 11.5 H.A end 291 2461 a 291 2461 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.42) cvn /DEST pdfmark end 291 2461 a 305 2319 a Fn( )398 2358 y FA(1)371 2383 y Fn(X)371 2561 y Fu(k)q Fo(=1)505 2461 y Fq(S)561 2418 y Fo(\()p Fu(n)p Fo(\))555 2486 y Fu(k)673 2461 y Fr(sin)13 b Fv(k)s Fp(M)925 2319 y Fn(!)g( )1109 2358 y FA(1)1082 2383 y Fn(X)1070 2558 y Fu(m)p Fo(=1)1227 2461 y Fv(G)1292 2473 y Fu(m)1369 2461 y Fr(sin)h Fv(m)p Fp(M)1649 2319 y Fn(!)1764 2461 y Fr(=)1906 2358 y FA(1)1879 2383 y Fn(X)1878 2561 y Fu(k)q Fo(=1)2051 2358 y FA(1)2024 2383 y Fn(X)2013 2558 y Fu(m)p Fo(=1)2180 2405 y Fr(1)p 2180 2442 42 4 v 2180 2518 a(2)2231 2461 y Fq(S)2287 2418 y Fo(\()p Fu(n)p Fo(\))2281 2486 y Fu(k)2385 2461 y Fv(G)2450 2473 y Fu(m)2527 2461 y Fr(\(cos)o([)p Fv(k)22 b Fq(\000)c Fv(m)p Fr(])p Fp(M)g Fq(\000)g Fr(cos)o([)p Fv(k)j Fr(+)d Fv(m)p Fr(])p Fp(M)p Fr(\))1764 2740 y(=)1905 2636 y FA(1)1878 2661 y Fn(X)1881 2838 y Fu(j)s Fo(=0)2012 2740 y Fv(P)2077 2697 y Fu(S)s(S;)p Fo([)p Fu(n)p Fo(])2065 2763 y Fu(j)2278 2740 y Fr(cos)13 b Fv(j)5 b Fp(M)14 b Fv(:)p Black 1367 w FB(\(42\))p Black -150 2998 a(The)28 b(co)r(e\036cien)n(ts)f Fv(P)506 2955 y Fu(S)s(S;)p Fo([)p Fu(n)p Fo(])494 3021 y Fu(j)720 2998 y FB(\(\020SS\021)34 b(stands)28 b(for)f(the)h(pro)r(duct)f(of)h(t)n(w)n(o)e(sin)i(series\)) f(are)f(de\034ned)i(as)f(follo)n(ws.)789 3223 y SDict begin H.S end 789 3223 a 789 3223 a SDict begin 11.5 H.A end 789 3223 a 789 3223 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.43) cvn /DEST pdfmark end 789 3223 a Fv(P)854 3180 y Fu(S)s(S;)p Fo([)p Fu(n)p Fo(])842 3245 y(0)1090 3223 y Fr(:=)1238 3167 y(1)p 1238 3204 42 4 v 1238 3280 a(2)1330 3119 y FA(1)1304 3144 y Fn(X)1303 3323 y Fu(k)q Fo(=1)1438 3223 y Fq(S)1494 3180 y Fo(\()p Fu(n)p Fo(\))1488 3248 y Fu(k)1591 3223 y Fv(G)1656 3235 y Fu(k)1711 3223 y Fv(;)p Black 2203 w FB(\(43\))p Black 789 3634 a Fv(P)854 3591 y Fu(S)s(S;)p Fo([)p Fu(n)p Fo(])842 3657 y Fu(j)1090 3634 y Fr(:=)1238 3578 y(1)p 1238 3615 V 1238 3691 a(2)1303 3364 y Fn(8)1303 3439 y(>)1303 3464 y(>)1303 3489 y(>)1303 3514 y(>)1303 3538 y(<)1303 3688 y(>)1303 3713 y(>)1303 3738 y(>)1303 3763 y(>)1303 3788 y(:)1404 3530 y FA(1)1377 3555 y Fn(X)1377 3734 y Fu(k)q Fo(=1)1511 3442 y Fn(0)1511 3588 y(B)1511 3641 y(@)1584 3634 y Fq(S)1640 3591 y Fo(\()p Fu(n)p Fo(\))1634 3659 y Fu(k)1738 3634 y Fv(G)1803 3646 y Fu(k)q Fo(+)p Fu(j)1944 3634 y Fr(+)18 b Fq(S)2083 3591 y Fo(\()p Fu(n)p Fo(\))2077 3659 y Fu(k)2181 3634 y Fv(G)2246 3646 y Fu(k)q FA(\000)p Fu(j)2027 3694 y Fn(|)p 2064 3694 97 10 v 97 w({z)p 2235 3694 V 97 w(})2085 3773 y Fo(for)27 b Fu(k)q(>j)2370 3442 y Fn(1)2370 3588 y(C)2370 3641 y(A)2461 3634 y Fq(\000)2546 3526 y Fu(j)s FA(\000)p Fo(1)2544 3555 y Fn(X)2544 3734 y Fu(k)q Fo(=1)2678 3634 y Fq(S)2734 3591 y Fo(\()p Fu(n)p Fo(\))2728 3659 y Fu(k)2832 3634 y Fv(G)2897 3646 y Fu(j)s FA(\000)p Fu(k)2544 3777 y Fn(|)p 2581 3777 164 10 v 164 w({z)p 2819 3777 V 164 w(})2671 3856 y Fo(for)g Fu(j)s(>)p Fo(1)3021 3364 y Fn(9)3021 3439 y(>)3021 3464 y(>)3021 3489 y(>)3021 3514 y(>)3021 3538 y(=)3021 3688 y(>)3021 3713 y(>)3021 3738 y(>)3021 3763 y(>)3021 3788 y(;)3122 3634 y Fv(:)p Black 792 w FB(\(44\))p Black -67 3995 a(The)h(second)f(quan)n(tit)n(y) g(consists)g(of)h(a)f Fr(sin)h FB(and)g(a)f Fr(cos)g FB(series.)37 b(F)-7 b(actor)27 b(1)g(is)h(the)g(decomp)r(osition)f(of) h Fv(A)p Fr(\()p Fv(u)p Fr(\))3302 3965 y FA(\000)p Fu(n)3427 3995 y FB(in)g Fp(M)g FB(and)g(factor)f(2)-150 4091 y(is)g(\025)h (again)e(\025)h(the)h(non-secular)e(part)h(of)g(the)h(exp)r(onen)n (tial)f(of)h(the)g Fv(v)s Fr(\()p Fv(l)r Fr(\))p FB(-term.)272 4310 y SDict begin H.S end 272 4310 a 272 4310 a SDict begin 11.5 H.A end 272 4310 a 272 4310 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.45) cvn /DEST pdfmark end 272 4310 a 286 4168 a Fn( )379 4207 y FA(1)352 4232 y Fn(X)352 4410 y Fu(k)q Fo(=1)486 4310 y Fq(A)552 4267 y Fo(\()p Fu(n)p Fo(\))552 4335 y Fu(k)663 4310 y Fr(cos)13 b Fv(k)s Fp(M)925 4168 y Fn(!)g( )1109 4207 y FA(1)1082 4232 y Fn(X)1070 4407 y Fu(m)p Fo(=1)1227 4310 y Fv(G)1292 4322 y Fu(m)1369 4310 y Fr(sin)h Fv(m)p Fp(M)1648 4168 y Fn(!)1764 4310 y Fr(=)1888 4254 y(1)p 1888 4291 42 4 v 1888 4367 a(2)1981 4207 y FA(1)1954 4232 y Fn(X)1954 4410 y Fu(k)q Fo(=1)2127 4207 y FA(1)2100 4232 y Fn(X)2088 4407 y Fu(m)p Fo(=1)2245 4310 y Fq(A)2311 4267 y Fo(\()p Fu(n)p Fo(\))2311 4335 y Fu(k)2422 4310 y Fv(G)2487 4322 y Fu(m)2564 4310 y Fr(\()q(sin[)p Fv(k)21 b Fr(+)d Fv(m)p Fr(])p Fp(M)g Fq(\000)g Fr(sin[)p Fv(k)j Fq(\000)d Fv(m)p Fr(])p Fp(M)p Fr(\))937 4589 y(=)1052 4485 y FA(1)1025 4510 y Fn(X)1027 4687 y Fu(j)s Fo(=1)1159 4589 y Fv(P)1224 4546 y Fu(C)t(S;)p Fo([)p Fu(n)p Fo(])1212 4612 y Fu(j)1432 4589 y Fr(sin)c Fv(j)5 b Fp(M)13 b Fv(:)p Black 2223 w FB(\(45\))p Black -150 4847 a(As)28 b(w)n(ell)f(as)g(for)g Fv(P)435 4804 y Fu(S)s(S;)p Fo([)p Fu(n)p Fo(])423 4870 y Fu(j)622 4847 y FB(,)g(w)n(e)g(pro)n(vide)g(the)h(de\034nition)g(of)f(the)h Fv(P)1904 4804 y Fu(C)t(S;)p Fo([)p Fu(n)p Fo(])1892 4870 y Fu(j)2126 4847 y FB(as)f(com)n(binations)g(of)g(the)h(elemen)n (ts)f(of)h(the)g(t)n(w)n(o)f(series.)714 5205 y SDict begin H.S end 714 5205 a 714 5205 a SDict begin 11.5 H.A end 714 5205 a 714 5205 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.46) cvn /DEST pdfmark end 714 5205 a Fv(P)779 5161 y Fu(C)t(S;)p Fo([)p Fu(n)p Fo(])767 5228 y Fu(j)996 5205 y Fr(:=)1117 5148 y(1)p 1117 5185 42 4 v 1117 5261 a(2)1182 4935 y Fn(8)1182 5009 y(>)1182 5034 y(>)1182 5059 y(>)1182 5084 y(>)1182 5109 y(<)1182 5258 y(>)1182 5283 y(>)1182 5308 y(>)1182 5333 y(>)1182 5358 y(:)1259 5097 y Fu(j)s FA(\000)p Fo(1)1256 5126 y Fn(X)1256 5304 y Fu(k)q Fo(=1)1391 5112 y Fn(\020)1440 5205 y Fq(A)1506 5161 y Fo(\()p Fu(n)p Fo(\))1506 5230 y Fu(k)1604 5205 y Fv(G)1669 5217 y Fu(j)s FA(\000)p Fu(k)1792 5112 y Fn(\021)1256 5348 y(|)p 1293 5348 219 10 v 219 w({z)p 1586 5348 V 219 w(})1438 5427 y Fo(for)g Fu(j)s(>)p Fo(1)1856 5205 y Fq(\000)1962 5101 y FA(1)1935 5126 y Fn(X)1935 5304 y Fu(k)q Fo(=1)2069 5013 y Fn(0)2069 5159 y(B)2069 5212 y(@)2142 5205 y Fq(A)2208 5161 y Fo(\()p Fu(n)p Fo(\))2208 5230 y Fu(k)2305 5205 y Fv(G)2370 5217 y Fu(k)q FA(\000)p Fu(j)2142 5264 y Fn(|)p 2179 5264 102 10 v 102 w({z)p 2355 5264 V 102 w(})2204 5344 y Fo(for)h Fu(k)q(>j)2508 5205 y Fq(\000A)2639 5161 y Fo(\()p Fu(n)p Fo(\))2639 5230 y Fu(k)2736 5205 y Fv(G)2801 5217 y Fu(k)q Fo(+)p Fu(j)2923 5013 y Fn(1)2923 5159 y(C)2923 5212 y(A)2996 4935 y(9)2996 5009 y(>)2996 5034 y(>)2996 5059 y(>)2996 5084 y(>)2996 5109 y(=)2996 5258 y(>)2996 5283 y(>)2996 5308 y(>)2996 5333 y(>)2996 5358 y(;)3098 5205 y Fv(:)p Black 816 w FB(\(46\))p Black Black Black eop end end %%Page: 8 8 TeXDict begin HPSdict begin 8 7 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.8) cvn /DEST pdfmark end -158 -515 a Black 4043 -299 a FB(8)p Black -150 -83 a(The)35 b(t)n(w)n(o)f(ab)r(o)n(v)n(e)f(summation)i(form)n (ulas)e(will)i(also)f(b)r(e)h(pro)n(v)n(en)e(in)i(App)r(endix)2442 -83 y SDict begin H.S end 2442 -83 a Black FB(D)p Black 2505 -140 a SDict begin H.R end 2505 -140 a 2505 -83 a 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Fv(k)i(>)d(j)5 b FB(\021)35 b(ensure)-150 203 y(that,)30 b(for)f(example,)h(the)g(\034rst)f(term)g(in)h(\(4.19\)) f(is)g(not)h(able)f(to)g(con)n(tribute)g(to)h(the)f(\034rst)h(harmonic) e(in)i Fv(l)h FB(and)e(that)h(the)g(indices)f(of)-150 299 y(the)f(decomp)r(osition)f(co)r(e\036cien)n(ts)g(do)g(not)h(b)r (ecome)f(zero,)g(or)f(else,)i(these)f(elemen)n(ts)h(are)e(set)i(to)f (zero)g(automatically)-7 b(.)-67 394 y(The)30 b(next)h(step)f(is)g(to)g (sum)g(up)h(and)f(summarise)f(the)h(terms)g(app)r(earing)f(in)h(the)h (m)n(ultip)r(ole)f(momen)n(ts.)45 b(Using)30 b(Eq.)45 b(\()3756 394 y SDict begin H.S end 3756 394 a Black FB(22)p Black 3839 341 a SDict begin H.R end 3839 341 a 3839 394 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.22) cvn H.B /ANN pdfmark end 3839 394 a FB(\),)31 b(they)-150 490 y(also)26 b(con)n(tain)h(absolute)g(static)h(elemen)n(ts)f(and)g (will)h(b)r(e)g(listed)g(b)r(elo)n(w.)109 822 y SDict begin H.S 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1118 y Fr(1)j(+)i(\()p Fq(\000)p Fv(iq)s Fp(k)p Fr(\))1667 1014 y FA(1)1641 1039 y Fn(X)1643 1216 y Fu(j)s Fo(=1)1774 1118 y Fv(G)1839 1130 y Fu(j)1888 1118 y Fr(sin)14 b Fv(j)5 b Fp(M)2134 976 y Fn(#)2196 1026 y(\020)2246 1118 y Fr(1)18 b(+)g Fv(b)2425 1075 y Fo(\()p Fu(n)p Fo(\))2425 1140 y(0)2521 1026 y Fn(\021)2589 1118 y Fr(+)2672 976 y Fn(")2748 1014 y FA(1)2721 1039 y Fn(X)2723 1216 y Fu(j)s Fo(=1)2854 1118 y Fq(A)2920 1075 y Fo(\()p Fu(n)p Fo(\))2920 1141 y Fu(j)3032 1118 y Fr(cos)13 b Fv(j)5 b Fp(M)3287 976 y Fn(#)1120 1414 y Fr(+\()p Fq(\000)p Fv(iq)s Fp(k)p Fr(\))1474 1310 y FA(1)1447 1335 y Fn(X)1449 1512 y Fu(j)s Fo(=1)1581 1414 y Fv(P)1646 1371 y Fu(C)t(S;)p Fo([)p Fu(n)p Fo(])1634 1437 y Fu(j)1854 1414 y Fr(sin)14 b Fv(j)5 b Fp(M)2100 1272 y Fn(\))o(\()2233 1414 y Fv(e)2272 1380 y FA(\000)p Fo(\()p Fu(iq)r Fg(k)p Fo(\))p Fg(M)2546 1272 y Fn(\))1006 1710 y Fr(=)1120 1568 y Fn(\()1187 1618 y(\020)1237 1710 y Fr(1)18 b(+)g Fv(b)1416 1667 y Fo(\()p Fu(n)p Fo(\))1416 1732 y(0)1512 1618 y Fn(\021)1580 1710 y Fr(+)g(\()p Fq(\000)p Fv(iq)s Fp(k)p Fr(\))1952 1606 y FA(1)1925 1631 y Fn(X)1928 1808 y Fu(j)s Fo(=1)2059 1618 y Fn(h)2098 1710 y Fr(\(1)h(+)f Fv(b)2310 1667 y Fo(\()p Fu(n)p Fo(\))2310 1732 y(0)2406 1710 y Fr(\))p Fv(G)2503 1722 y Fu(j)2557 1710 y Fr(+)g Fv(P)2705 1667 y Fu(C)t(S;)p Fo([)p Fu(n)p Fo(])2693 1733 y Fu(j)2900 1618 y Fn(i)2953 1710 y Fr(sin)c Fv(j)5 b Fp(M)1120 2006 y Fr(+)1225 1902 y FA(1)1199 1927 y Fn(X)1201 2104 y Fu(j)s Fo(=1)1332 2006 y Fq(A)1398 1963 y Fo(\()p Fu(n)p Fo(\))1398 2029 y Fu(j)1510 2006 y Fr(cos)13 b Fv(j)5 b Fp(M)1765 1864 y Fn(\))o(\()1898 2006 y Fv(e)1937 1972 y FA(\000)p Fo(\()p Fu(iq)r Fg(k)p Fo(\))p Fg(M)2211 1864 y Fn(\))2292 2006 y Fv(:)p Black 1622 w FB(\(47\))p Black -150 2272 a(W)-7 b(e)28 b(will)g(p)r(erform)f(the)h(equiv)-5 b(alen)n(t)27 b(steps)g(for)340 2525 y SDict begin H.S end 340 2525 a 340 2525 a SDict begin 11.5 H.A end 340 2525 a 340 2525 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.48) cvn /DEST pdfmark end 340 2525 a 507 2469 a Fr(sin)14 b Fv(u)p 350 2506 478 4 v 350 2582 a Fr(\(1)k Fq(\000)g Fv(e)c Fr(cos)e Fv(u)p Fr(\))782 2558 y Fu(n)851 2525 y Fv(e)890 2491 y Fo(\()p FA(\000)p Fu(iq)r Fg(k)p Fo(\))p Fu(v)1122 2499 y Fm(e)1150 2510 y(t)1236 2525 y Fr(=)1351 2383 y Fn(\()1444 2421 y FA(1)1417 2446 y Fn(X)1420 2623 y Fu(j)s Fo(=1)1551 2525 y Fq(S)1607 2482 y Fo(\()p Fu(n)p Fo(\))1601 2548 y Fu(j)1719 2525 y Fr(sin)h Fv(j)5 b Fp(M)1964 2383 y Fn(\)\()2098 2525 y Fr(1)18 b(+)g(\()p Fq(\000)p Fv(iq)s Fp(k)p Fr(\))2541 2421 y FA(1)2514 2446 y Fn(X)2503 2622 y Fu(m)p Fo(=1)2660 2525 y Fv(G)2725 2537 y Fu(m)2802 2525 y Fr(sin)13 b Fv(m)p Fp(M)3081 2383 y Fn(\)\()3215 2525 y Fv(e)3254 2491 y Fo(\()p FA(\000)p Fu(iq)r Fg(k)p Fo(\))p Fg(M)3528 2383 y Fn(\))1236 2821 y Fr(=)1351 2679 y Fn(\()1444 2717 y FA(1)1417 2742 y Fn(X)1420 2919 y Fu(j)s Fo(=1)1551 2821 y Fq(S)1607 2778 y Fo(\()p Fu(n)p Fo(\))1601 2844 y Fu(j)1719 2821 y Fr(sin)g Fv(j)5 b Fp(M)18 b Fr(+)g(\()p Fq(\000)p Fv(iq)s Fp(k)p Fr(\))2354 2717 y FA(1)2327 2742 y Fn(X)2330 2919 y Fu(j)s Fo(=0)2461 2821 y Fv(P)2526 2778 y Fu(S)s(S;)p Fo([)p Fu(n)p Fo(])2514 2844 y Fu(j)2727 2821 y Fr(cos)13 b Fv(j)5 b Fp(M)2982 2679 y Fn(\))o(\()3115 2821 y Fv(e)3154 2787 y Fo(\()p FA(\000)p Fu(iq)r Fg(k)p Fo(\))p Fg(M)3428 2679 y Fn(\))3509 2821 y Fv(:)p Black 405 w FB(\(48\))p Black -150 3087 a(This)28 b(will)f(b)r(e)h(shortened)f(b)n(y)g(writing) 278 3365 y SDict begin H.S end 278 3365 a 278 3365 a SDict begin 11.5 H.A end 278 3365 a 278 3365 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.49) cvn /DEST pdfmark end 278 3365 a 506 3309 a Fr(1)p 288 3346 478 4 v 288 3422 a(\(1)18 b Fq(\000)g Fv(e)c Fr(cos)f Fv(u)p Fr(\))721 3398 y Fu(n)789 3365 y Fv(e)828 3330 y Fo(\()p FA(\000)p Fu(iq)r Fg(k)p Fo(\))p Fu(v)1060 3338 y Fm(e)1088 3349 y(t)1186 3365 y Fr(=)1312 3195 y Fn(8)1312 3269 y(<)1312 3419 y(:)1386 3365 y Fr(\()1447 3344 y(\026)1418 3365 y Fq(A)1484 3377 y Fu(c)1518 3365 y Fr(\))1550 3322 y Fo([)p Fu(n;q)r Fo(])1550 3387 y(0)1704 3365 y Fr(+)1814 3261 y FA(1)1787 3286 y Fn(X)1789 3463 y Fu(j)s Fo(=1)1907 3365 y Fr(\()1968 3344 y(\026)1939 3365 y Fq(A)2005 3377 y Fu(c)2039 3365 y Fr(\))2071 3322 y Fo([)p Fu(n;q)r Fo(])2071 3388 y Fu(j)2220 3365 y Fr(cos)g Fv(j)5 b Fp(M)18 b Fr(+)2603 3261 y FA(1)2576 3286 y Fn(X)2579 3463 y Fu(j)s Fo(=1)2696 3365 y Fr(\()2757 3344 y(\026)2728 3365 y Fq(A)2794 3377 y Fu(s)2830 3365 y Fr(\))2862 3322 y Fo([)p Fu(n;q)r Fo(])2862 3388 y Fu(j)3012 3365 y Fr(sin)13 b Fv(j)5 b Fp(M)3257 3195 y Fn(9)3257 3269 y(=)3257 3419 y(;)3345 3365 y Fv(e)3384 3330 y FA(\000)p Fu(iq)r Fg(kM)3619 3365 y Fv(;)p Black 295 w FB(\(49\))p Black 445 3641 a Fr(sin)14 b Fv(u)p 288 3678 V 288 3754 a Fr(\(1)k Fq(\000)g Fv(e)c Fr(cos)f Fv(u)p Fr(\))721 3730 y Fu(n)789 3697 y Fv(e)828 3663 y Fo(\()p FA(\000)p Fu(iq)r Fg(k)p Fo(\))p Fu(v)1060 3671 y Fm(e)1088 3682 y(t)1186 3697 y Fr(=)1312 3527 y Fn(8)1312 3601 y(<)1312 3751 y(:)1386 3697 y Fr(\()1437 3676 y(\026)1418 3697 y Fq(S)1468 3709 y Fu(c)1502 3697 y Fr(\))1534 3654 y Fo([)p Fu(n;q)r Fo(])1534 3719 y(0)1688 3697 y Fr(+)1798 3593 y FA(1)1771 3618 y Fn(X)1773 3795 y Fu(j)s Fo(=1)1891 3697 y Fr(\()1942 3676 y(\026)1923 3697 y Fq(S)1973 3709 y Fu(c)2007 3697 y Fr(\))2039 3654 y Fo([)p Fu(n;q)r Fo(])2039 3720 y Fu(j)2188 3697 y Fr(cos)g Fv(j)5 b Fp(M)18 b Fr(+)2571 3593 y FA(1)2544 3618 y Fn(X)2547 3795 y Fu(j)s Fo(=1)2664 3697 y Fr(\()2716 3676 y(\026)2696 3697 y Fq(S)2746 3709 y Fu(s)2782 3697 y Fr(\))2814 3654 y Fo([)p Fu(n;q)r Fo(])2814 3720 y Fu(j)2963 3697 y Fr(sin)c Fv(j)5 b Fp(M)3209 3527 y Fn(9)3209 3601 y(=)3209 3751 y(;)3296 3697 y Fv(e)3335 3663 y FA(\000)p Fu(iq)r Fg(kM)3571 3697 y Fv(;)p Black 343 w FB(\(50\))p Black 823 3954 a Fr(\()884 3933 y(\026)855 3954 y Fq(A)921 3966 y Fu(s)957 3954 y Fr(\))989 3911 y Fo([)p Fu(n;q)r Fo(])989 3977 y Fu(j)s(>)p Fo(0)1175 3954 y Fr(:=)49 b(\()p Fq(\000)p Fv(iq)s Fp(k)p Fr(\))1574 3862 y Fn(h)1613 3954 y Fr(\(1)18 b(+)g Fv(b)1824 3911 y Fo(\()p Fu(n)p Fo(\))1824 3976 y(0)1921 3954 y Fr(\))p Fv(G)2018 3966 y Fu(j)2072 3954 y Fr(+)g Fv(P)2220 3911 y Fu(C)t(S;)p Fo([)p Fu(n)p Fo(])2208 3977 y Fu(j)2415 3862 y Fn(i)2481 3954 y Fv(;)p Black 1433 w FB(\(51\))p Black 825 4137 a Fr(\()886 4116 y(\026)857 4137 y Fq(A)923 4149 y Fu(c)957 4137 y Fr(\))989 4094 y Fo([)p Fu(n;q)r Fo(])989 4159 y(0)1175 4137 y Fr(:=)1312 4045 y Fn(h)1351 4137 y Fr(\(1)g(+)g Fv(b)1562 4094 y Fo(\()p Fu(n)p Fo(\))1562 4159 y(0)1659 4137 y Fr(\))1691 4045 y Fn(i)1758 4137 y Fv(;)p Black 2156 w FB(\(52\))p Black 825 4311 a Fr(\()886 4290 y(\026)857 4311 y Fq(A)923 4323 y Fu(c)957 4311 y Fr(\))989 4268 y Fo([)p Fu(n;q)r Fo(])989 4334 y Fu(j)s(>)p Fo(0)1175 4311 y Fr(:=)49 b Fq(A)1378 4268 y Fo(\()p Fu(n)p Fo(\))1378 4334 y Fu(j)1489 4311 y Fv(;)p Black 2425 w FB(\(53\))p Black 841 4465 a Fr(\()892 4444 y(\026)873 4465 y Fq(S)923 4477 y Fu(c)957 4465 y Fr(\))989 4422 y Fo([)p Fu(n;q)r Fo(])989 4488 y Fu(j)s(>)p Fo(0)1175 4465 y Fr(:=)g(\()p Fq(\000)p Fv(iq)s Fp(k)p Fr(\))p Fv(P)1625 4422 y Fu(S)s(S;)p Fo([)p Fu(n)p Fo(])1613 4488 y Fu(j)1826 4465 y Fv(;)p Black 2088 w FB(\(54\))p Black 839 4619 a Fr(\()891 4598 y(\026)871 4619 y Fq(S)921 4631 y Fu(s)957 4619 y Fr(\))989 4576 y Fo([)p Fu(n;q)r Fo(])989 4643 y Fu(j)s(>)p Fo(0)1175 4619 y Fr(:=)g Fq(S)1368 4576 y Fo(\()p Fu(n)p Fo(\))1362 4643 y Fu(j)1479 4619 y Fv(:)p Black 2435 w FB(\(55\))p Black -67 4798 a(W)-7 b(e)29 b(are)f(no)n(w)g(in)i(the)f(luc)n(ky)f(p)r (osition)h(to)g(decomp)r(ose)f(all)g(p)r(oin)n(t)i(particle)e(con)n (tributions)g(to)h(the)g(radiation)e(\034eld)j(amplitude)f(in)-150 4909 y(terms)e(of)h(the)g(ab)r(o)n(v)n(e)e(de\034ned)i Fr(\()901 4888 y(\026)872 4909 y Fq(A)938 4921 y Fu(s)974 4909 y Fr(\))1006 4866 y Fo([)p Fu(n;q)r Fo(])1006 4932 y Fu(j)1141 4909 y FB(,)g Fr(\()1253 4888 y(\026)1224 4909 y Fq(A)1290 4921 y Fu(c)1324 4909 y Fr(\))1356 4866 y Fo([)p Fu(n;q)r Fo(])1356 4932 y Fu(j)1492 4909 y FB(,)f Fr(\()1594 4888 y(\026)1574 4909 y Fq(S)1624 4921 y Fu(s)1660 4909 y Fr(\))1692 4866 y Fo([)p Fu(n;q)r Fo(])1692 4932 y Fu(j)1828 4909 y FB(,)g(and)h Fr(\()2091 4888 y(\026)2072 4909 y Fq(S)2122 4921 y Fu(c)2156 4909 y Fr(\))2188 4866 y Fo([)p Fu(n;q)r Fo(])2188 4932 y Fu(j)2323 4909 y FB(.)37 b(F)-7 b(or)27 b(simplicit)n(y)-7 b(,)28 b(w)n(e)f(in)n(tro)r(duce)1162 5145 y SDict begin H.S end 1162 5145 a 1162 5145 a SDict begin 11.5 H.A end 1162 5145 a 1162 5145 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.56) cvn /DEST pdfmark end 1162 5145 a 1390 5088 a Fr(1)p 1172 5125 478 4 v 1172 5201 a(\(1)18 b Fq(\000)g Fv(e)c Fr(cos)e Fv(u)p Fr(\))1604 5178 y Fu(n)1659 5145 y Fv(e)1698 5110 y FA(\000)p Fu(iq)r Fg(k)p Fu(v)1878 5118 y Fm(e)1906 5129 y(t)1992 5145 y Fr(=:)50 b Fq(F)2190 5160 y Fo([)p Fu(nq)r Fo(])2305 5145 y Fr(\()p Fv(u)p Fr(\))14 b Fv(e)2470 5110 y FA(\000)p Fu(iq)r Fg(kM)2706 5145 y Fv(;)p Black 1208 w FB(\(56\))p Black 1329 5311 a Fr(sin)f Fv(u)p 1171 5348 V 1171 5424 a Fr(\(1)19 b Fq(\000)f Fv(e)c Fr(cos)e Fv(u)p Fr(\))1604 5400 y Fu(n)1659 5367 y Fv(e)1698 5333 y FA(\000)p Fu(iq)r Fg(k)p Fu(v)1878 5341 y Fm(e)1906 5352 y(t)1992 5367 y Fr(=:)50 b Fq(F)2190 5382 y Fu(S)s Fo([)p Fu(nq)r Fo(])2349 5367 y Fr(\()p Fv(u)p Fr(\))14 b Fv(e)2514 5333 y FA(\000)p Fu(iq)r Fg(kM)2750 5367 y Fv(;)p Black 1164 w FB(\(57\))p Black Black Black eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.9) cvn /DEST pdfmark end -158 -515 a Black 4043 -299 a FB(9)p Black -150 -83 a(and)26 b(use)h(them)g(from)f(no)n(w)g(on)g(in)h(the)g(m)n (ultip)r(oles,)f(where)g Fv(n)h FB(and)f Fv(q)k FB(will)d(also)e(b)r(e) 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Fg(k)p Fu(v)2292 260 y Fm(e)2320 271 y(t)857 519 y Fr(=)49 b Fv(e)1010 484 y FA(\000)p Fu(im\036)1184 492 y Fl(0)1234 451 y Fn(\002)1269 519 y Fv(e)1308 484 y FA(\000)p Fu(imv)1475 492 y Fm(e)1503 503 y(t)1553 519 y Fv(f)1594 531 y Fu(ml)1696 519 y Fr(+)18 b Fq(O)r Fr(\()p Fv(\017)1913 484 y Fo(2)1951 519 y Fr(\))1983 451 y Fn(\003)2046 519 y Fv(e)2085 484 y FA(\000)p Fu(im)p Fg(kM)2347 352 y Fn(0)2347 501 y(@)2419 519 y Fr(1)g(+)g(\()p Fq(\000)p Fv(im)p Fp(k)p Fr(\))2884 415 y FA(1)2857 440 y Fn(X)2860 616 y Fu(j)s Fo(=1)2991 519 y Fv(G)3056 531 y Fu(j)3105 519 y Fr(sin)c Fv(j)5 b Fp(M)p Fr(\))3383 352 y Fn(1)3383 501 y(A)857 793 y Fr(=)49 b Fv(e)1010 758 y FA(\000)p Fu(im\036)1184 766 y Fl(0)1234 714 y Fn(X)1279 890 y Fu(j)1368 725 y Fn(\002)1403 793 y Fv(\024)1451 808 y Fo([)p Fu(j)s(m)p Fo(])1596 793 y Fq(F)1656 808 y Fo([)p Fu(j)s(m)p Fo(])1787 793 y Fr(\()p Fv(u)p Fr(\))19 b(+)i(~)-45 b Fv(\024)2049 808 y Fo([)p Fu(j)s(m)p Fo(])2194 793 y Fq(F)2254 808 y Fu(S)s Fo([)p Fu(j)s(m)p 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Fo(\))720 1653 y Fu(n)840 1643 y FB(,)g Fv(:)14 b(:)g(:)50 b FB(and)43 b Fr(~)-50 b Fv(\013)1270 1600 y Fo(\()p Fu(m)p Fo(\))1270 1653 y Fu(n)1386 1643 y FB(,)1459 1621 y Fr(~)1447 1643 y Fv(\014)1498 1600 y Fo(\()p Fu(m)p Fo(\))1494 1653 y Fu(n)1613 1643 y FB(,)38 b Fv(:)14 b(:)g(:)50 b FB(for)35 b(the)h(di\033eren)n(t)g(m)n(ultip)r(ole)h(t)n(yp)r(es,)g(where)f Fv(m)g FB(denotes)f(again)g(the)-150 1739 y(magnetic)27 b(n)n(um)n(b)r(er)g(and)h Fv(n)f FB(is)h(the)g(same)f(exp)r(onen)n(t)g (lab)r(el)h(as)e(in)i(Defs.)38 b(\()2188 1739 y SDict begin H.S end 2188 1739 a Black FB(56)p Black 2271 1685 a SDict begin H.R end 2271 1685 a 2271 1739 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.56) cvn H.B /ANN pdfmark end 2271 1739 a FB(\).)f(and)28 b(\()2557 1739 y SDict begin H.S end 2557 1739 a Black FB(57)p Black 2640 1685 a SDict begin H.R end 2640 1685 a 2640 1739 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.56) cvn H.B /ANN pdfmark end 2640 1739 a FB(\).)183 1975 y SDict begin H.S end 183 1975 a 183 1975 a SDict begin 11.5 H.A end 183 1975 a 183 1975 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.59) cvn /DEST pdfmark end 183 1975 a 209 1902 a Fo(\(2\))227 1983 y Fv(I)291 1987 y Fo(22)473 1975 y Fr(=)112 b(8)692 1845 y Fn(r)p 774 1845 112 4 v 784 1919 a Fr(2)p Fv(\031)p 784 1956 92 4 v 809 2032 a Fr(5)900 1975 y Fv(E)19 b(\026)14 b(e)1083 1940 y FA(\000)p Fo(2)1083 1996 y Fu(t)1185 1975 y Fv(e)1224 1941 y FA(\000)p Fo(2)p Fu(i)p Fg(kM)1447 1975 y Fv(e)1486 1941 y FA(\000)p Fu(i)p Fo(2)p Fu(\036)1634 1949 y Fl(0)1684 1833 y Fn(\()1795 1871 y Fo(5)1752 1896 y Fn(X)1751 2075 y Fu(k)q Fo(=0)1886 1975 y Fv(\013)1939 1990 y Fo([)p Fu(k)q Fo(2])2050 1975 y Fq(F)2110 1990 y Fo([)p Fu(k)q Fo(2])2222 1975 y Fr(\()p Fv(u)p Fr(\))k(+)2479 1871 y Fo(5)2436 1896 y Fn(X)2435 2075 y Fu(k)q Fo(=1)2578 1975 y Fr(~)-50 b Fv(\013)2623 1990 y Fo([)p Fu(k)q Fo(2])2734 1975 y Fq(F)2794 1990 y Fu(S)s Fo([)p Fu(k)q Fo(2])2950 1975 y Fr(\()p Fv(u)p Fr(\))3062 1833 y Fn(\))3142 1975 y Fv(;)p Black 772 w FB(\(59\))p Black 200 2226 a Fv(\013)253 2241 y Fo([02])462 2226 y Fr(:=)650 2159 y Fn(\002)684 2226 y Fr(2)18 b Fq(\000)g Fv(e)866 2192 y Fo(2)866 2247 y Fu(t)903 2159 y Fn(\003)956 2226 y Fr(+)g Fv(\017)1073 2192 y Fo(2)1155 2170 y Fr(1)p 1134 2207 84 4 v 1134 2283 a(14)1227 2226 y Fq(j)p Fv(E)5 b Fq(j)1353 2159 y Fn(\002)1388 2226 y Fr(\(9)p Fv(\021)21 b Fq(\000)d Fr(3\))p Fv(e)1720 2192 y Fo(2)1720 2247 y Fu(t)1775 2226 y Fr(+)g(94)p Fv(\021)j Fq(\000)d Fr(442)2213 2159 y Fn(\003)2274 2226 y Fv(;)p Black 1640 w FB(\(60\))p Black 200 2426 a Fv(\013)253 2441 y Fo([12])462 2426 y Fr(:=)650 2359 y Fn(\002)684 2426 y Fv(e)723 2392 y Fo(2)723 2446 y Fu(t)779 2426 y Fq(\000)g Fr(2)904 2359 y Fn(\003)956 2426 y Fr(+)g Fv(\017)1073 2392 y Fo(2)1141 2370 y Fr(1)p 1120 2407 V 1120 2483 a(14)1213 2426 y Fq(j)p Fv(E)5 b Fq(j)1339 2359 y Fn(\002)1374 2426 y Fr(\(285)p Fv(\021)20 b Fq(\000)e Fr(781\))p Fv(e)1873 2392 y Fo(2)1873 2446 y Fu(t)1927 2426 y Fq(\000)g Fr(346)p Fv(\021)j Fr(+)d(1450)2449 2359 y Fn(\003)2509 2426 y Fv(;)p Black 1405 w FB(\(61\))p Black 200 2626 a Fv(\013)253 2641 y Fo([22])462 2626 y Fr(:=)100 b(2)706 2559 y Fn(\000)743 2626 y Fv(e)782 2591 y Fo(2)782 2646 y Fu(t)837 2626 y Fq(\000)18 b Fr(1)962 2559 y Fn(\001)1019 2626 y Fr(+)g Fv(\017)1136 2591 y Fo(2)1186 2626 y Fq(j)p Fv(E)5 b Fq(j)1329 2570 y Fr(1)p 1308 2607 V 1308 2683 a(21)1415 2559 y Fn(\002)1450 2626 y Fr(84\(3)p Fv(\021)20 b Fq(\000)e Fr(8\))p Fv(e)1865 2591 y Fo(4)1865 2646 y Fu(t)1921 2626 y Fr(+)g(\(3713)e Fq(\000)i Fr(1458)p Fv(\021)s Fr(\))p Fv(e)2586 2591 y Fo(2)2586 2646 y Fu(t)2640 2626 y Fr(+)g(981)p Fv(\021)i Fq(\000)e Fr(2861)3161 2559 y Fn(\003)3222 2626 y Fv(;)p Black 692 w FB(\(62\))p Black 200 2836 a Fv(\013)253 2851 y Fo([32])462 2836 y Fr(:=)100 b(2)706 2769 y Fn(\002)740 2836 y Fv(e)779 2802 y Fo(2)779 2857 y Fu(t)834 2836 y Fq(\000)18 b Fr(1)959 2769 y Fn(\003)993 2785 y Fo(2)1049 2836 y Fr(+)g Fv(\017)1166 2802 y Fo(2)1203 2836 y Fq(j)p Fv(E)5 b Fq(j)1315 2719 y Fn(\024)1359 2836 y Fq(\000)1454 2780 y Fr(1)p 1434 2817 V 1434 2893 a(21)1540 2769 y Fn(\000)1578 2836 y Fv(e)1617 2802 y Fo(2)1617 2857 y Fu(t)1673 2836 y Fq(\000)18 b Fr(1)1798 2769 y Fn(\001)13 b(\000)1887 2836 y Fr(\(663)p Fv(\021)21 b Fq(\000)d Fr(1537\))p Fv(e)2429 2802 y Fo(2)2429 2857 y Fu(t)2482 2836 y Fq(\000)g Fr(1005)p Fv(\021)i Fr(+)e(1861)3045 2769 y Fn(\001)3081 2719 y(\025)3139 2836 y Fv(;)p Black 775 w FB(\(63\))p Black 200 3069 a Fv(\013)253 3084 y Fo([42])462 3069 y Fr(:=)100 b Fv(\017)684 3034 y Fo(2)721 3069 y Fq(j)p Fv(E)5 b Fq(j)847 2952 y Fn(\024)921 3013 y Fr(2)p 900 3050 V 900 3126 a(21)993 3069 y(\(219)p Fv(\021)21 b Fq(\000)d Fr(262\))1468 3002 y Fn(\000)1505 3069 y Fv(e)1544 3034 y Fo(2)1544 3089 y Fu(t)1599 3069 y Fq(\000)g Fr(1)1724 3002 y Fn(\001)1762 3018 y Fo(2)1799 3069 y Fr(\))1831 2952 y Fn(\025)1903 3069 y Fv(;)p Black 2011 w FB(\(64\))p Black 200 3301 a Fv(\013)253 3316 y Fo([52])462 3301 y Fr(:=)100 b Fv(\017)684 3267 y Fo(2)721 3301 y Fq(j)p Fv(E)5 b Fq(j)847 3184 y Fn(\024)900 3245 y Fr(4)p 900 3282 42 4 v 900 3358 a(7)952 3301 y(\(3)p Fv(\021)21 b Fq(\000)d Fr(1\))1259 3234 y Fn(\000)1297 3301 y Fv(e)1336 3267 y Fo(2)1336 3322 y Fu(t)1391 3301 y Fq(\000)g Fr(1)1516 3234 y Fn(\001)1554 3250 y Fo(3)1591 3184 y Fn(\025)1663 3301 y Fv(;)p Black 2251 w FB(\(65\))p Black 208 3559 a Fr(~)-50 b Fv(\013)253 3574 y Fo([12])462 3559 y Fr(:=)650 3442 y Fn(\024)694 3559 y Fr(2)p Fv(ie)804 3571 y Fu(t)832 3457 y Fn(q)p 915 3457 219 4 v 102 x Fr(1)18 b Fq(\000)g Fv(e)1097 3530 y Fo(2)1097 3579 y Fu(t)1133 3442 y Fn(\025)1196 3559 y Fr(+)g Fv(\017)1313 3524 y Fo(2)1364 3417 y Fn(")1422 3497 y Fv(i)p Fq(j)p Fv(E)5 b Fq(j)p Fv(e)1602 3509 y Fu(t)1644 3430 y Fn(\000)1682 3497 y Fr(\(109)18 b Fq(\000)g Fr(19)p Fv(\021)s Fr(\))p Fv(e)2140 3467 y Fo(2)2140 3518 y Fu(t)2195 3497 y Fr(+)g(47)p Fv(\021)i Fq(\000)e Fr(221)2632 3430 y Fn(\001)p 1422 3540 1248 4 v 1874 3629 a Fr(7)1916 3556 y Fn(p)p 1998 3556 219 4 v 1998 3629 a Fr(1)g Fq(\000)g Fv(e)2180 3601 y Fo(2)2180 3650 y Fu(t)2679 3417 y Fn(#)2755 3559 y Fv(;)p Black 1159 w FB(\(66\))p Black 208 3841 a Fr(~)-50 b Fv(\013)253 3856 y Fo([22])462 3841 y Fr(:=)100 b Fv(\017)684 3807 y Fo(2)735 3699 y Fn(")783 3841 y Fq(\000)p Fv(i)887 3779 y Fr(2)p Fq(j)p Fv(E)5 b Fq(j)o Fv(e)1079 3791 y Fu(t)1122 3712 y Fn(\000)1160 3779 y Fr(2\(3)p Fv(\021)21 b Fq(\000)d Fr(8\))p Fv(e)1534 3749 y Fo(4)1534 3800 y Fu(t)1589 3779 y Fr(+)g(\(46)f Fq(\000)h Fr(15)p Fv(\021)s Fr(\))p Fv(e)2087 3749 y Fo(2)2087 3800 y Fu(t)2142 3779 y Fr(+)g(9\()p Fv(\021)k Fq(\000)c Fr(4\))2519 3712 y Fn(\001)p 887 3822 1671 4 v 1570 3838 a(p)p 1653 3838 219 4 v 73 x Fr(1)g Fq(\000)g Fv(e)1835 3883 y Fo(2)1835 3932 y Fu(t)2567 3699 y Fn(#)2643 3841 y Fv(;)p Black 1271 w FB(\(67\))p Black 208 4098 a Fr(~)-50 b Fv(\013)253 4113 y Fo([32])462 4098 y Fr(:=)650 4006 y Fn(h)689 4098 y Fq(\000)p Fr(2)p Fv(ie)864 4110 y Fu(t)905 4031 y Fn(\000)943 4098 y Fr(1)18 b Fq(\000)g Fv(e)1125 4064 y Fo(2)1125 4119 y Fu(t)1162 4031 y Fn(\001)1200 4047 y Fo(3)p Fu(=)p Fo(2)1305 4006 y Fn(i)1362 4098 y Fr(+)g Fv(i\017)1508 4064 y Fo(2)1545 3981 y Fn(\024)1619 4042 y Fr(1)p 1599 4079 84 4 v 1599 4155 a(21)1692 4098 y Fq(j)p Fv(E)5 b Fq(j)p Fv(e)1843 4110 y Fu(t)1872 3996 y Fn(q)p 1955 3996 219 4 v 102 x Fr(1)18 b Fq(\000)g Fv(e)2137 4070 y Fo(2)2137 4119 y Fu(t)2187 4031 y Fn(\000)2226 4098 y Fr(\(1531)e Fq(\000)i Fr(645)p Fv(\021)s Fr(\))p Fv(e)2766 4064 y Fo(2)2766 4119 y Fu(t)2821 4098 y Fr(+)g(603)p Fv(\021)i Fq(\000)e Fr(1349)3342 4031 y Fn(\001)3378 3981 y(\025)3436 4098 y Fv(;)p Black 478 w FB(\(68\))p Black 208 4331 a Fr(~)-50 b Fv(\013)253 4346 y Fo([42])462 4331 y Fr(:=)100 b Fv(i\017)713 4296 y Fo(2)749 4214 y Fn(\024)793 4331 y Fq(\000)888 4275 y Fr(2)p 868 4312 84 4 v 868 4388 a(21)961 4331 y Fq(j)p Fv(E)5 b Fq(j)p Fr(\(201)p Fv(\021)20 b Fq(\000)e Fr(256\))p Fv(e)1572 4343 y Fu(t)1614 4263 y Fn(\000)1652 4331 y Fr(1)g Fq(\000)g Fv(e)1834 4296 y Fo(2)1834 4351 y Fu(t)1871 4263 y Fn(\001)1909 4280 y Fo(3)p Fu(=)p Fo(2)2013 4214 y Fn(\025)2071 4331 y Fv(;)p Black 1843 w FB(\(69\))p Black 208 4563 a Fr(~)-50 b Fv(\013)253 4578 y Fo([52])462 4563 y Fr(:=)100 b Fv(i\017)713 4529 y Fo(2)763 4446 y Fn(\024)817 4507 y Fr(4)p 817 4544 42 4 v 817 4620 a(7)868 4563 y Fq(j)p Fv(E)5 b Fq(j)q Fr(\(3)p Fv(\021)21 b Fq(\000)d Fr(1\))p Fv(e)1313 4575 y Fu(t)1356 4496 y Fn(\000)1394 4563 y Fr(1)g Fq(\000)g Fv(e)1576 4529 y Fo(2)1576 4584 y Fu(t)1612 4496 y Fn(\001)1664 4529 y Fo(5)p Fu(=)p Fo(2)1769 4446 y Fn(\025)1840 4563 y Fv(;)p Black 2074 w FB(\(70\))p Black -150 4670 4235 4 v 209 4722 a Fo(\(2\))227 4803 y Fv(I)291 4807 y Fo(21)473 4795 y Fr(=)112 b(0)14 b Fv(;)p Black 3208 w FB(\(71\))p Black -150 4844 V 209 4899 a Fo(\(2\))227 4980 y Fv(I)291 4984 y Fo(20)473 4972 y Fr(=)112 b Fq(\000)p Fr(16)799 4851 y Fn(r)p 880 4851 103 4 v 907 4916 a Fv(\031)p 890 4953 84 4 v 890 5029 a Fr(15)983 4972 y Fv(E)5 b(\026)1099 4880 y Fn(n)1155 4972 y Fv(\013)1208 4987 y Fo([00])1330 4972 y Fq(F)1390 4987 y Fo([00])1497 4972 y Fr(\()p Fv(u)p Fr(\))19 b(+)f Fv(\013)1764 4987 y Fo([10])1886 4972 y Fq(F)1946 4987 y Fo([10])2053 4972 y Fr(\()p Fv(u)p Fr(\))h(+)f Fv(\013)2320 4987 y Fo([20])2442 4972 y Fq(F)2502 4987 y Fo([20])2609 4972 y Fr(\()p Fv(u)p Fr(\))h(+)f Fv(\013)2876 4987 y Fo([30])2998 4972 y Fq(F)3058 4987 y Fo([30])3165 4972 y Fr(\()p Fv(u)p Fr(\))3277 4880 y Fn(o)3347 4972 y Fv(;)p Black 567 w FB(\(72\))p Black 200 5189 a Fv(\013)253 5204 y Fo([00])462 5189 y Fr(:=)100 b(1)18 b Fq(\000)823 5133 y Fr(3)p 803 5170 V 803 5246 a(14)896 5189 y Fq(j)p Fv(E)5 b Fq(j)p Fv(\017)1042 5155 y Fo(2)1079 5189 y Fr(\(3)p Fv(\021)21 b Fq(\000)d Fr(1\))c Fv(;)p Black 2528 w FB(\(73\))p Black 200 5389 a Fv(\013)253 5404 y Fo([10])462 5389 y Fr(:=)680 5333 y(1)p 660 5370 V 660 5446 a(14)753 5389 y Fq(j)p Fv(E)5 b Fq(j)p Fv(\017)899 5355 y Fo(2)936 5389 y Fr(\(51)p Fv(\021)21 b Fq(\000)d Fr(115\))f Fq(\000)h Fr(1)c Fv(;)p Black 2403 w FB(\(74\))p Black Black Black eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.10) cvn /DEST pdfmark end -158 -515 a Black 4001 -299 a FB(10)p Black 200 -56 a Fv(\013)253 -41 y Fo([20])462 -56 y Fr(:=)660 -113 y(1)p 660 -75 42 4 v 660 1 a(7)711 -56 y Fq(j)p Fv(E)5 b Fq(j)p Fv(\017)857 -91 y Fo(2)894 -56 y Fr(\(4)19 b Fq(\000)f Fr(19)p Fv(\021)s Fr(\))c Fv(;)p Black 2670 w FB(\(75\))p Black 200 143 a Fv(\013)253 158 y Fo([30])462 143 y Fr(:=)660 87 y(2)p 660 124 V 660 200 a(7)711 143 y Fq(j)p Fv(E)5 b Fq(j)p Fv(\017)857 109 y Fo(2)894 143 y Fr(\()p Fv(\021)22 b Fq(\000)c Fr(26\))1202 76 y Fn(\000)1239 143 y Fv(e)1278 109 y Fo(2)1278 164 y Fu(t)1334 143 y Fq(\000)g Fr(1)1459 76 y Fn(\001)1524 143 y Fv(;)p Black 2390 w FB(\(76\))p Black -150 229 4235 4 v 208 281 a Fo(\(2\))220 361 y Fv(S)291 365 y Fo(22)473 353 y Fr(=)112 b(0)14 b Fv(;)p Black 3208 w FB(\(77\))p Black -150 402 V 208 458 a Fo(\(2\))220 539 y Fv(S)291 543 y Fo(21)473 531 y Fr(=)660 474 y(32)p 660 511 84 4 v 680 588 a(3)753 409 y Fn(r)p 836 409 71 4 v 846 474 a Fv(\031)p 846 511 51 4 v 850 588 a Fr(5)906 452 y Fn(p)p 989 452 249 4 v 79 x Fr(1)k Fq(\000)g Fv(e)1171 543 y Fu(t)1200 507 y Fo(2)1237 531 y Fv(\021)s Fr(\()p Fq(\000)p Fv(E)5 b Fr(\))1476 489 y Fo(3)p Fu(=)p Fo(2)1581 531 y Fr(\()p Fv(m)1686 543 y Fo(1)1742 531 y Fq(\000)18 b Fv(m)1898 543 y Fo(2)1935 531 y Fr(\))c Fv(e)2020 496 y FA(\000)p Fu(i)p Fg(kM)2209 531 y Fv(e)2248 496 y FA(\000)p Fu(i\036)2363 504 y Fl(0)2418 531 y Fq(\002)650 641 y Fn(n)705 734 y Fv(\014)752 749 y Fo([21])860 734 y Fq(F)920 749 y Fo([21])1028 734 y Fr(\()p Fv(u)p Fr(\))k(+)g Fv(\014)1288 749 y Fo([31])1396 734 y Fq(F)1456 749 y Fo([31])1564 734 y Fr(\()p Fv(u)p Fr(\))g(+)1789 712 y(~)1777 734 y Fv(\014)1824 749 y Fo([31])1932 734 y Fq(F)1992 749 y Fu(S)s Fo([31])2144 734 y Fr(\()p Fv(u)p Fr(\))2256 641 y Fn(o)2325 734 y Fv(;)p Black 1589 w FB(\(78\))p Black 206 930 a Fv(\014)253 945 y Fo([21])462 930 y Fr(:=)100 b Fq(\000)737 874 y Fr(1)p 725 911 68 4 v 725 987 a Fv(e)764 999 y Fu(t)816 930 y Fv(;)p Black 3098 w FB(\(79\))p Black 206 1143 a Fv(\014)253 1158 y Fo([31])462 1143 y Fr(:=)673 1087 y(1)p 660 1124 V 660 1200 a Fv(e)699 1212 y Fu(t)756 1143 y Fq(\000)18 b Fv(e)878 1155 y Fu(t)921 1143 y Fv(;)p Black 2993 w FB(\(80\))p Black 218 1332 a Fr(~)206 1354 y Fv(\014)253 1369 y Fo([31])462 1354 y Fr(:=)100 b Fq(\000)p Fv(i)744 1252 y Fn(q)p 826 1252 219 4 v 826 1354 a Fr(1)18 b Fq(\000)g Fv(e)1008 1325 y Fo(2)1008 1374 y Fu(t)1059 1354 y Fv(;)p Black 2855 w FB(\(81\))p Black -150 1426 4235 4 v 208 1478 a Fo(\(2\))220 1559 y Fv(S)291 1563 y Fo(20)473 1551 y Fr(=)112 b(0)14 b Fv(;)p Black 3208 w FB(\(82\))p Black -150 1600 V 209 1677 a Fo(\(3\))227 1758 y Fv(I)291 1762 y Fo(33)473 1750 y Fr(=)112 b(8)706 1633 y Fn(\022)776 1693 y Fr(2)p Fv(\031)p 776 1731 92 4 v 780 1807 a Fr(21)878 1633 y Fn(\023)949 1627 y Fl(1)p 949 1636 29 3 v 949 1670 a(2)1005 1750 y Fv(\021)s Fr(\()p Fv(m)1154 1762 y Fo(1)1210 1750 y Fq(\000)18 b Fv(m)1366 1762 y Fo(2)1403 1750 y Fr(\)\()p Fq(\000)p Fv(E)5 b Fr(\))1630 1715 y Fo(3)p Fu(=)p Fo(2)1735 1750 y Fv(e)1774 1715 y FA(\000)p Fu(i)p Fo(3)p Fg(kM)1997 1750 y Fv(e)2036 1715 y FA(\000)p Fu(i)p Fo(3)p Fu(\036)2184 1723 y Fl(0)2248 1608 y Fn(\()2359 1646 y Fo(5)2315 1671 y Fn(X)2315 1849 y Fu(k)q Fo(=0)2450 1750 y Fv(\015)2493 1765 y Fo([)p Fu(k)q Fo(3])2604 1750 y Fq(F)2664 1765 y Fo([)p Fu(k)q Fo(3])2775 1750 y Fr(\()p Fv(u)p Fr(\))19 b(+)3033 1646 y Fo(5)2989 1671 y Fn(X)2989 1849 y Fu(k)q Fo(=1)3127 1750 y Fr(~)-45 b Fv(\015)3167 1765 y Fo([)p Fu(k)q Fo(3])3278 1750 y Fq(F)3338 1765 y Fu(S)s Fo([)p Fu(k)q Fo(3])3493 1750 y Fr(\()p Fv(u)p Fr(\))3605 1608 y Fn(\))3686 1750 y Fv(;)p Black 228 w FB(\(83\))p Black 210 2032 a Fv(\015)253 2047 y Fo([03])462 2032 y Fr(:=)660 1971 y Fv(i)689 1898 y Fn(p)p 771 1898 219 4 v 771 1971 a Fr(1)18 b Fq(\000)g Fv(e)953 1942 y Fo(2)953 1991 y Fu(t)1004 1903 y Fn(\000)1042 1971 y Fv(e)1081 1941 y Fo(2)1081 1991 y Fu(t)1136 1971 y Fq(\000)g Fr(4)1261 1903 y Fn(\001)p 660 2013 640 4 v 941 2089 a Fv(e)980 2060 y Fo(3)980 2109 y Fu(t)1323 2032 y Fv(;)p Black 2591 w FB(\(84\))p Black 210 2284 a Fv(\015)253 2299 y Fo([13])462 2284 y Fr(:=)100 b Fq(\000)725 2222 y Fv(i)754 2149 y Fn(p)p 836 2149 219 4 v 836 2222 a Fr(1)18 b Fq(\000)g Fv(e)1018 2194 y Fo(2)1018 2243 y Fu(t)1069 2155 y Fn(\000)1107 2222 y Fv(e)1146 2192 y Fo(2)1146 2243 y Fu(t)1201 2222 y Fq(\000)g Fr(4)1326 2155 y Fn(\001)p 725 2265 640 4 v 1006 2341 a Fv(e)1045 2312 y Fo(3)1045 2361 y Fu(t)1387 2284 y Fv(;)p Black 2527 w FB(\(85\))p Black 210 2536 a Fv(\015)253 2551 y Fo([23])462 2536 y Fr(:=)660 2474 y Fv(i)703 2407 y Fn(\000)740 2474 y Fr(12)g Fq(\000)g Fr(7)p Fv(e)1006 2444 y Fo(2)1006 2495 y Fu(t)1042 2407 y Fn(\001)1094 2401 y(p)p 1177 2401 219 4 v 73 x Fr(1)g Fq(\000)g Fv(e)1359 2445 y Fo(2)1359 2495 y Fu(t)p 660 2516 737 4 v 969 2592 a Fr(2)p Fv(e)1050 2564 y Fo(3)1050 2613 y Fu(t)1420 2536 y Fv(;)p Black 2494 w FB(\(86\))p Black 210 2787 a Fv(\015)253 2802 y Fo([33])462 2787 y Fr(:=)660 2726 y Fv(i)689 2653 y Fn(p)p 771 2653 219 4 v 771 2726 a Fr(1)g Fq(\000)g Fv(e)953 2697 y Fo(2)953 2746 y Fu(t)1004 2659 y Fn(\000)1042 2726 y Fv(e)1081 2696 y Fo(2)1081 2746 y Fu(t)1136 2726 y Fq(\000)g Fr(1)1261 2659 y Fn(\001)c(\000)1351 2726 y Fv(e)1390 2696 y Fo(2)1390 2746 y Fu(t)1445 2726 y Fr(+)k(4)1570 2659 y Fn(\001)p 660 2768 949 4 v 1075 2844 a Fr(2)p Fv(e)1156 2816 y Fo(3)1156 2865 y Fu(t)1632 2787 y Fv(;)p Black 2282 w 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1051 y Fo(22)355 1039 y Fr(=)219 b Fv(e)678 1009 y FA(\000)p Fo(2)p Fu(i\036)826 1017 y Fl(0)863 1039 y Fv(e)902 1009 y FA(\000)p Fo(2)p Fu(i)p Fg(kM)1293 1039 y Fr(8)1335 909 y Fn(r)p 1417 909 112 4 v 1427 983 a Fr(2)p Fv(\031)p 1427 1020 92 4 v 1452 1096 a Fr(5)1543 1039 y Fv(E)19 b(\026)14 b(e)1726 1004 y FA(\000)p Fo(2)1726 1060 y Fu(t)p Black 3895 1039 a FB(\(152\))p Black 1293 1145 a Fn( )1385 1184 y FA(1)1359 1208 y Fn(X)1361 1385 y Fu(j)s Fo(=1)1492 1287 y Fr(sin)g Fv(j)5 b Fp(M)1751 1145 y Fn(\()1862 1184 y Fo(5)1819 1208 y Fn(X)1818 1387 y Fu(k)q Fo(=0)1953 1287 y Fv(\013)2006 1302 y Fo([)p Fu(k)q Fo(2])2118 1287 y Fr(\()2178 1266 y(\026)2150 1287 y Fq(A)2216 1299 y Fu(s)2252 1287 y Fr(\))2284 1244 y Fo([)p Fu(k)q(;)p Fo(2])2284 1311 y Fu(j)2434 1287 y Fr(+)2561 1184 y Fo(5)2517 1208 y Fn(X)2517 1387 y Fu(k)q Fo(=1)2660 1287 y Fr(~)-51 b Fv(\013)2704 1302 y Fo([)p Fu(k)q Fo(2])2816 1287 y Fr(\()2867 1266 y(\026)2848 1287 y Fq(S)2898 1299 y Fu(s)2934 1287 y Fr(\))2966 1244 y Fo([)p Fu(k)q(;)p Fo(2])2966 1311 y Fu(j)3097 1145 y Fn(\))3178 1287 y Fv(;)1293 1588 y Fr(+)1398 1484 y FA(1)1371 1509 y Fn(X)1374 1686 y Fu(j)s Fo(=0)1505 1588 y Fr(cos)13 b Fv(j)5 b Fp(M)1773 1446 y Fn(\()1884 1484 y Fo(5)1841 1509 y Fn(X)1840 1688 y Fu(k)q Fo(=0)1975 1588 y Fv(\013)2028 1603 y Fo([)p Fu(k)q Fo(2])2140 1588 y Fr(\()2200 1567 y(\026)2172 1588 y Fq(A)2238 1600 y Fu(c)2272 1588 y Fr(\))2304 1545 y Fo([)p Fu(k)q(;)p Fo(2])2304 1611 y Fu(j)2454 1588 y Fr(+)2581 1484 y Fo(5)2537 1509 y Fn(X)2537 1688 y Fu(k)q Fo(=1)2680 1588 y Fr(~)-50 b Fv(\013)2725 1603 y Fo([)p Fu(k)q Fo(2])2836 1588 y Fr(\()2887 1567 y(\026)2868 1588 y Fq(S)2918 1600 y Fu(c)2953 1588 y Fr(\))2985 1545 y Fo([)p Fu(k)q(;)p Fo(2])2985 1611 y Fu(j)3116 1446 y Fn(\)!)3263 1588 y Fv(;)p -150 1733 4235 4 v 129 1785 a Fo(\(2\))147 1866 y Fv(I)211 1870 y Fo(21)355 1858 y Fr(=)422 b(0)14 b Fv(;)p Black 2974 w FB(\(153\))p Black -150 1907 V 129 1984 a Fo(\(2\))147 2065 y Fv(I)211 2069 y Fo(20)355 2057 y Fr(=)237 b Fq(\000)p Fr(16)806 1935 y Fn(r)p 887 1935 103 4 v 914 2001 a Fv(\031)p 897 2038 84 4 v 897 2114 a Fr(15)990 2057 y Fv(E)5 b(\026)1293 1915 y Fn( )1385 1953 y FA(1)1359 1978 y Fn(X)1361 2155 y Fu(j)s Fo(=1)1492 2057 y Fr(sin)14 b Fv(j)5 b Fp(M)1751 1915 y Fn(\()1862 1953 y Fo(3)1819 1978 y Fn(X)1818 2157 y Fu(k)q Fo(=0)1953 2057 y Fv(\013)2006 2072 y Fo([)p Fu(k)q Fo(2])2118 2057 y Fr(\()2178 2036 y(\026)2150 2057 y Fq(A)2216 2069 y Fu(s)2252 2057 y Fr(\))2284 2014 y Fo([)p Fu(k)q(;)p Fo(2])2284 2080 y Fu(j)2415 1915 y Fn(\))2501 2057 y Fr(+)2610 1953 y FA(1)2584 1978 y Fn(X)2586 2155 y Fu(j)s Fo(=0)2717 2057 y Fr(cos)13 b Fv(j)5 b Fp(M)2986 1915 y Fn(\()3096 1953 y Fo(5)3053 1978 y Fn(X)3053 2157 y Fu(k)q Fo(=0)3187 2057 y Fv(\013)3240 2072 y Fo([)p Fu(k)q Fo(2])3352 2057 y Fr(\()3413 2036 y(\026)3384 2057 y Fq(A)3450 2069 y Fu(c)3484 2057 y Fr(\))3516 2014 y Fo([)p Fu(k)q(;)p Fo(2])3516 2080 y Fu(j)3648 1915 y Fn(\)!)3794 2057 y Fv(;)p Black 78 w FB(\(154\))p Black -150 2203 4235 4 v 128 2254 a Fo(\(2\))141 2335 y Fv(S)211 2339 y Fo(22)355 2327 y Fr(=)422 b(0)14 b Fv(;)p Black 2974 w FB(\(155\))p Black -150 2376 V 128 2432 a Fo(\(2\))141 2512 y Fv(S)211 2516 y Fo(21)355 2504 y Fr(=)241 b Fv(e)700 2474 y FA(\000)p Fu(i\036)815 2482 y Fl(0)851 2504 y Fv(e)890 2474 y FA(\000)p Fu(i)12 b Fg(k)f(M)1303 2448 y Fr(32)p 1303 2485 84 4 v 1324 2561 a(3)1396 2383 y Fn(r)p 1479 2383 71 4 v 1489 2448 a Fv(\031)p 1489 2485 51 4 v 1493 2561 a Fr(5)1549 2402 y Fn(q)p 1632 2402 219 4 v 102 x Fr(1)18 b Fq(\000)g Fv(e)1814 2476 y Fo(2)1814 2525 y Fu(t)1851 2504 y Fv(\021)s Fr(\()p Fq(\000)p Fv(E)5 b Fr(\))2090 2463 y Fo(3)p Fu(=)p Fo(2)2195 2504 y Fr(\()p Fv(m)2300 2516 y Fo(1)2356 2504 y Fq(\000)18 b Fv(m)2512 2516 y Fo(2)2549 2504 y Fr(\))p Fq(\002)p Black 1249 w FB(\(156\))p Black 1293 2620 a Fn( )1385 2658 y FA(1)1359 2683 y Fn(X)1361 2860 y Fu(j)s Fo(=1)1492 2762 y Fr(sin)c Fv(j)5 b Fp(M)1751 2620 y Fn(\()1862 2658 y Fo(3)1819 2683 y Fn(X)1818 2861 y Fu(k)q Fo(=2)1953 2762 y Fv(\014)2000 2777 y Fo([)p Fu(k)q Fo(1])2111 2762 y Fr(\()2172 2741 y(\026)2143 2762 y Fq(A)2209 2774 y Fu(s)2246 2762 y Fr(\))2278 2718 y Fo([)p Fu(k)q(;)p Fo(1])2278 2785 y Fu(j)2427 2762 y Fr(+)2522 2740 y(~)2511 2762 y Fv(\014)2558 2777 y Fo([31])2665 2762 y Fr(\()2717 2741 y(\026)2697 2762 y Fq(S)2747 2774 y Fu(s)2783 2762 y Fr(\))2815 2718 y Fo([1)p Fu(;)p Fo(1])2815 2785 y Fu(j)2943 2620 y Fn(\))1293 3062 y Fr(+)1398 2958 y FA(1)1371 2983 y Fn(X)1374 3160 y Fu(j)s Fo(=0)1505 3062 y Fr(cos)13 b Fv(j)5 b Fp(M)1773 2920 y Fn(\()1884 2958 y Fo(3)1841 2983 y Fn(X)1840 3162 y Fu(k)q Fo(=2)1975 3062 y Fv(\014)2022 3077 y Fo([)p Fu(k)q Fo(1])2133 3062 y Fr(\()2194 3041 y(\026)2165 3062 y Fq(A)2231 3074 y Fu(c)2266 3062 y Fr(\))2298 3019 y Fo([)p Fu(k)q(;)p Fo(1])2298 3085 y Fu(j)2448 3062 y Fr(+)2543 3040 y(~)2531 3062 y Fv(\014)2578 3077 y Fo([31])2686 3062 y Fr(\()2737 3041 y(\026)2718 3062 y Fq(S)2768 3074 y Fu(c)2802 3062 y Fr(\))2834 3019 y Fo([)p Fu(k)q(;)p Fo(1])2834 3085 y Fu(j)2966 2920 y Fn(\))o(!)3112 3062 y Fv(;)p -150 3208 4235 4 v 128 3260 a Fo(\(2\))141 3340 y Fv(S)211 3344 y Fo(20)355 3332 y Fr(=)422 b(0)14 b Fv(;)p Black 2974 w FB(\(157\))p Black -150 3381 V 129 3446 a Fo(\(3\))147 3526 y Fv(I)211 3530 y Fo(33)355 3518 y Fr(=)208 b Fv(e)667 3488 y FA(\000)p Fo(3)p Fu(i\036)815 3496 y Fl(0)851 3518 y Fv(e)890 3488 y FA(\000)p Fo(3)p Fu(i)12 b Fg(k)f(M)1293 3518 y Fr(8)1335 3388 y Fn(r)p 1417 3388 112 4 v 1427 3462 a Fr(2)p Fv(\031)p 1427 3499 92 4 v 1431 3575 a Fr(21)1529 3518 y(\()p Fq(\000)p Fv(E)5 b Fr(\))1724 3484 y Fo(3)p Fu(=)p Fo(2)1843 3518 y Fr(\()p Fv(m)1948 3530 y Fo(1)2003 3518 y Fq(\000)18 b Fv(m)2159 3530 y Fo(2)2197 3518 y Fr(\))c Fv(\021)s Fq(\002)p Black 1543 w FB(\(158\))p Black 1293 3625 a Fn( )1385 3663 y FA(1)1359 3688 y Fn(X)1361 3865 y Fu(j)s Fo(=1)1492 3767 y Fr(sin)g Fv(j)5 b Fp(M)1738 3625 y Fn(\()1848 3663 y Fo(5)1805 3688 y Fn(X)1805 3867 y Fu(k)q Fo(=0)1939 3767 y Fv(\015)1982 3779 y Fu(k)q Fo(3)2056 3767 y Fr(\()2117 3746 y(\026)2088 3767 y Fq(A)2154 3779 y Fu(s)2190 3767 y Fr(\))2222 3732 y Fo([)p Fu(k)q(;)p Fo(3])2372 3767 y Fr(+)2499 3663 y Fo(5)2456 3688 y Fn(X)2455 3867 y Fu(k)q Fo(=1)2593 3767 y Fr(~)-45 b Fv(\015)2633 3779 y Fu(k)q Fo(3)2707 3767 y Fr(\()2758 3746 y(\026)2739 3767 y Fq(S)2789 3779 y Fu(s)2825 3767 y Fr(\))2857 3732 y Fo([)p Fu(k)q Fo(3])2969 3625 y Fn(\))1293 4067 y Fr(+)1384 3963 y FA(1)1358 3988 y Fn(X)1361 4165 y Fu(j)s Fo(=0)1491 4067 y Fr(cos)13 b Fv(j)5 b Fp(M)1746 3925 y Fn(\()1856 3963 y Fo(5)1813 3988 y Fn(X)1813 4167 y Fu(k)q Fo(=0)1947 4067 y Fv(\015)1990 4079 y Fu(k)q Fo(3)2064 4067 y Fr(\()2125 4046 y(\026)2096 4067 y Fq(A)2162 4079 y Fu(c)2197 4067 y Fr(\))2229 4033 y Fo([)p Fu(k)q(;)p Fo(3])2379 4067 y Fr(+)2505 3963 y Fo(5)2462 3988 y Fn(X)2462 4167 y Fu(k)q Fo(=1)2599 4067 y Fr(~)-45 b Fv(\015)2639 4079 y Fu(k)q Fo(3)2713 4067 y Fr(\()2764 4046 y(\026)2745 4067 y Fq(S)2795 4079 y Fu(c)2829 4067 y Fr(\))2861 4033 y Fo([)p Fu(k)q Fo(3])2973 3925 y Fn(\)!)3120 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Fu(s)2190 4772 y Fr(\))2222 4738 y Fo([)p Fu(k)q(;)p Fo(1])2372 4772 y Fr(+)2499 4668 y Fo(3)2456 4693 y Fn(X)2455 4872 y Fu(k)q Fo(=1)2593 4772 y Fr(~)-45 b Fv(\015)2633 4784 y Fu(k)q Fo(1)2707 4772 y Fr(\()2758 4751 y(\026)2739 4772 y Fq(S)2789 4784 y Fu(s)2825 4772 y Fr(\))2857 4738 y Fo([)p Fu(k)q Fo(1])2969 4630 y Fn(\))1293 5072 y Fr(+)1384 4968 y FA(1)1358 4993 y Fn(X)1361 5170 y Fu(j)s Fo(=0)1491 5072 y Fr(cos)13 b Fv(j)5 b Fp(M)1746 4930 y Fn(\()1856 4968 y Fo(3)1813 4993 y Fn(X)1813 5172 y Fu(k)q Fo(=0)1947 5072 y Fv(\015)1990 5084 y Fu(k)q Fo(1)2064 5072 y Fr(\()2125 5051 y(\026)2096 5072 y Fq(A)2162 5084 y Fu(c)2197 5072 y Fr(\))2229 5038 y Fo([)p Fu(k)q(;)p Fo(1])2379 5072 y Fr(+)2505 4968 y Fo(3)2462 4993 y Fn(X)2462 5172 y Fu(k)q Fo(=1)2599 5072 y Fr(~)-45 b Fv(\015)2639 5084 y Fu(k)q Fo(1)2713 5072 y Fr(\()2764 5051 y(\026)2745 5072 y Fq(S)2795 5084 y Fu(c)2829 5072 y Fr(\))2861 5038 y Fo([)p Fu(k)q Fo(1])2973 4930 y Fn(\)!)3120 5072 y Fv(;)p -150 5218 4235 4 v 129 5270 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-150 1623 4235 4 v 129 1687 a Fo(\(4\))147 1768 y Fv(I)211 1772 y Fo(44)355 1760 y Fr(=)208 b Fv(e)667 1730 y FA(\000)p Fo(4)p Fu(i\036)815 1738 y Fl(0)851 1760 y Fv(e)890 1730 y FA(\000)p Fo(4)p Fu(i)12 b Fg(k)f(M)1303 1704 y Fr(4)p 1303 1741 42 4 v 1303 1817 a(9)1354 1629 y Fn(r)p 1437 1629 112 4 v 1447 1704 a Fr(2)p Fv(\031)p 1447 1741 92 4 v 1472 1817 a Fr(7)1549 1760 y Fv(E)1615 1726 y Fo(2)1652 1760 y Fv(\026)p Fr(\(1)19 b Fq(\000)f Fr(3)p Fv(\021)s Fr(\))p Fq(\002)p Black 1834 w FB(\(166\))p Black 1293 1866 a Fn( )1385 1904 y FA(1)1359 1929 y Fn(X)1361 2106 y Fu(j)s Fo(=1)1492 2008 y Fr(sin)c Fv(j)5 b Fp(M)1738 1866 y Fn(\()1848 1904 y Fo(7)1805 1929 y Fn(X)1805 2108 y Fu(k)q Fo(=0)1939 2008 y Fv(\020)1975 2020 y Fu(k)q Fo(4)2049 2008 y Fr(\()2110 1987 y(\026)2081 2008 y Fq(A)2147 2020 y Fu(s)2183 2008 y Fr(\))2215 1974 y Fo([)p Fu(k)q(;)p Fo(4])2365 2008 y Fr(+)2492 1904 y Fo(7)2449 1929 y Fn(X)2448 2108 y Fu(k)q Fo(=1)2590 1986 y Fr(~)2583 2008 y Fv(\020)2619 2020 y Fu(k)q Fo(4)2693 2008 y Fr(\()2745 1987 y(\026)2725 2008 y Fq(S)2775 2020 y Fu(s)2811 2008 y Fr(\))2843 1974 y Fo([)p Fu(k)q Fo(4])2955 1866 y Fn(\))1293 2308 y Fr(+)1384 2205 y FA(1)1358 2229 y Fn(X)1361 2406 y Fu(j)s Fo(=0)1491 2308 y Fr(cos)13 b Fv(j)5 b Fp(M)1746 2166 y Fn(\()1856 2205 y Fo(7)1813 2229 y Fn(X)1813 2408 y Fu(k)q Fo(=0)1947 2308 y Fv(\020)1983 2320 y Fu(k)q Fo(4)2057 2308 y Fr(\()2118 2287 y(\026)2089 2308 y Fq(A)2155 2320 y Fu(c)2190 2308 y Fr(\))2222 2274 y Fo([)p Fu(k)q(;)p Fo(4])2372 2308 y Fr(+)2499 2205 y Fo(7)2455 2229 y Fn(X)2455 2408 y Fu(k)q Fo(=1)2597 2286 y Fr(~)2589 2308 y Fv(\020)2625 2320 y Fu(k)q Fo(4)2700 2308 y Fr(\()2751 2287 y(\026)2732 2308 y Fq(S)2782 2320 y Fu(c)2816 2308 y Fr(\))2848 2274 y Fo([)p Fu(k)q Fo(4])2960 2166 y Fn(\)!)3106 2308 y Fv(;)p -150 2454 4235 4 v 129 2506 a Fo(\(4\))147 2587 y Fv(I)211 2591 y Fo(43)355 2579 y Fr(=)422 b(0)14 b Fv(;)p Black 2974 w FB(\(167\))p Black -150 2628 V 129 2680 a Fo(\(4\))147 2761 y Fv(I)211 2765 y Fo(42)355 2753 y Fr(=)208 b Fv(e)667 2722 y FA(\000)p Fo(2)p Fu(i\036)815 2730 y Fl(0)851 2753 y Fv(e)890 2722 y FA(\000)p Fo(2)p Fu(i)12 b Fg(k)f(M)1324 2696 y Fr(8)p 1303 2734 84 4 v 1303 2810 a(63)1396 2680 y Fq(p)p 1465 2680 92 4 v 73 x Fr(2)p Fv(\031)s(E)1623 2718 y Fo(2)1660 2753 y Fv(\026)p Fr(\(1)18 b Fq(\000)g Fr(3)p Fv(\021)s Fr(\))p Fq(\002)p Black 1827 w FB(\(168\))p Black 1293 2850 a Fn( )1385 2889 y FA(1)1359 2913 y Fn(X)1361 3090 y Fu(j)s Fo(=1)1492 2992 y Fr(sin)c Fv(j)5 b Fp(M)1738 2850 y Fn(\()1848 2889 y Fo(5)1805 2913 y Fn(X)1805 3092 y Fu(k)q Fo(=0)1939 2992 y Fv(\020)1975 3004 y Fu(k)q Fo(2)2049 2992 y Fr(\()2110 2971 y(\026)2081 2992 y Fq(A)2147 3004 y Fu(s)2183 2992 y Fr(\))2215 2958 y Fo([)p Fu(k)q(;)p Fo(2])2365 2992 y Fr(+)2492 2889 y Fo(5)2449 2913 y Fn(X)2448 3092 y Fu(k)q Fo(=1)2590 2970 y Fr(~)2583 2992 y Fv(\020)2619 3004 y Fu(k)q Fo(2)2693 2992 y Fr(\()2745 2971 y(\026)2725 2992 y Fq(S)2775 3004 y Fu(s)2811 2992 y Fr(\))2843 2958 y Fo([)p Fu(k)q Fo(2])2955 2850 y Fn(\))1293 3293 y Fr(+)1384 3189 y FA(1)1358 3214 y Fn(X)1361 3391 y Fu(j)s Fo(=0)1491 3293 y Fr(cos)13 b Fv(j)5 b Fp(M)1746 3151 y Fn(\()1856 3189 y Fo(5)1813 3214 y Fn(X)1813 3392 y Fu(k)q Fo(=0)1947 3293 y Fv(\020)1983 3305 y Fu(k)q Fo(2)2057 3293 y Fr(\()2118 3272 y(\026)2089 3293 y Fq(A)2155 3305 y Fu(c)2190 3293 y Fr(\))2222 3258 y Fo([)p Fu(k)q(;)p Fo(2])2372 3293 y Fr(+)2499 3189 y Fo(5)2455 3214 y Fn(X)2455 3392 y Fu(k)q Fo(=1)2597 3271 y Fr(~)2589 3293 y Fv(\020)2625 3305 y Fu(k)q Fo(2)2700 3293 y Fr(\()2751 3272 y(\026)2732 3293 y Fq(S)2782 3305 y Fu(c)2816 3293 y Fr(\))2848 3258 y Fo([)p Fu(k)q Fo(2])2960 3151 y Fn(\)!)3106 3293 y Fv(;)p -150 3438 4235 4 v 129 3490 a Fo(\(4\))147 3571 y Fv(I)211 3575 y Fo(40)355 3563 y Fr(=)422 b(0)14 b Fv(;)p Black 2974 w FB(\(169\))p Black -150 3612 V 129 3689 a Fo(\(4\))147 3770 y Fv(I)211 3774 y Fo(40)355 3762 y Fr(=)547 3729 y Fo(8)p 530 3743 67 4 v 530 3790 a(21)606 3697 y Fn(p)p 689 3697 61 4 v 699 3729 a Fu(\031)p 699 3743 41 4 v 703 3790 a Fo(5)750 3762 y Fq(j)p Fv(E)5 b Fq(j)862 3732 y Fo(2)900 3762 y Fv(\026)p Fr(\(1)18 b Fq(\000)g Fr(3)p Fv(\021)s Fr(\))1293 3620 y Fn( )1385 3658 y FA(1)1359 3683 y Fn(X)1361 3860 y Fu(j)s Fo(=1)1492 3762 y Fr(sin)c Fv(j)5 b Fp(M)1738 3620 y Fn(\()1848 3658 y Fo(3)1805 3683 y Fn(X)1805 3862 y Fu(k)q Fo(=0)1939 3762 y Fv(\020)1975 3774 y Fu(k)q Fo(0)2049 3762 y Fr(\()2110 3741 y(\026)2081 3762 y Fq(A)2147 3774 y Fu(s)2183 3762 y Fr(\))2215 3719 y Fo([)p Fu(k)q Fo(0])2215 3785 y Fu(j)2327 3620 y Fn(\))2413 3762 y Fr(+)2522 3658 y FA(1)2496 3683 y Fn(X)2498 3860 y Fu(j)s Fo(=1)2629 3762 y Fr(cos)13 b Fv(j)5 b Fp(M)2884 3620 y Fn(\()2995 3658 y Fo(3)2951 3683 y Fn(X)2951 3862 y Fu(k)q Fo(=0)3085 3762 y Fv(\020)3121 3774 y Fu(k)q Fo(0)3196 3762 y Fr(\()3256 3741 y(\026)3228 3762 y Fq(A)3294 3774 y Fu(c)3328 3762 y Fr(\))3360 3719 y Fo([)p Fu(k)q Fo(0])3360 3785 y Fu(j)3472 3620 y Fn(\)!)3618 3762 y Fv(:)p Black 254 w FB(\(170\))p Black -150 4028 a(The)28 b(F)-7 b(ourier)27 b(transformation)g(of)h(the)g(ab)r(o)n(v)n (e)f(gra)n(vitational)f(w)n(a)n(v)n(eforms)g(is)i(easily)f(done.)38 b(W)-7 b(e)29 b(tak)n(e)e(the)i(F)-7 b(ourier)27 b(transformation)-150 4123 y(of)h(the)f Fr(sin)14 b Fv(j)5 b Fp(M)27 b FB(and)h Fr(cos)13 b Fv(j)5 b Fp(M)27 b FB(terms,)415 4359 y SDict begin H.S end 415 4359 a 415 4359 a SDict begin 11.5 H.A end 415 4359 a 415 4359 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.171) cvn /DEST pdfmark end 415 4359 a 485 4303 a Fr(1)p 425 4340 161 4 v 425 4357 a Fq(p)p 494 4357 92 4 v 68 x Fr(2)p Fv(\031)610 4246 y Fn(Z)693 4267 y Fo(+)p FA(1)656 4435 y(\0001)828 4359 y Fv(e)867 4325 y FA(\000)p Fu(im)p Fg(kM)1129 4359 y Fr(sin)14 b Fv(j)5 b Fp(M)13 b Fv(e)1427 4325 y Fu(i!)r(t)1524 4359 y Fr(d)p Fv(t)50 b Fr(=)f Fv(i)1793 4238 y Fn(r)p 1876 4238 71 4 v 1886 4303 a Fv(\031)p 1886 4340 51 4 v 1890 4416 a Fr(2)1946 4359 y Fv(\016)s Fr(\()p Fv(j)5 b(n)18 b Fr(+)h Fp(k)p Fv(mn)f Fq(\000)g Fv(!)s Fr(\))g Fq(\000)h Fv(i)2701 4238 y Fn(r)p 2783 4238 71 4 v 2793 4303 a Fv(\031)p 2793 4340 51 4 v 2797 4416 a Fr(2)2853 4359 y Fv(\016)s Fr(\()p Fv(j)5 b(n)19 b Fq(\000)f Fp(k)p Fv(mn)g Fr(+)h Fv(!)s Fr(\))14 b Fv(;)p Black 380 w FB(\(171\))p Black 476 4547 a Fr(1)p 416 4584 161 4 v 416 4601 a Fq(p)p 485 4601 92 4 v 69 x Fr(2)p Fv(\031)601 4490 y Fn(Z)684 4511 y Fo(+)p FA(1)647 4679 y(\0001)819 4603 y Fv(e)858 4569 y FA(\000)p Fu(im)p Fg(kM)1120 4603 y Fr(cos)f Fv(j)5 b Fp(M)13 b Fv(e)1427 4569 y Fu(i!)r(t)1524 4603 y Fr(d)p Fv(t)50 b Fr(=)1764 4482 y Fn(r)p 1847 4482 71 4 v 1857 4547 a Fv(\031)p 1857 4584 51 4 v 1861 4660 a Fr(2)1917 4603 y Fv(\016)s Fr(\()p Fv(j)5 b(n)19 b Fr(+)f Fp(k)p Fv(mn)g Fq(\000)g Fv(!)s Fr(\))h(+)2643 4482 y Fn(r)p 2726 4482 71 4 v 2736 4547 a Fv(\031)p 2736 4584 51 4 v 2740 4660 a Fr(2)2796 4603 y Fv(\016)s Fr(\()p Fv(j)5 b(n)19 b Fq(\000)f Fp(k)p Fv(mn)g Fr(+)g Fv(!)s Fr(\))c Fv(;)p Black 438 w FB(\(172\))p Black -150 4834 a(and)27 b(apply)g(this)h(to)f(the)h(sums)f(o)n(v)n(er)f Fv(j)5 b FB(.)37 b(Ho)n(w)26 b(man)n(y)h(terms)g(of)h(the)f(in\034nite)i(sum)e (com)n(binations)f(are)h(necessary)e(to)i(determine)h(the)-150 4930 y(m)n(ultip)r(oles)23 b(up)h(to)f(a)g(certain)g(arbitrary)e(order) h(in)h(the)h(eccen)n(tricit)n(y)e(of)h(the)h(binary)e(system)h(is)h (discussed)e(in)i(detail)f(in)h(App)r(endix)4007 4930 y SDict begin H.S end 4007 4930 a Black FB(F)p Black 4061 4873 a SDict begin H.R end 4061 4873 a 4061 4930 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section*.21) cvn H.B /ANN pdfmark end 4061 4930 a FB(.)-150 5025 y(Note)h(that)h(in)g(Eqs.)36 b(\()545 5025 y SDict begin H.S end 545 5025 a Black FB(153)p Black 670 4972 a SDict begin H.R end 670 4972 a 670 5025 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.5.152) cvn H.B /ANN pdfmark end 670 5025 a FB(\))25 b(to)g(\()858 5025 y SDict begin H.S end 858 5025 a Black FB(170)p Black 983 4972 a SDict begin H.R end 983 4972 a 983 5025 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.5.152) cvn H.B /ANN pdfmark end 983 5025 a FB(\),)h(the)g(expansion)e(co)r(e\036cien) n(ts)h(are)g(not)g(longer)f(dep)r(enden)n(t)i(on)f Fp(M)p FB(.)36 b(They)25 b(are)f(functions)i(of)f Fv(e)4032 5037 y Fu(t)4061 5025 y FB(,)-150 5121 y Fv(n)f FB(and)f(the)h (\020magnetic\021)29 b(n)n(um)n(b)r(er)24 b Fv(m)f FB(and)h(include)g (m)n(ultiple)g(sums)f(as)g(w)n(e)h(k)n(eep)f(in)h(mind)g(the)g (previous)e(sections.)35 b(In)24 b(1pN-accurate)-150 5216 y(orbital)k(dynamics,)i(they)f(are)f(constan)n(ts.)41 b(Due)30 b(to)f(RR,)h(they)f(will)h(b)r(ecome)f(slo)n(wly)f(v)-5 b(arying)28 b(functions)h(of)h(time.)42 b(All)30 b(the)f(terms,)-150 5312 y(where)c(this)h(condition)f(is)h(satis\034ed)f(and)g(whic)n(h)h (will)g(app)r(ear)e(together)h(with)h(an)f(explicit)h(phase)f(factor,)g (can)h(b)r(e)g(treated)f(with)h(the)-150 5407 y(help)i(of)f(the)h(SP)-7 b(A.)29 b(All)f(other)f(ones)g(whic)n(h)g(do)h(not)f(ha)n(v)n(e)g(a)g (phase)g(factor)f(ha)n(v)n(e)h(to)g(b)r(e)h(treated)f(individually)-7 b(.)p Black Black eop end end %%Page: 15 15 TeXDict begin HPSdict begin 15 14 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.15) cvn /DEST pdfmark end -158 -515 a Black 4001 -299 a FB(15)p Black -141 -83 a SDict begin H.S end -141 -83 a -141 -83 a SDict begin 10.5 H.A end -141 -83 a -141 -83 a SDict begin [/View [/XYZ H.V]/Dest (section*.6) cvn /DEST pdfmark end -141 -83 a Fs(VI.)88 b(THE)30 b(EFFECT)f(OF)g(RADIA)-7 b(TION)30 b(REA)n(CTION)g(AND)f(THE)h(ST)-7 b(A)g(TIONAR)g(Y)30 b(PHASE)g(APPR)n(O)n(XIMA)-7 b(TION)1553 4 y(OF)30 b(THE)g(GW)g(FIELD) 1498 209 y SDict begin H.S end 1498 209 a 1498 209 a SDict begin 10.5 H.A end 1498 209 a 1498 209 a SDict begin [/View [/XYZ H.V]/Dest (section*.7) cvn /DEST pdfmark end 1498 209 a Fs(A.)87 b(Radiativ)n(e)30 b(dynamics)-67 423 y FB(Ha)n(ving)22 b(the)i(conserv)-5 b(ativ)n(e)21 b(ev)n(olution)h(under)h(con)n(trol,)f(w)n(e)h(can)g(apply)f(the)i(F)-7 b(ourier)22 b(decomp)r(osition)g(to)h(the)h(radiativ)n(e)d(ev)n (olution.)-150 518 y(This)28 b(is)g(done)f(b)n(y)h(separating)e(the)i (full)h(orbital)e(ev)n(olution)g(in)n(to)h(t)n(w)n(o)f(time)h(scales:) 37 b(the)28 b Fw(orbital)h FB(and)f(the)g Fw(r)l(e)l(active)h FB(timescale.)37 b(The)-150 614 y(latter)24 b(is)g(\025)f(for)h(our)f (calculation)g(\025)h(assumed)f(to)h(b)r(e)h(m)n(uc)n(h)f(larger)e (than)i(the)h(orbital)e(scale.)35 b(This)24 b(is)g(equiv)-5 b(alen)n(t)23 b(to)h(the)h(assumption)-150 709 y(that)h(the)h(rate)e (of)i(c)n(hange)d(of)j(the)f(orbital)f(frequency)h(is)g(small)g (measured)f(o)n(v)n(er)f(one)i(orbit.)36 b(The)26 b(time)h(dep)r (endence)f(of)h(the)f(orbital)-150 805 y(frequency)34 b(is)g(closely)f(related)h(to)g(the)h(loss)e(of)i(energy)e(and)h (orbital)g(angular)e(momen)n(tum)j(due)f(to)h(the)f(GW)h(emission.)57 b(P)n(eters)-150 900 y(and)29 b(Mathews)f([)390 900 y SDict begin H.S end 390 900 a FB(13)474 900 y SDict begin 11.5 H.L end 474 900 a 474 900 a SDict begin [/Subtype /Link/Dest (cite.Peters:Mathews:1963) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 474 900 a -1 w FB(,)525 900 y SDict begin H.S end 525 900 a FB(14)609 900 y SDict begin 11.5 H.L end 609 900 a 609 900 a SDict begin [/Subtype /Link/Dest (cite.Peters:1964) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 609 900 a -1 w FB(])h(prop)r(osed)e(a)i(relativ)n(ely)e (simple)i(mo)r(del)g(of)f(the)h(binary)f(inspiral)g(in)h(the)g(appro)n (ximation)e(of)i(slo)n(w)f(motion)-150 996 y(and)34 b(w)n(eak)g(gra)n (vitational)e(in)n(teraction)h(for)h(arbitrary)f(eccen)n(tricities)g Fr(0)h Fq(\024)h Fv(e)f(<)g Fr(1)p FB(,)i(whic)n(h)e(has)g(b)r(een)h (standard)f(for)g(some)g(time.)-150 1091 y(Blanc)n(het,)f(Damour,)f(Iy) n(er,)h(and)e(Thorne)h(made)g(endea)n(v)n(ors)d(to)j(obtain)g (higher-order)d(corrections)i(to)h(the)g(quadrup)r(ole)f(form)n(ula) -150 1187 y([)-127 1187 y SDict begin H.S end -127 1187 a FB(1)-85 1187 y SDict begin 11.5 H.L end -85 1187 a -85 1187 a SDict begin [/Subtype /Link/Dest (cite.Blanchet:Damour:1986) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end -85 1187 a FB(,)-24 1187 y SDict begin H.S end -24 1187 a FB(10)60 1187 y SDict begin 11.5 H.L end 60 1187 a 60 1187 a SDict begin [/Subtype /Link/Dest (cite.Thorne:1980) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 60 1187 a -1 w FB(,)120 1187 y SDict begin H.S end 120 1187 a FB(15)204 1187 y SDict begin 11.5 H.L end 204 1187 a 204 1187 a SDict begin [/Subtype /Link/Dest (cite.Blanchet:1987) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 204 1187 a -1 w FB(,)265 1187 y SDict begin H.S end 265 1187 a FB(16)349 1187 y SDict begin 11.5 H.L end 349 1187 a 349 1187 a SDict begin [/Subtype /Link/Dest (cite.Damour:Iyer:1991) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 349 1187 a -1 w FB(].)69 b(The)38 b(far-zone)f(\035uxes)h(of)g(energy)g(and)g (angular)e(momen)n(tum)j(to)f(2pN)g(order)f(can)h(b)r(e)h(found)f(in)h ([)3715 1187 y SDict begin H.S end 3715 1187 a FB(2)3757 1187 y SDict begin 11.5 H.L end 3757 1187 a 3757 1187 a SDict begin [/Subtype /Link/Dest (cite.Blanchet:Damour:Iyer:1995) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 3757 1187 a -1 w FB(].)70 b(These)-150 1282 y(corrections)23 b(ha)n(v)n(e)g(sho)n(wn)h(up)h(to)g(b)r(e)g (unrenounceable)f(for)g(the)h(data)g(analysis)e(comm)n(unit)n(y)-7 b(.)36 b(F)-7 b(or)24 b(the)h(time)g(b)r(eing,)h(w)n(e)e(will)h (restrict)-150 1378 y(ourselv)n(es)j(to)i(the)h(P)n(eters-Mathews)e(mo) r(del)i(for)e(an)h(exemplary)g(calculation.)44 b(Higher-order)27 b(corrections)i(can)h(b)r(e)g(included)h(in)g(a)-150 1473 y(forthcoming)c(publication.)-67 1569 y(In)h(the)g(conserv)-5 b(ativ)n(e)26 b(case,)h Fv(n)h FB(and)g Fv(e)1127 1581 y Fu(t)1184 1569 y FB(w)n(ere)f(constan)n(ts)f(of)i(motion,)g(but)g (when)g(the)h(orbit)e(shrinks)g(due)h(to)g(RR,)g(b)r(oth)g(elemen)n(ts) -150 1664 y(will)g(follo)n(w)f(coupled)g(EOM)h(connected)f(to)g(the)h (loss)f(of)h(orbital)e(energy)h(and)g(angular)f(momen)n(tum)i([)3113 1664 y SDict begin H.S end 3113 1664 a FB(17)3197 1664 y SDict begin 11.5 H.L end 3197 1664 a 3197 1664 a SDict begin [/Subtype /Link/Dest (cite.Damour:Gopakumar:Iyer:2004) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 3197 1664 a -1 w FB(,)3247 1664 y SDict begin H.S end 3247 1664 a FB(18)3331 1664 y SDict begin 11.5 H.L end 3331 1664 a 3331 1664 a SDict begin [/Subtype /Link/Dest (cite.Konigsdorffer:Gopakumar:2006) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 3331 1664 a -1 w FB(],)1694 1834 y SDict begin H.S end 1694 1834 a 1694 1834 a SDict begin 11.5 H.A end 1694 1834 a 1694 1834 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.173) cvn /DEST pdfmark end 1694 1834 a 13 w Fr(_)-36 b Fv(n)50 b Fr(=)f Fq(N)12 b Fr(\()p Fv(n;)i(e)2146 1846 y Fu(t)2175 1834 y Fr(\))g Fv(;)p Black 1651 w FB(\(173\))p Black 1688 1954 a Fr(_)-35 b Fv(e)1715 1966 y Fu(t)1794 1954 y Fr(=)49 b Fq(E)7 b Fr(\()p Fv(n;)14 b(e)2117 1966 y Fu(t)2146 1954 y Fr(\))g Fv(;)p Black 1680 w FB(\(174\))p Black -150 2124 a(sym)n(b)r(olically)-7 b(.)44 b(Note)30 b(that)h(ab)r(o)n(v) n(e)e(Eqs.)45 b(hold)30 b(after)g(a)n(v)n(eraging)d(o)n(v)n(er)i(one)h (orbit.)44 b(Th)n(us,)31 b Fv(n)f FB(will)h(not)f(simply)g(b)r(e)h Fp(M)c Fr(=)g Fv(n)p Fr(\()p Fv(t)21 b Fq(\000)f Fv(t)3992 2136 y Fo(0)4029 2124 y Fr(\))p FB(,)-150 2220 y(but)28 b(will)g(satisfy)1570 2447 y SDict begin H.S end 1570 2447 a 1570 2447 a SDict begin 11.5 H.A end 1570 2447 a 1570 2447 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.175) cvn /DEST pdfmark end 1570 2447 a Fp(M)p Fr(\()p Fv(t)p Fr(\))50 b(=)1920 2334 y Fn(Z)2003 2354 y Fu(t)1966 2522 y(t)1991 2530 y Fl(0)2046 2447 y Fv(n)p Fr(\()p Fv(t)2158 2412 y FA(0)2181 2447 y Fr(\)d)p Fv(t)2289 2412 y FA(0)2327 2447 y Fv(;)p Black 1545 w FB(\(175\))p Black 1604 2619 a Fr(_)1570 2640 y Fp(M)p Fr(\()p Fv(t)p Fr(\))g(=)g Fv(n)p Fr(\()p Fv(t)p Fr(\))14 b Fv(:)p Black 1794 w FB(\(176\))p Black -150 2810 a(The)28 b(F)-7 b(ourier-domain)26 b(w)n(a)n(v)n(e)h(forms)g(will)i(not)f(longer)f(follo)n(w)g(Eqs.)39 b(\()2060 2810 y SDict begin H.S end 2060 2810 a Black FB(153)p Black 2184 2757 a SDict begin H.R end 2184 2757 a 2184 2810 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.5.152) cvn H.B /ANN pdfmark end 2184 2810 a FB(\))29 b(\025)f(\()2347 2810 y SDict begin H.S end 2347 2810 a Black FB(170)p Black 2471 2757 a SDict begin H.R end 2471 2757 a 2471 2810 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.5.152) cvn H.B /ANN pdfmark end 2471 2810 a FB(\))h(and)f(will)g(require)f(some)h (more)f(considerations.)-150 2906 y(W)-7 b(e)23 b(will)g(apply)g(the)g (SP)-7 b(A)24 b(at)f(the)g(\034rst)g(order)e(to)i(appro)n(ximate)e(the) j(frequency-domain)d(w)n(a)n(v)n(e)g(form)i(\034rst)g(to)f(eac)n(h)g (single)h(frequency)-150 3001 y(separately)i(and)h(will)h(later)f(sum)g (up)h(all)f(the)h(terms)f(of)h(the)f(discrete)g(decomp)r(osition)g(as)g (a)g(virtue)g(of)h(the)f(linearit)n(y)g(of)g(the)h(F)-7 b(ourier)-150 3097 y(in)n(tegral)26 b(with)i(resp)r(ect)g(to)f(the)h (in)n(tegrand.)1448 3371 y SDict begin H.S end 1448 3371 a 1448 3371 a SDict begin 10.5 H.A end 1448 3371 a 1448 3371 a SDict begin [/View [/XYZ H.V]/Dest (section*.8) cvn /DEST pdfmark end 1448 3371 a Fs(B.)87 b(SP)-7 b(A)30 b(of)f(the)h(GW)g(signal)-67 3585 y FB(Supp)r(ose)e(an)f(in)n(tegral)f (of)i(the)g(form)1360 3812 y SDict begin H.S end 1360 3812 a 1360 3812 a SDict begin 11.5 H.A end 1360 3812 a 1360 3812 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.177) cvn /DEST pdfmark end 1360 3812 a 1361 3790 a Fr(~)1360 3812 y Fv(h)o Fr(\()p Fv(f)9 b Fr(\))24 b(=)1632 3699 y Fn(Z)1715 3719 y Fo(+)p FA(1)1678 3888 y(\0001)1851 3812 y Fq(A)p Fr(\()p Fv(t)p Fr(\))14 b Fv(e)2064 3778 y Fu(i)p Fo(\(2)e Fu(\031)r(f)7 b(t)p FA(\000)p Fu(\036)p Fo(\()p Fu(t)p Fo(\)\))2461 3812 y Fr(d)p Fv(t)14 b(:)1321 b FB(\(177\))-150 4054 y(Then,)32 b(ha)n(ving)d(found)i(a)f(stationary) f(p)r(oin)n(t)i Fv(t)1320 4023 y FA(\003)1358 4054 y FB(,)h(where)e(the)h(phase)f(de\034ned)h(as)f Fr(\010)e(:=)f(2)14 b Fv(\031)s(f)29 b Fq(\000)20 b Fv(\036)p Fr(\()p Fv(t)p Fr(\))p Fv(t)32 b FB(has)e(zero)f(ascen)n(t,)3725 4033 y Fr(_)3707 4054 y(\010\()p Fv(t)3829 4066 y Fo(0)3866 4054 y Fr(\))g(=)f(0)p FB(,)-150 4149 y(with)g(the)g(assumption)f(of)h (the)f(righ)n(t)g(b)r(eha)n(vior)f(at)i(the)g(b)r(oundary)-7 b(,)27 b(the)h(in)n(tegral)e(tak)n(es)h(the)g(form)1187 4445 y SDict begin H.S end 1187 4445 a 1187 4445 a SDict begin 11.5 H.A end 1187 4445 a 1187 4445 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.178) cvn /DEST pdfmark end 1187 4445 a 1188 4424 a Fr(~)1187 4445 y Fv(h)p Fr(\()p Fv(f)9 b Fr(\))23 b Fq(\031)1459 4373 y(p)p 1529 4373 92 4 v 1529 4445 a Fr(2)p Fv(\031)1666 4389 y(A)p Fr(\()p Fv(t)1790 4359 y FA(\003)1828 4389 y Fr(\))p 1630 4426 266 4 v 1630 4443 a Fn(q)p 1713 4443 183 4 v 1724 4525 a Fr(\177)1713 4546 y Fv(\036)q Fr(\()p Fv(t)1825 4522 y FA(\003)1863 4546 y Fr(\))1906 4445 y Fv(e)1945 4411 y FA(\000)p Fu(i)p Fo(\()p Fu(\036)p Fo(\()p Fu(t)2137 4386 y Ff(\003)2171 4411 y Fo(\))p FA(\000)p Fo(2)p Fu(\031)r(f)7 b(t)2387 4386 y Ff(\003)2422 4411 y Fo(+)p Fu(\031)r(=)p Fo(4\))2625 4445 y Fv(:)p Black 1247 w FB(\(178\))p Black -150 4729 a(In)28 b(the)g(case)e(of)i(our)f(GW,)h(the)g(in)n(tegral)e (turns)i(out)f(to)h(b)r(e)555 5060 y SDict begin H.S end 555 5060 a 555 5060 a SDict begin 11.5 H.A end 555 5060 a 555 5060 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.179) cvn /DEST pdfmark end 555 5060 a 613 4959 a Fo(\()p Fu(n)p Fo(\))579 5038 y Fr(~)578 5060 y Fv(h)p Fr(\()p Fv(f)9 b Fr(\))740 5085 y Fu(nm)894 5060 y Fr(=)1078 5004 y(1)p 1018 5041 161 4 v 1018 5057 a Fq(p)p 1088 5057 92 4 v 1088 5126 a Fr(2)p Fv(\031)1189 5060 y(e)1228 5026 y FA(\000)p Fu(mi\036)1402 5034 y Fl(0)1453 4947 y Fn(Z)1536 4967 y Fo(+)p FA(1)1499 5136 y(\0001)1698 4956 y(1)1671 4981 y Fn(X)1673 5158 y Fu(j)s Fo(=0)1805 5060 y Fr(\()p Fv(S)1888 5072 y Fu(j)1950 5060 y Fr(sin)14 b Fv(j)5 b Fp(M)18 b Fr(+)g Fv(C)2356 5072 y Fu(j)2419 5060 y Fr(cos)13 b Fv(j)5 b Fp(M)p Fr(\))28 b Fv(e)2773 5026 y FA(\000)p Fu(im)p Fg(kM)3035 5060 y Fv(e)3074 5026 y Fu(i)11 b Fo(2)p Fu(\031)r(f)c(t)3264 5060 y Fr(d)p Fv(t)14 b(;)p Black 518 w FB(\(179\))p Black 545 5263 a Fo(\()p Fu(n)p Fo(\))511 5342 y Fr(~)510 5364 y Fv(h)p Fr(\()p Fv(f)9 b Fr(\))672 5389 y Fu(C;nm)894 5364 y Fr(=)1078 5308 y(1)p 1018 5345 161 4 v 1018 5361 a Fq(p)p 1088 5361 92 4 v 1088 5430 a Fr(2)p Fv(\031)1189 5364 y(e)1228 5329 y FA(\000)p Fu(mi\036)1402 5337 y Fl(0)1439 5251 y Fn(Z)1522 5271 y Fo(+)p FA(1)1485 5439 y(\0001)1653 5308 y Fv(C)1712 5320 y Fu(j)p 1653 5345 95 4 v 1680 5421 a Fr(2)1771 5296 y Fn(\000)1809 5364 y Fv(e)1848 5329 y Fu(ij)s Fg(M)p FA(\000)p Fu(im)p Fg(kM)2240 5364 y Fr(+)18 b Fv(e)2362 5329 y FA(\000)p Fu(ij)s Fg(M)p FA(\000)p Fu(im)p Fg(kM)2787 5296 y Fn(\001)2853 5364 y Fv(e)2892 5329 y Fu(i)p Fo(2)p Fu(\031)r(f)7 b(t)3057 5364 y Fr(d)p Fv(t)p Black Black eop end end %%Page: 16 16 TeXDict begin HPSdict begin 16 15 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.16) cvn /DEST pdfmark end -158 -515 a Black 4001 -299 a FB(16)p Black 894 -38 a Fr(=)1078 -94 y(1)p 1018 -57 161 4 v 1018 -40 a Fq(p)p 1088 -40 92 4 v 1088 29 a Fr(2)p Fv(\031)1189 -38 y(e)1228 -72 y FA(\000)p Fu(mi\036)1402 -64 y Fl(0)1439 -151 y Fn(Z)1522 -130 y Fo(+)p FA(1)1485 38 y(\0001)1653 -94 y Fv(C)1712 -82 y Fu(j)p 1653 -57 95 4 v 1680 19 a Fr(2)1771 -130 y Fn(\020)1821 -38 y Fv(e)1860 -72 y Fu(i)p Fo(\()p Fu(j)s Fg(M)p Fo(+2)p Fu(\031)r(f)7 b(t)p FA(\000)p Fu(m)p Fg(kM)p Fo(\))2469 -38 y Fr(+)18 b Fv(e)2591 -72 y Fu(i)p Fo(\()p FA(\000)p Fu(j)s 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b Fq(\000)g Fv(m)p Fp(kM)p Fr(\))28 b FB(will)f(con)n(tribute,)h(since)f(the)h (phase)f Fr(\010)3450 683 y Fu(mj)3571 671 y FB(de\034ned)h(as)1464 849 y SDict begin H.S end 1464 849 a 1464 849 a SDict begin 11.5 H.A end 1464 849 a 1464 849 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.181) cvn /DEST pdfmark end 1464 849 a Fr(\010)1524 861 y Fu(mj)1641 849 y Fr(:=)23 b(2)p Fv(\031)s(f)9 b(t)18 b Fq(\000)g Fv(j)5 b Fp(M)18 b Fq(\000)g Fv(m)p Fp(kM)1425 b FB(\(181\))-150 1028 y(has)27 b(a)g(stationary)f(p) r(oin)n(t)i(at)f Fv(t)810 998 y FA(\003)876 1028 y FB(where)1410 1206 y SDict begin H.S end 1410 1206 a 1410 1206 a SDict begin 11.5 H.A end 1410 1206 a 1410 1206 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.182) cvn /DEST pdfmark end 1410 1206 a 1429 1185 a Fr(_)1410 1206 y(\010)1470 1218 y Fu(mj)1614 1206 y Fr(=)49 b(2)p Fv(\031)s(f)27 b Fq(\000)18 b Fv(j)5 b(n)19 b Fq(\000)f Fv(m)p Fp(k)p Fv(n)23 b Fr(=)f(0)14 b Fv(:)p Black 1371 w FB(\(182\))p Black -150 1399 a(Here)35 b(w)n(e)g(note)g(that)577 1377 y Fr(_)563 1399 y Fp(k)i Fr(=)e Fq(O)r Fr(\()p Fv(\017)884 1369 y Fo(7)922 1399 y Fr(\))g FB(and)g(will)h(b)r(e)g(consisten)n(tly) e(neglected.)60 b(W)-7 b(e)35 b(w)n(an)n(t)g(to)g(compute)g Fv(t)3168 1369 y FA(\003)3242 1399 y FB(and)g(its)g(con)n(tribution)g (in)-150 1494 y(the)d(follo)n(wing)f(lines.)51 b(The)32 b(question)f(arises)g(ho)n(w)g(to)h(\034nd)h(out)f Fv(t)1965 1464 y FA(\003)2035 1494 y FB(without)h(solving)e(Eqs.)50 b(\()2871 1494 y SDict begin H.S end 2871 1494 a Black FB(173)p Black 2996 1441 a SDict begin H.R end 2996 1441 a 2996 1494 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.173) cvn H.B /ANN pdfmark end 2996 1494 a FB(\))32 b(and)g(\()3258 1494 y SDict begin H.S end 3258 1494 a Black FB(174)p Black 3383 1441 a SDict begin H.R end 3383 1441 a 3383 1494 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.173) cvn H.B /ANN pdfmark end 3383 1494 a FB(\))h(n)n(umerically)-7 b(.)49 b(It)32 b(is)-150 1590 y(answ)n(ered)26 b(in)i(a)f(simple)h (manner.)36 b(W)-7 b(e)28 b(searc)n(h)e(for)h(the)h(solution)f Fv(n)c Fr(=)f Fv(n)p Fr(\()p Fv(e)2225 1602 y Fu(t)2254 1590 y Fv(;)14 b(e)2330 1602 y Fu(t)p Fo(0)2392 1590 y Fr(\))p FB(,)28 b Fv(e)2514 1602 y Fu(t)p Fo(0)2604 1590 y FB(b)r(eing)g(the)g(eccen)n(tricit)n(y)e(at)i Fv(t)23 b Fr(=)f Fv(t)3683 1602 y Fo(0)3721 1590 y FB(,)27 b(of)1782 1816 y SDict begin H.S end 1782 1816 a 1782 1816 a SDict begin 11.5 H.A end 1782 1816 a 1782 1816 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.183) cvn /DEST pdfmark end 1782 1816 a 1801 1759 a Fr(d)p Fv(n)p 1792 1797 115 4 v 1792 1873 a Fr(d)p Fv(e)1877 1885 y Fu(t)1939 1816 y Fr(=)2059 1759 y(_)-36 b Fv(n)p 2037 1797 68 4 v 2049 1873 a Fr(_)h Fv(e)2076 1885 y Fu(t)2129 1816 y Fv(;)1743 b FB(\(183\))-150 2039 y(insert)27 b(it)h(in)n(to)g(Eq.)37 b(\()525 2039 y SDict begin H.S end 525 2039 a Black FB(174)p Black 650 1985 a SDict begin H.R end 650 1985 a 650 2039 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.173) cvn H.B /ANN pdfmark end 650 2039 a FB(\),)1608 2217 y SDict begin H.S end 1608 2217 a 1608 2217 a SDict begin 11.5 H.A end 1608 2217 a 1608 2217 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.184) cvn /DEST pdfmark end 1608 2217 a 12 w Fr(_)-35 b Fv(e)1647 2229 y Fu(t)1726 2217 y Fr(=)49 b Fq(E)7 b Fr(\()p Fv(n)p Fr(\()p Fv(e)2044 2229 y Fu(t)2074 2217 y Fr(\))p Fv(;)14 b(e)2182 2229 y Fu(t)2211 2217 y Fv(;)g(e)2287 2229 y Fu(t)p Fo(0)2349 2217 y Fr(\))g Fv(;)p Black 1477 w FB(\(184\))p Black 1502 2337 a Fq(\))23 b Fv(e)1647 2349 y Fu(t)1726 2337 y Fr(=)49 b Fv(e)1879 2349 y Fu(t)1908 2337 y Fr(\()p Fv(e)1979 2349 y Fu(t)p Fo(0)2042 2337 y Fv(;)14 b(t)k Fq(\000)g Fv(t)2240 2349 y Fo(0)2277 2337 y Fr(\))c Fv(;)p Black 1549 w FB(\(185\))p Black -150 2516 a(and)23 b(in)n(v)n(ert)f (Eq.)36 b(\()431 2516 y SDict begin H.S end 431 2516 a Black FB(185)p Black 556 2463 a SDict begin H.R end 556 2463 a 556 2516 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.184) cvn H.B /ANN pdfmark end 556 2516 a FB(\))23 b(to)g(\034nd)h Fr(\()p Fv(t)10 b Fq(\000)g Fv(t)1047 2528 y Fo(0)1083 2516 y Fr(\))24 b FB(as)e(a)h(function)h(of)f Fv(e)1751 2528 y Fu(t)1803 2516 y FB(and)g Fv(e)1999 2528 y Fu(t)p Fo(0)2061 2516 y FB(,)h(where)f(for)f(the)i(stationary)d (p)r(oin)n(t)j(the)f(v)-5 b(alue)23 b Fv(e)3595 2486 y FA(\003)3595 2537 y Fu(t)3656 2516 y FB(corresp)r(onds)-150 2611 y(to)k(the)h(solution)f(of)h(the)g(equation)1029 2790 y SDict begin H.S end 1029 2790 a 1029 2790 a SDict begin 11.5 H.A end 1029 2790 a 1029 2790 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.186) cvn /DEST pdfmark end 1029 2790 a Fr(0)23 b(=)g(2)p Fv(\031)s(f)k Fq(\000)18 b Fv(j)5 b(n)p Fr(\()p Fv(e)1585 2756 y FA(\003)1585 2810 y Fu(t)1622 2790 y Fv(;)14 b(e)1698 2802 y Fu(t)p Fo(0)1760 2790 y Fr(\))19 b Fq(\000)f Fv(m)p Fp(k)p Fr(\()p Fv(n)p Fr(\()p Fv(e)2170 2756 y FA(\003)2170 2810 y Fu(t)2209 2790 y Fv(;)c(e)2285 2802 y Fu(t)p Fo(0)2346 2790 y Fr(\))p Fv(;)g(e)2454 2756 y FA(\003)2454 2810 y Fu(t)2492 2790 y Fr(\))g Fv(n)p Fr(\()p Fv(e)2659 2756 y FA(\003)2659 2810 y Fu(t)2698 2790 y Fv(;)g(e)2774 2802 y Fu(t)p Fo(0)2835 2790 y Fr(\))g Fv(:)991 b FB(\(186\))-150 2968 y(W)-7 b(e)32 b(see)g(that)g Fv(e)359 2938 y FA(\003)359 2989 y Fu(t)429 2968 y FB(and)f(th)n(us)h Fv(t)811 2938 y FA(\003)881 2968 y FB(dep)r(ends)g(only)g(on)f(the)i(magnetic)e(n)n(um) n(b)r(er)h Fv(m)p FB(,)h(the)f(summation)f(index)h Fv(j)5 b FB(,)33 b(the)g(eccen)n(tricit)n(y)d(and)-150 3064 y(mean)i(motion)f(at)h Fv(t)497 3076 y Fo(0)534 3064 y FB(,)h Fv(e)629 3076 y Fu(t)p Fo(0)723 3064 y FB(and)f Fv(n)939 3076 y Fo(0)976 3064 y FB(,)h(and)e(of)h(course)f(the)h (frequency)f Fv(f)9 b FB(.)49 b(The)32 b Fr(sin)g FB(term)g(of)g(all)f (those)h(calculations)e(will)i(ha)n(v)n(e)f(the)-150 3159 y(same)c(stationary)f(p)r(oin)n(t)i Fv(t)700 3129 y FA(\003)738 3159 y FB(,)g(suc)n(h)f(that)h(the)g(full)g(F)-7 b(ourier-domain)25 b(w)n(a)n(v)n(e)h(form)h(will)h(ha)n(v)n(e)f(the)h (follo)n(wing)e(app)r(earance:)p -49 3258 4135 4 v -49 4480 4 1222 v 313 3544 a SDict begin H.S end 313 3544 a 313 3544 a SDict begin 11.5 H.A end 313 3544 a 313 3544 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.187) cvn /DEST pdfmark end 313 3544 a -101 x Fo(\()p Fu(n)p Fo(\))336 3523 y Fr(~)335 3544 y Fv(h)d Fr(\()p Fv(f)9 b Fr(\))520 3556 y Fu(n;m>)p Fo(0)790 3544 y Fr(=)61 b Fv(\017)950 3510 y Fo(4)997 3488 y Fv(G)p 997 3525 66 4 v 998 3601 a(R)1113 3441 y FA(1)1086 3466 y Fn(X)1089 3642 y Fu(j)s Fo(=0)1230 3488 y Fr(1)p 1230 3525 42 4 v 1230 3601 a(2)1295 3452 y Fn(\020)1345 3544 y Fv(iS)1430 3510 y FA(\003)1425 3567 y Fu(m)p Fo(\()p Fu(j)s(>)p Fo(0\))1673 3544 y Fr(+)18 b(\(1)h(+)f Fv(\016)1969 3556 y Fo(0)p Fu(j)2037 3544 y Fr(\))p Fv(C)2134 3510 y FA(\003)2128 3565 y Fu(mj)2222 3452 y Fn(\021)2286 3378 y(2)2286 3527 y(4)2538 3488 y Fr(1)p 2351 3525 415 4 v 2351 3542 a Fn(q)p 2434 3542 332 4 v 2445 3617 a Fr(\177)2434 3639 y Fv(\036)2483 3651 y Fu(mj)2577 3639 y Fr(\()p Fv(t)2639 3610 y FA(\003)2639 3662 y Fu(mj)2733 3639 y Fr(\))2776 3544 y Fv(e)2815 3509 y Fu(i)2838 3515 y Fr(\()2870 3509 y Fo(\010)2917 3517 y Fm(mj)3000 3509 y Fo(\()p Fu(t)3051 3483 y Ff(\003)3051 3525 y Fm(mj)3133 3509 y Fo(\))p FA(\000)p Fu(\031)r(=)p Fo(4)3319 3515 y Fr(\))3355 3378 y Fn(3)3355 3527 y(5)3424 3544 y Fv(e)3463 3510 y FA(\000)p Fu(im\036)3637 3518 y Fl(0)3687 3544 y Fv(;)p Black 131 w FB(\(187\))p Black 314 3790 a Fo(\()p Fu(n)p Fo(\))337 3870 y Fr(~)336 3891 y Fv(h)23 b Fr(\()p Fv(f)9 b Fr(\))521 3903 y Fu(n;m)p Fo(=0)790 3891 y Fr(=)61 b Fv(\017)950 3857 y Fo(4)997 3835 y Fv(G)p 997 3872 66 4 v 998 3948 a(R)1113 3788 y FA(1)1086 3813 y Fn(X)1089 3989 y Fu(j)s Fo(=1)1230 3835 y Fr(1)p 1230 3872 42 4 v 1230 3948 a(2)1295 3824 y Fn(\000)1333 3891 y Fv(iS)1418 3857 y FA(\003)1413 3912 y Fo(0)p Fu(j)1499 3891 y Fr(+)18 b Fv(C)1647 3857 y FA(\003)1641 3912 y Fo(0)p Fu(j)1710 3824 y Fn(\001)1762 3725 y(2)1762 3874 y(4)1988 3835 y Fr(1)p 1827 3872 364 4 v 1827 3889 a Fn(q)p 1910 3889 281 4 v 1921 3964 a Fr(\177)1910 3986 y Fv(\036)1959 3998 y Fo(0)p Fu(j)2027 3986 y Fr(\()p Fv(t)2089 3957 y FA(\003)2089 4009 y Fo(0)p Fu(j)2158 3986 y Fr(\))2200 3891 y Fv(e)2239 3856 y Fu(i)2262 3862 y Fr(\()2294 3856 y Fo(\010)2341 3864 y Fl(0)p Fm(j)2401 3856 y Fo(\()p Fu(t)2452 3831 y Ff(\003)2452 3872 y Fl(0)p Fm(j)2511 3856 y Fo(\))p FA(\000)p Fu(\031)r(=)p Fo(4)2697 3862 y Fr(\))2734 3725 y Fn(3)2734 3874 y(5)2817 3891 y Fv(;)p Black 1001 w FB(\(188\))p Black 408 4124 a Fr(\177)397 4146 y Fv(\036)446 4158 y Fu(mj)540 4146 y Fr(\()p Fv(t)602 4111 y FA(\003)602 4166 y Fu(mj)697 4146 y Fr(\))50 b(:=)62 b(_)-36 b Fv(n)14 b Fr(\()p Fv(j)23 b Fr(+)18 b Fv(m)p Fp(k)p Fr(\))d Fq(j)1345 4158 y Fu(t)p Fo(=)p Fu(t)1446 4138 y Ff(\003)1446 4177 y Fm(mj)p Black 3841 4146 a FB(\(189\))p Black 584 4281 a Fv(S)640 4246 y FA(\003)635 4301 y Fu(mj)779 4281 y Fr(:=)49 b Fv(S)967 4293 y Fu(mj)1061 4281 y Fq(j)1084 4293 y Fu(t)p Fo(=)p Fu(t)1185 4273 y Ff(\003)1185 4312 y Fm(mj)1285 4281 y Fv(;)p Black 2533 w FB(\(190\))p Black 576 4416 a Fv(C)641 4381 y FA(\003)635 4436 y Fu(mj)779 4416 y Fr(:=)g Fv(C)975 4428 y Fu(mj)1069 4416 y Fq(j)1092 4428 y Fu(t)p Fo(=)p Fu(t)1193 4408 y Ff(\003)1193 4447 y Fm(mj)1293 4416 y Fv(:)p Black 2525 w FB(\(191\))p Black 4082 4480 4 1222 v -49 4483 4135 4 v -150 4658 a(The)32 b(addend)h Fv(\016)358 4670 y Fo(0)p Fu(j)458 4658 y FB(app)r(ears)e(only)h(at)g (one)g(sp)r(ecial)g(place)g(in)h(Eq.)52 b(\()2021 4658 y SDict begin H.S end 2021 4658 a Black FB(187)p Black 2145 4605 a SDict begin H.R end 2145 4605 a 2145 4658 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.187) cvn H.B /ANN pdfmark end 2145 4658 a FB(\))33 b(b)r(ecause)f(in)h(Eq.)51 b(\()2830 4658 y SDict begin H.S end 2830 4658 a Black FB(180)p Black 2955 4605 a SDict begin H.R end 2955 4605 a 2955 4658 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.179) cvn H.B /ANN pdfmark end 2955 4658 a FB(\))33 b(b)r(oth)g(terms,) g(the)g(\034rst)f Fw(and)h FB(the)-150 4753 y(second,)25 b(con)n(tribute)h(to)f(the)h Fr(cos)f FB(terms)g(for)g Fv(j)j Fr(=)23 b(0)i FB(and)g(the)h Fr(sin)g FB(terms)f(cancel)g(eac)n (h)g(other.)36 b(F)-7 b(or)25 b(computing)g(the)h(energy)f Fv(E)30 b FB(whic)n(h)-150 4849 y(is)c(a)g(usual)g(prefactor)g(with)h (some)e(exp)r(onen)n(t)i(in)f(all)h(the)f(m)n(ultip)r(oles,)h(use)f(of) h(Eq.)37 b(\()2523 4849 y SDict begin H.S end 2523 4849 a Black FB(5)p Black 2565 4796 a SDict begin H.R end 2565 4796 a 2565 4849 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.1) cvn H.B /ANN pdfmark end 2565 4849 a FB(\))26 b(has)g(to)h(b)r(e)f(made)h(when)f(the)h(\020stationary\021)k Fv(e)4055 4861 y Fu(t)-150 4944 y FB(and)24 b Fv(n)g FB(are)f(determined.)36 b(This)24 b(appro)n(ximation)e(is)i (justi\034ed)g(as)g(w)n(e)f(simply)h(use)g(the)h(leading-order)c(SP)-7 b(A)25 b(and)f(is)g(v)-5 b(alid)24 b(in)g(a)g(regime)-150 5040 y(where)k(the)g(orbit)g(ev)n(olv)n(es)e Fw(slow)t(ly)k FB(to)n(w)n(ards)c(coalescence.)37 b(It)28 b(is,)g(th)n(us,)h(an)f (appro)n(ximation)e(around)h(the)h(p)r(oin)n(t)h(\()p Fv(n)3544 5052 y Fo(0)3581 5040 y Fv(;)14 b(e)3657 5052 y Fu(t)p Fo(0)3719 5040 y FB(\))28 b(with)h(the)-150 5135 y(aforemen)n(tioned)d(requiremen)n(ts.)-67 5231 y(These)j(are)f(the)i(ideas)e(so)h(far.)41 b(Here)29 b(are)f(the)i(explicit)f(equations)g(up)g(to)g(1pN)g(in)h(conserv)-5 b(ativ)n(e)27 b(and)i(only)g(to)g(leading-order)e(in)-150 5326 y(radiativ)n(e)f(dynamics,)h(where)g Fv(n)h FB(is)f(unscaled)g (\(that)h(means)f(it)h(has)f(unit)i(second)2438 5296 y FA(\000)p Fo(1)2526 5326 y FB(\),)p Black Black eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.17) cvn /DEST pdfmark end -158 -515 a Black 4001 -299 a FB(17)p Black 960 143 a SDict begin H.S end 960 143 a 960 143 a SDict begin 11.5 H.A end 960 143 a 960 143 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.192) cvn /DEST pdfmark end 960 143 a 13 w Fr(_)-36 b Fv(n)50 b Fr(=)22 b Fq(N)12 b Fr(\()p Fv(n;)i(e)1385 155 y Fu(t)1414 143 y Fr(\))51 b(=)22 b Fv(\017)1618 109 y Fo(5)1665 82 y Fv(n)1715 52 y Fo(2)1752 82 y Fv(\021)1810 15 y Fn(\000)1848 82 y Fr(37)p Fv(e)1971 52 y Fo(4)1971 103 y Fu(t)2026 82 y Fr(+)c(292)p Fv(e)2274 52 y Fo(2)2274 103 y Fu(t)2328 82 y Fr(+)g(96)2495 15 y Fn(\001)2545 82 y Fr(\()p Fv(Gm)2715 94 y Fu(t)2745 82 y Fv(n)p Fr(\))2827 52 y Fo(5)p Fu(=)p Fo(3)p 1665 124 1267 4 v 2070 202 a Fr(5)c(\()o(1)k Fq(\000)g Fv(e)2339 173 y Fo(2)2339 222 y Fu(t)2376 202 y Fr(\))2422 178 y Fo(7)p Fu(=)p Fo(2)2955 143 y Fv(;)p Black 917 w FB(\(192\))p Black 954 391 a Fr(_)-35 b Fv(e)981 403 y Fu(t)1074 391 y Fr(=)23 b Fq(E)7 b Fr(\()p Fv(n;)14 b(e)1371 403 y Fu(t)1400 391 y Fr(\))65 b(=)22 b Fq(\000)p Fv(\017)1683 357 y Fo(5)1730 330 y Fv(n\021)s(e)1863 342 y Fu(t)1905 262 y Fn(\000)1943 330 y Fr(121)p Fv(e)2108 299 y Fo(2)2108 350 y Fu(t)2162 330 y Fr(+)c(304)2371 262 y Fn(\001)2422 330 y Fr(\()p Fv(Gm)2592 342 y Fu(t)2621 330 y Fv(n)p Fr(\))2703 299 y Fo(5)p Fu(=)p Fo(3)p 1730 372 1079 4 v 2020 449 a Fr(15)c(\()o(1)k Fq(\000)g Fv(e)2331 421 y Fo(2)2331 470 y Fu(t)2368 449 y Fr(\))2414 425 y Fo(5)p Fu(=)p Fo(2)2832 391 y Fv(:)p Black 1040 w FB(\(193\))p Black -150 619 a(In)27 b(terms)g(of)g(com)n(bined)g(quan)n(tities,)g (related)f(form)n(ulas)g(for)h(the)g(simple)h(P)n(eters)e(and)h (Matthews)g(approac)n(h)e(ha)n(v)n(e)h(b)r(een)i(published)-150 714 y(b)n(y)k(Pierro)g(and)g(Pin)n(to)h([)645 714 y SDict begin H.S end 645 714 a FB(19)729 714 y SDict begin 11.5 H.L end 729 714 a 729 714 a SDict begin [/Subtype /Link/Dest (cite.Pierro:Pinto:1996) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 729 714 a -1 w FB(])g(and)f(App)r(ell's)h(2-v)-5 b(ariable)31 b(h)n(yp)r (ergeometric)f(function)j Fr(App)r(ellF)2876 734 y Fo(1)2946 714 y FB(has)f(come)g(to)h(use.)51 b(W)-7 b(e)33 b(will)g(k)n(eep)-150 810 y(our)28 b(o)n(wn)g(expressions)e(for)i Fv(n)h FB(and)f Fv(e)1014 822 y Fu(t)1072 810 y FB(and)g(will)h(express)e(unscaled)h (elapsed)g(times)h(as)f(functions)h(of)f(the)h(latter.)40 b(Note)28 b(that)h(b)n(y)g(an)-150 905 y(appropriate)d(scaling)g(the)i (factor)f Fv(G)14 b(m)1103 917 y Fu(t)1160 905 y FB(is)27 b(re-absorb)r(ed)f(in)i(the)g(time)g(unit.)-67 1001 y(The)g(solution)f (to)g Fr(d)p Fv(n=)p Fr(d)p Fv(e)742 1013 y Fu(t)794 1001 y Fr(=)22 b(\(d)p Fv(n=)p Fr(d)p Fv(t)p Fr(\))p Fv(=)p Fr(\(d)p Fv(e)1318 1013 y Fu(t)1347 1001 y Fv(=)p Fr(d)p Fv(t)p Fr(\))28 b FB(reads)622 1281 y SDict begin H.S end 622 1281 a 622 1281 a SDict begin 11.5 H.A end 622 1281 a 622 1281 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.194) cvn /DEST pdfmark end 622 1281 a Fv(n)p Fr(\()p Fv(n)754 1293 y Fo(0)792 1281 y Fv(;)14 b(e)868 1293 y Fu(t)896 1281 y Fv(;)g(e)972 1293 y Fu(t)p Fo(0)1034 1281 y Fr(\))24 b(=)e Fv(n)1227 1293 y Fo(0)1292 1139 y Fn( )1368 1152 y(\000)1406 1220 y Fv(e)1445 1232 y Fu(t)1474 1189 y Fo(2)1529 1220 y Fq(\000)c Fr(1)1654 1152 y Fn(\001)p 1368 1262 325 4 v 1376 1338 a Fr(\()p Fv(e)1447 1309 y Fo(2)1447 1360 y Fu(t)p Fo(0)1527 1338 y Fq(\000)g Fr(1\))1702 1139 y Fn(!)1768 1156 y Fo(3)p Fu(=)p Fo(2)1886 1164 y Fn(\022)1957 1225 y Fv(e)1996 1237 y Fu(t)p Fo(0)p 1957 1262 101 4 v 1973 1338 a Fv(e)2012 1350 y Fu(t)2068 1164 y Fn(\023)2129 1181 y Fo(18)p Fu(=)p Fo(19)2327 1164 y Fn(\022)2398 1225 y Fr(121)p Fv(e)2563 1237 y Fu(t)p Fo(0)2624 1195 y(2)2679 1225 y Fr(+)g(304)p 2398 1262 489 4 v 2415 1338 a(121)p Fv(e)2580 1350 y Fu(t)2607 1314 y Fo(2)2663 1338 y Fr(+)g(304)2897 1164 y Fn(\023)2958 1178 y Fo(1305)p Fu(=)p Fo(2299)3289 1281 y Fv(;)583 b FB(\(194\))-150 1533 y(with)28 b Fv(e)78 1545 y Fu(t)p Fo(0)167 1533 y FB(and)f Fv(n)378 1545 y Fo(0)442 1533 y FB(as)g(the)g(v)-5 b(alue)27 b(of)g Fv(e)1033 1545 y Fu(t)1089 1533 y FB(and)g Fv(n)g FB(at)g(the)h (initial)f(instan)n(t)g(of)g(time)h Fv(t)2402 1545 y Fo(0)2439 1533 y FB(,)f(resp)r(ectiv)n(ely)-7 b(.)36 b(The)27 b(elapsed)g(time)h(as)e(a)h(function)-150 1629 y(of)h Fv(e)-16 1641 y Fu(t)40 1629 y FB(and)f Fv(e)240 1641 y Fu(t)p Fo(0)330 1629 y FB(reads)746 1862 y SDict begin H.S end 746 1862 a 746 1862 a SDict begin 11.5 H.A end 746 1862 a 746 1862 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.195) cvn /DEST pdfmark end 746 1862 a Fv(t)19 b Fq(\000)f Fv(t)908 1874 y Fo(0)1006 1862 y Fr(=)61 b Fv(\017)1166 1828 y FA(\000)p Fo(5)1516 1806 y Fr(9519)1684 1776 y Fo(1181)p Fu(=)p Fo(2299)p 1279 1843 944 4 v 1279 1921 a Fr(22)1363 1897 y Fo(2173)p Fu(=)p Fo(2299)1665 1921 y Fv(c)1701 1933 y Fo(0)1738 1897 y(8)p Fu(=)p Fo(3)1842 1921 y Fv(\021)s Fr(\()p Fv(Gm)2056 1933 y Fu(t)2086 1921 y Fr(\))2118 1897 y Fo(5)p Fu(=)p Fo(3)2251 1862 y Fq(\002)1132 1982 y Fn(\032)1194 2099 y Fv(e)1233 2111 y Fu(t)p Fo(0)1295 2064 y(48)p Fu(=)p Fo(19)1466 2099 y Fr(App)r(ellF)1760 2119 y Fo(1)1811 1982 y Fn(\022)1882 2042 y Fr(24)p 1882 2080 84 4 v 1882 2156 a(19)1975 2099 y(;)2022 2042 y(3)p 2022 2080 42 4 v 2022 2156 a(2)2073 2099 y Fv(;)14 b Fq(\000)2185 2042 y Fr(1181)p 2185 2080 167 4 v 2185 2156 a(2299)2361 2099 y(;)2408 2042 y(43)p 2408 2080 84 4 v 2408 2156 a(19)2501 2099 y(;)g Fv(e)2577 2111 y Fu(t)p Fo(0)2639 2064 y(2)2676 2099 y Fv(;)g Fq(\000)2788 2042 y Fr(121)p Fv(e)2953 2054 y Fu(t)p Fo(0)3013 2012 y(2)p 2787 2080 263 4 v 2856 2156 a Fr(304)3060 1982 y Fn(\023)1178 2334 y Fq(\000)k Fv(e)1300 2346 y Fu(t)1329 2300 y Fo(48)p Fu(=)p Fo(19)1500 2334 y Fr(App)r(ellF)1794 2355 y Fo(1)1845 2217 y Fn(\022)1916 2278 y Fr(24)p 1916 2315 84 4 v 1916 2391 a(19)2009 2334 y(;)2056 2278 y(3)p 2056 2315 42 4 v 2056 2391 a(2)2107 2334 y Fv(;)c Fq(\000)2219 2278 y Fr(1181)p 2219 2315 167 4 v 2219 2391 a(2299)2395 2334 y(;)2442 2278 y(43)p 2442 2315 84 4 v 2442 2391 a(19)2535 2334 y(;)g Fv(e)2611 2346 y Fu(t)2639 2300 y Fo(2)2677 2334 y Fv(;)g Fq(\000)2789 2278 y Fr(121)p Fv(e)2954 2290 y Fu(t)2980 2248 y Fo(2)p 2788 2315 230 4 v 2841 2391 a Fr(304)3028 2217 y Fn(\023\033)3165 2334 y Fv(:)p Black 707 w FB(\(195\))p Black 872 2602 a Fv(c)908 2614 y Fo(0)995 2602 y Fr(:=)1142 2540 y Fv(e)1181 2497 y Fo(18)p Fu(=)p Fo(19)1181 2562 y Fu(t)p Fo(0)1365 2473 y Fn(\000)1403 2540 y Fr(121)p Fv(e)1568 2510 y Fo(2)1568 2561 y Fu(t)p Fo(0)1647 2540 y Fr(+)k(304)1856 2473 y Fn(\001)1893 2490 y Fo(1305)p Fu(=)p Fo(2299)2210 2540 y Fv(n)2260 2552 y Fo(0)p 1142 2582 1155 4 v 1513 2682 a Fr(\(1)g Fq(\000)g Fv(e)1727 2653 y Fo(2)1727 2704 y Fu(t)p Fo(0)1789 2682 y Fr(\))1821 2636 y Fo(3)p Fu(=)p Fo(2)2320 2602 y Fv(:)p Black 1552 w FB(\(196\))p Black -150 2852 a(A)28 b(c)n(hec)n(k)f(will)g(sho)n(w)g(that)h(the)g(righ)n (t)f(hand)g(side)h(has)f(the)h(dimension)f(of)g(time.)999 3128 y SDict begin H.S end 999 3128 a 999 3128 a SDict begin 10.5 H.A end 999 3128 a 999 3128 a SDict begin [/View [/XYZ H.V]/Dest (section*.9) cvn /DEST pdfmark end 999 3128 a Fs(C.)88 b(Solution)32 b(to)e(the)f(SP)-7 b(A)30 b(condition)i(equation,)d Fe(j)d Fd(6)p Fc(=)21 b(0)-67 3342 y FB(It)28 b(sho)n(w)n(ed)e(up)h(that)h(it)g(is)f(easier)f(to)h (solv)n(e)f(Eq.)37 b(\()1501 3342 y SDict begin H.S end 1501 3342 a Black FB(186)p Black 1626 3289 a SDict begin H.R end 1626 3289 a 1626 3342 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.186) cvn H.B /ANN pdfmark end 1626 3342 a FB(\))28 b(for)e Fv(n)i FB(instead)f(of)g Fv(e)2309 3354 y Fu(t)2365 3342 y FB(and)g(then)h(to)f(express)f Fv(e)3144 3354 y Fu(t)3191 3342 y Fq(\000)18 b Fv(e)3313 3354 y Fu(t)p Fo(0)3402 3342 y FB(in)28 b(terms)f(of)g Fv(n)18 b Fq(\000)f Fv(n)4024 3354 y Fo(0)4061 3342 y FB(.)-150 3437 y(Expressed)27 b(fully)h(in)g(terms)f(of)h Fv(n)p FB(,)f(it)h(reads)1326 3616 y SDict begin H.S end 1326 3616 a 1326 3616 a SDict begin 11.5 H.A end 1326 3616 a 1326 3616 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.197) cvn /DEST pdfmark end 1326 3616 a Fr(0)23 b(=)f(2)p Fv(\031)s(f)27 b Fq(\000)18 b Fv(j)5 b(n)1810 3582 y FA(\003)1867 3616 y Fq(\000)18 b Fv(m)p Fp(k)p Fr(\()p Fv(n)2155 3582 y FA(\003)2193 3616 y Fv(;)c(e)2269 3628 y Fu(t)2298 3616 y Fr(\()p Fv(n)2380 3582 y FA(\003)2418 3616 y Fr(\)\))p Fv(n)2532 3582 y FA(\003)2585 3616 y Fv(:)1287 b FB(\(197\))-150 3794 y(It)28 b(is)f(solv)n(ed)g(in)h(t)n(w) n(o)e(steps:)37 b(\034rst,)28 b(solv)n(e)e(the)i(Newtonian)f(and)h (second:)36 b(solv)n(e)26 b(the)i(1pN)g(equation)e(with)j(the)e(help)h (of)g(step)g(1,)945 4016 y SDict begin H.S end 945 4016 a 945 4016 a SDict begin 11.5 H.A end 945 4016 a 945 4016 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.198) cvn /DEST pdfmark end 945 4016 a Fr(Step)g(I)23 b(:)557 b Fv(n)1817 3986 y FA(\003)1817 4039 y Fo(N)1955 4016 y Fr(=)2052 3960 y(2)p Fv(\031)s(f)p 2052 3997 142 4 v 2103 4073 a(j)2217 4016 y(;)p Black 1655 w FB(\(198\))p Black 912 4240 a Fr(Step)28 b(I)r(I)c(:)522 b Fv(n)1782 4210 y FA(\003)1782 4263 y Fo(1pN)1955 4240 y Fr(=)2052 4184 y(2)p Fv(\031)s(f)27 b Fq(\000)18 b Fv(m)p Fp(k)p Fr(\()p Fv(n)2500 4154 y FA(\003)2500 4207 y Fo(N)2554 4184 y Fv(;)c(e)2630 4196 y Fu(t)2658 4184 y Fr(\()p Fv(n)2740 4154 y FA(\003)2740 4207 y Fo(N)2794 4184 y Fr(\)\))g Fv(n)2922 4154 y FA(\003)2922 4207 y Fo(N)p 2052 4221 923 4 v 2494 4297 a Fv(j)2999 4240 y(:)p Black 873 w FB(\(199\))p Black -150 4467 a(Step)33 b(I)r(I)g(drops)e(out)h (in)h(case)f Fv(m)f Fr(=)f(0)p FB(.)51 b(The)33 b(Newtonian)f (\020stationary)e(eccen)n(tricit)n(y\021)38 b(for)32 b(the)h(frequency)f Fv(f)9 b FB(,)33 b(i.e.)52 b Fv(e)3589 4479 y Fu(t)3617 4467 y Fr(\()p Fv(n)3699 4437 y FA(\003)3699 4490 y Fo(N)3753 4467 y Fr(\))p FB(,)34 b(can)e(b)r(e)-150 4563 y(found)26 b(n)n(umerically)e(or)h(with)g(the)h(help)g(of)f(a)g(p) r(erturbativ)n(e)g(solution)g(sc)n(heme.)35 b(It)26 b(is)f(a)g(rather)g (n)n(umerical)f(issue)h(to)g(apply)g(\034xp)r(oin)n(t-)-150 4658 y(metho)r(d-lik)n(e)g(iterativ)n(e)g(algorithms)g(a)g(la)h(Dan)n (b)n(y)f(and)h(Burkhards')e(metho)r(d)i([)2370 4658 y SDict begin H.S end 2370 4658 a FB(20)2454 4658 y SDict begin 11.5 H.L end 2454 4658 a 2454 4658 a SDict begin [/Subtype /Link/Dest (cite.Danby:Burkhardt:1983) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 2454 4658 a -1 w FB(])g(to)g(solv)n(e)e(the)j(Kepler)e(equation,)g(and) h(a)f(detailed)-150 4754 y(analysis)i(could)i(indicate)f(ho)n(w)h(man)n (y)f(steps)g(are)g(necessary)f(and)h(reasonable)f(to)n(w)n(ards)g(the)i (solution.)39 b(An)n(yw)n(a)n(y)-7 b(,)28 b(w)n(e)g(promised)g(to)-150 4849 y(giv)n(e)j(analytical)g(results)g(and)h(this)g(will)g(b)r(e)h (done)e(b)r(elo)n(w.)50 b(W)-7 b(e)32 b(are)f(a)n(w)n(are)f(that)i (there)g(ma)n(y)f(exist)g(b)r(etter)i(algorithms)d(and)i(refer)-150 4944 y(the)26 b(reader)e(to)h(the)h(common)f(literature)g(of)g(appro)n (ximativ)n(e)f(solving)g(metho)r(ds.)36 b(The)26 b(function)g(to)f(b)r (e)h(in)n(v)n(erted)f(for)g Fv(e)3639 4956 y Fu(t)3682 4944 y Fq(\000)14 b Fv(e)3800 4956 y Fu(t)p Fo(0)3887 4944 y FB(is)26 b(the)-150 5040 y(follo)n(wing,)521 5316 y SDict begin H.S end 521 5316 a 521 5316 a SDict begin 11.5 H.A end 521 5316 a 521 5316 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.200) cvn /DEST pdfmark end 521 5316 a Fv(n)p Fr(\()p Fv(n)653 5328 y Fo(0)690 5316 y Fv(;)14 b(e)766 5328 y Fu(t)795 5316 y Fv(;)g(e)871 5328 y Fu(t)p Fo(0)933 5316 y Fr(\))61 b(=)g Fv(n)1202 5328 y Fo(0)1267 5174 y Fn( )1343 5187 y(\000)1381 5254 y Fv(e)1420 5266 y Fu(t)1449 5224 y Fo(2)1504 5254 y Fq(\000)18 b Fr(1)1629 5187 y Fn(\001)p 1343 5297 325 4 v 1351 5373 a Fr(\()p Fv(e)1422 5344 y Fo(2)1422 5395 y Fu(t)p Fo(0)1502 5373 y Fq(\000)g Fr(1\))1677 5174 y Fn(!)1743 5191 y Fo(3)p Fu(=)p Fo(2)1861 5199 y Fn(\022)1932 5260 y Fv(e)1971 5272 y Fu(t)p Fo(0)p 1932 5297 101 4 v 1948 5373 a Fv(e)1987 5385 y Fu(t)2043 5199 y Fn(\023)2104 5216 y Fo(18)p Fu(=)p Fo(19)2302 5199 y Fn(\022)2373 5260 y Fr(121)p Fv(e)2538 5272 y Fu(t)p Fo(0)2599 5229 y(2)2654 5260 y Fr(+)g(304)p 2373 5297 489 4 v 2390 5373 a(121)p Fv(e)2555 5385 y Fu(t)2582 5349 y Fo(2)2638 5373 y Fr(+)g(304)2872 5199 y Fn(\023)2933 5213 y Fo(1305)p Fu(=)p Fo(2299)p Black 3895 5316 a FB(\(200\))p Black Black Black eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.18) cvn /DEST pdfmark end -158 -515 a Black 4001 -299 a FB(18)p Black 1026 -83 a Fr(=)61 b Fv(n)1202 -71 y Fo(0)1258 -83 y Fr(+)18 b Fv(g)1384 -117 y Fo(\(1\))1472 -83 y Fr(\()p Fv(e)1543 -71 y Fu(t)1591 -83 y Fq(\000)g Fv(e)1713 -71 y Fu(t)p Fo(0)1775 -83 y Fr(\))h(+)g(~)-43 b Fv(\017)13 b(g)1999 -117 y Fo(\(2\))2088 -83 y Fr(\()p Fv(e)2159 -71 y Fu(t)2207 -83 y Fq(\000)18 b Fv(e)2329 -71 y Fu(t)p Fo(0)2391 -83 y Fr(\))2423 -117 y Fo(2)2479 -83 y Fr(+)g(~)-42 b Fv(\017)2596 -117 y Fo(2)2647 -83 y Fv(g)2690 -117 y Fo(\(3\))2778 -83 y Fr(\()p Fv(e)2849 -71 y Fu(t)2897 -83 y Fq(\000)18 b Fv(e)3019 -71 y Fu(t)p Fo(0)3081 -83 y Fr(\))3113 -117 y Fo(3)3169 -83 y Fr(+)g Fv(:)c(:)g(:)27 b(;)p Black 496 w FB(\(201\))p Black 832 85 a Fv(g)875 50 y Fo(\()p Fu(p)p Fo(\))1015 85 y Fr(:=)1174 29 y(1)p 1162 66 65 4 v 1162 142 a Fv(p)p Fr(!)1271 29 y Fv(@)1320 -2 y Fu(p)p 1261 66 108 4 v 1261 142 a Fv(@)1310 102 y Fu(p)1305 151 y(e)1336 159 y Fm(t)1378 85 y Fv(n)p Fr(\()p Fv(n)1510 97 y Fo(0)1547 85 y Fv(;)14 b(e)1623 97 y Fu(t)1652 85 y Fv(;)g(e)1728 97 y Fu(t)p Fo(0)1790 85 y Fr(\))p Fq(j)1845 97 y Fu(e)1876 105 y Fm(t)p Fl(0)1951 85 y Fv(;)p Black 1921 w FB(\(202\))p Black -150 296 a(ha)n(ving)25 b(in)n(tro)r(duced)g(some)g(smallness)g(parameter)g Fr(~)-43 b Fv(\017)26 b FB(whic)n(h)f(will)h(b)r(e)g(set)g Fr(1)f FB(after)h(the)g(calculation.)35 b(The)26 b(solution)f (algorithm)f(reads)-150 392 y(\(de\034ning)k Fv(\024)f FB(as)g(the)h(di\033erence)g(of)f Fv(e)1021 362 y FA(\003)1021 412 y Fu(t)1087 392 y FB(and)g Fv(e)1287 404 y Fu(t)p Fo(0)1377 392 y FB(and)g(lea)n(ving)f(out)i(the)g(\020star\021\))986 570 y SDict begin H.S end 986 570 a 986 570 a SDict begin 11.5 H.A end 986 570 a 986 570 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.203) cvn /DEST pdfmark end 986 570 a Fv(\024)1034 536 y Fo([)p Fu(N)6 b Fo(])1184 570 y Fr(:=)49 b(\()p Fv(e)1392 582 y Fu(t)1440 570 y Fq(\000)18 b Fv(e)1562 582 y Fu(t)p Fo(0)1624 570 y Fr(\))1656 536 y Fo([)p Fu(N)6 b Fo(])1771 570 y Fv(;)p Black 2101 w FB(\(203\))p Black 1011 742 a Fv(\024)1059 708 y Fo([1])1195 742 y Fr(=)1331 686 y(\()p Fv(n)19 b Fq(\000)f Fv(n)1565 698 y Fo(0)1602 686 y Fr(\))p 1331 723 303 4 v 1417 801 a Fv(g)1460 777 y Fo(\(1\))1658 742 y Fv(;)p Black 2214 w FB(\(204\))p Black 1011 960 a Fv(\024)1059 926 y Fo([2])1195 960 y Fr(=)1376 904 y(1)p 1331 941 132 4 v 1331 1018 a Fv(g)1374 994 y Fo(\(1\))1487 868 y Fn(n)1542 960 y Fr(\()p Fv(n)h Fq(\000)f Fv(n)1776 972 y Fo(0)1813 960 y Fr(\))g Fq(\000)h Fr(~)-42 b Fv(\017)13 b(g)2037 926 y Fo(\(2\))2140 960 y Fr(\()p Fv(\024)2220 926 y Fo([1])2295 960 y Fr(\))2327 926 y Fo(2)2364 868 y Fn(o)2447 960 y Fv(;)p Black 1425 w FB(\(205\))p Black 986 1220 a Fv(\024)1034 1185 y Fo([)p Fu(N)6 b Fo(])1195 1220 y Fr(=)1376 1163 y(1)p 1331 1200 V 1331 1278 a Fv(g)1374 1254 y Fo(\(1\))1487 1078 y Fn(\()1554 1220 y Fr(\()p Fv(n)18 b Fq(\000)g Fv(n)1787 1232 y Fo(0)1824 1220 y Fr(\))h Fq(\000)1989 1116 y Fu(N)1958 1141 y Fn(X)1959 1316 y Fu(p)p Fo(=2)2093 1220 y Fr(~)-43 b Fv(\017)2126 1185 y Fu(p)p FA(\000)p Fo(1)2263 1220 y Fv(g)2306 1185 y Fo(\()p Fu(p)p Fo(\))2409 1220 y Fr(\()p Fv(\024)2489 1185 y Fo([)p Fu(N)6 b Fo(+1)p FA(\000)p Fu(p)p Fo(])2760 1220 y Fr(\))2792 1185 y Fu(p)2831 1078 y Fn(\))2925 1220 y Fv(;)p Black 947 w FB(\(206\))p Black -150 1478 a(with)26 b(some)f(curren)n(t)g(solution)g(order)f Fr([)p Fv(N)9 b Fr(])p FB(.)36 b(F)-7 b(or)25 b(con)n(v)n(enience,)g(w) n(e)g(will)h(giv)n(e)e(the)i(\034rst)f(four)h(orders)e(of)h Fv(e)3165 1490 y Fu(t)3208 1478 y Fq(\000)14 b Fv(e)3326 1490 y Fu(t)p Fo(0)3414 1478 y FB(in)26 b(terms)f(of)h Fv(n)14 b Fq(\000)g Fv(n)4024 1490 y Fo(0)4061 1478 y FB(.)-150 1573 y(With)28 b(the)g(de\034nitions)1387 1740 y SDict begin H.S end 1387 1740 a 1387 1740 a SDict begin 11.5 H.A end 1387 1740 a 1387 1740 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.207) cvn /DEST pdfmark end 1387 1740 a Fv(f)1428 1752 y Fo(1)1514 1740 y Fr(:=)50 b Fq(\000)p Fr(121)p Fv(e)1882 1706 y Fo(4)1882 1761 y Fk(t0)1961 1740 y Fq(\000)18 b Fr(183)p Fv(e)2209 1706 y Fo(2)2209 1761 y Fk(t0)2289 1740 y Fr(+)g(304)c Fv(;)p Black 1360 w FB(\(207\))p Black 1387 1860 a Fv(f)1428 1872 y Fo(2)1514 1860 y Fr(:=)50 b(37)p Fv(e)1775 1826 y Fo(4)1775 1881 y Fk(t0)1855 1860 y Fr(+)18 b(292)p Fv(e)2103 1826 y Fo(2)2103 1881 y Fk(t0)2183 1860 y Fr(+)g(96)c Fv(;)p Black 1508 w FB(\(208\))p Black -150 2027 a(w)n(e)27 b(obtain)419 2221 y SDict begin H.S end 419 2221 a 419 2221 a SDict begin 11.5 H.A end 419 2221 a 419 2221 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.209) cvn /DEST pdfmark end 419 2221 a Fv(\024)467 2186 y Fo([4])592 2221 y Fr(=)49 b Fq(\000)14 b Fr(\()p Fv(n)k Fq(\000)g Fv(n)1018 2233 y Fo(0)1055 2221 y Fr(\))1139 2164 y Fv(f)1180 2176 y Fo(1)1217 2164 y Fv(e)1256 2176 y Fk(t0)p 1125 2201 207 4 v 1125 2278 a Fr(3)p Fv(f)1208 2290 y Fo(2)1245 2278 y Fv(n)1295 2290 y Fo(0)706 2459 y Fq(\000)q Fr(~)-43 b Fv(\017)13 b Fr(\()q Fv(n)18 b Fq(\000)g Fv(n)1052 2471 y Fo(0)1089 2459 y Fr(\))1121 2417 y Fo(2)1182 2397 y Fv(f)1223 2409 y Fo(1)1260 2397 y Fv(e)1299 2409 y Fk(t0)1376 2330 y Fn(\000)1414 2397 y Fr(370)p Fv(e)1579 2367 y Fo(8)1579 2418 y Fk(t0)1659 2397 y Fr(+)g(34401)p Fv(e)1991 2367 y Fo(6)1991 2418 y Fk(t0)2070 2397 y Fq(\000)g Fr(131844)p Fv(e)2444 2367 y Fo(4)2444 2418 y Fk(t0)2522 2397 y Fq(\000)g Fr(26720)p Fv(e)2854 2367 y Fo(2)2854 2418 y Fk(t0)2933 2397 y Fq(\000)g Fr(56832)3226 2330 y Fn(\001)p 1182 2440 2080 4 v 2093 2516 a Fr(18)p Fv(f)2227 2487 y Fo(3)2218 2538 y(2)2263 2516 y Fv(n)2313 2487 y Fo(2)2313 2538 y(0)706 2716 y Fq(\000)q Fr(~)-43 b Fv(\017)805 2682 y Fo(2)869 2716 y Fr(\()q Fv(n)18 b Fq(\000)g Fv(n)1103 2728 y Fo(0)1140 2716 y Fr(\))1172 2675 y Fo(3)1223 2575 y Fn(\()1290 2716 y Fv(f)1331 2728 y Fo(1)1368 2716 y Fv(e)1407 2728 y Fk(t0)1470 2599 y Fn(\022)1121 2925 y Fr(410700)p Fv(e)1412 2891 y Fo(16)1412 2946 y Fk(t0)1498 2925 y Fr(+)g(76370220)p Fv(e)1956 2891 y Fo(14)1956 2946 y Fk(t0)1121 3046 y Fr(+)c(\(3257592723)g Fq(\000)k Fr(48470)p Fv(f)2000 3058 y Fo(2)2034 3046 y Fr(\))c Fv(e)2119 3011 y Fo(12)2119 3066 y Fk(t0)2208 3046 y Fq(\000)k Fr(3)c(\()o(4124011)p Fv(f)2713 3058 y Fo(2)2765 3046 y Fr(+)k(9090903688\))9 b Fv(e)3348 3011 y Fo(10)3348 3066 y Fk(t0)1121 3166 y Fr(+168)14 b(\()n(658637)p Fv(f)1649 3178 y Fo(2)1702 3166 y Fr(+)k(276828486\))10 b Fv(e)2244 3132 y Fo(8)2244 3186 y Fk(t0)2325 3166 y Fr(+)18 b(\(9406764288)c Fq(\000)k Fr(76265528)p Fv(f)3334 3178 y Fo(2)3367 3166 y Fr(\))c Fv(e)3452 3132 y Fo(6)3452 3186 y Fk(t0)1121 3286 y Fr(+768)g(\()n(131251)p Fv(f)1649 3298 y Fo(2)1702 3286 y Fr(+)k(61327636\))10 b Fv(e)2202 3252 y Fo(4)2202 3307 y Fk(t0)2283 3286 y Fr(+)18 b(18432)c(\()n(267)p Fv(f)2787 3298 y Fo(2)2841 3286 y Fr(+)k(494320\))11 b Fv(e)3258 3252 y Fo(2)3258 3307 y Fk(t0)1121 3486 y Fq(\000)p Fr(7274496)j(\()m(7)p Fv(f)1606 3498 y Fo(2)1660 3486 y Fq(\000)k Fr(1332\))1942 3369 y Fn(\023)2003 3344 y(\))706 3711 y Fq(\002)785 3644 y Fn(\000)823 3711 y Fr(162)p Fv(f)999 3677 y Fo(5)990 3732 y(2)1034 3711 y Fv(n)1084 3677 y Fo(3)1084 3732 y(0)1121 3644 y Fn(\001)1159 3660 y FA(\000)p Fo(1)706 3919 y Fq(\000)q Fr(~)-43 b Fv(\017)805 3885 y Fo(3)856 3919 y Fr(\()p Fv(n)18 b Fq(\000)g Fv(n)1089 3931 y Fo(0)1126 3919 y Fr(\))1159 3877 y Fo(4)1210 3777 y Fn(\()1277 3919 y Fv(f)1318 3931 y Fo(1)1354 3919 y Fv(e)1393 3931 y Fk(t0)1456 3802 y Fn(\022)706 4128 y Fr(+759795000)p Fv(e)1188 4093 y Fo(24)1188 4148 y Fk(t0)1121 4248 y Fr(+211927360500)p Fv(e)1729 4214 y Fo(22)1729 4268 y Fk(t0)1121 4368 y Fq(\000)p Fr(1850)c(\()n(96940)p Fv(f)1649 4380 y Fo(2)1702 4368 y Fq(\000)k Fr(10211818689\))9 b Fv(e)2327 4334 y Fo(20)2327 4389 y Fk(t0)1121 4489 y Fr(+)14 b(\(45946807490281)o(5)e Fq(\000)18 b Fr(62450686800)p Fv(f)2459 4501 y Fo(2)2491 4489 y Fr(\))c Fv(e)2576 4454 y Fo(18)2576 4509 y Fk(t0)1121 4609 y Fr(+90)1284 4542 y Fn(\000)1321 4609 y Fr(135716)p Fv(f)1623 4575 y Fo(2)1614 4630 y(2)1675 4609 y Fq(\000)k Fr(42030994737)p Fv(f)2261 4621 y Fo(2)2311 4609 y Fq(\000)g Fr(75142160154162)2981 4542 y Fn(\001)3027 4609 y Fv(e)3066 4575 y Fo(16)3066 4630 y Fk(t0)1121 4743 y Fr(+8)1242 4676 y Fn(\000)1279 4743 y Fr(755339319)p Fv(f)1707 4709 y Fo(2)1698 4764 y(2)1758 4743 y Fr(+)g(6763450991635)p Fv(f)2427 4755 y Fo(2)2477 4743 y Fr(+)g(31923339774273)o(90)3230 4676 y Fn(\001)3276 4743 y Fv(e)3315 4709 y Fo(14)3315 4764 y Fk(t0)1121 4877 y Fq(\000)p Fr(3)1242 4810 y Fn(\000)1279 4877 y Fr(31474158721)p Fv(f)1791 4843 y Fo(2)1782 4898 y(2)1841 4877 y Fr(+)g(56138837764240)o Fv(f)2552 4889 y Fo(2)2602 4877 y Fr(+)g(8745121825435)o(20)o(0)3355 4810 y Fn(\001)3401 4877 y Fv(e)3440 4843 y Fo(12)3440 4898 y Fk(t0)1121 5011 y Fr(+24)1284 4944 y Fn(\000)1321 5011 y Fr(7251390883)p Fv(f)1791 4977 y Fo(2)1782 5032 y(2)1841 5011 y Fr(+)g(4696313592840)p Fv(f)2510 5023 y Fo(2)2560 5011 y Fr(+)g(14389805163168)o(0)3272 4944 y Fn(\001)3318 5011 y Fv(e)3357 4977 y Fo(10)3357 5032 y Fk(t0)1121 5145 y Fq(\000)p Fr(8)1242 5078 y Fn(\000)1279 5145 y Fr(35529727041)p Fv(f)1791 5111 y Fo(2)1782 5166 y(2)1841 5145 y Fr(+)g(21737777348800)o Fv(f)2552 5157 y Fo(2)2602 5145 y Fr(+)g(5492015496046)o(08)o(0)3355 5078 y Fn(\001)3401 5145 y Fv(e)3440 5111 y Fo(8)3440 5166 y Fk(t0)1121 5279 y Fr(+384)1326 5212 y Fn(\000)1362 5279 y Fr(94956913)p Fv(f)1748 5245 y Fo(2)1739 5300 y(2)1800 5279 y Fq(\000)g Fr(19783158080)p Fv(f)2386 5291 y Fo(2)2436 5279 y Fq(\000)g Fr(3464902782848)o(0)3106 5212 y Fn(\001)3152 5279 y Fv(e)3191 5245 y Fo(6)3191 5300 y Fk(t0)1121 5413 y Fr(+30720)1410 5346 y Fn(\000)1445 5413 y Fr(2707169)p Fv(f)1789 5379 y Fo(2)1780 5434 y(2)1841 5413 y Fr(+)g(277826496)p Fv(f)2343 5425 y Fo(2)2394 5413 y Fq(\000)g Fr(683225807616)2981 5346 y Fn(\001)3027 5413 y Fv(e)3066 5379 y Fo(4)3066 5434 y Fk(t0)p Black Black eop end end %%Page: 19 19 TeXDict begin HPSdict begin 19 18 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.19) cvn /DEST pdfmark end -158 -515 a Black 4001 -299 a FB(19)p Black 1121 -83 a Fr(+18186240)1536 -150 y Fn(\000)1570 -83 y Fr(2319)p Fv(f)1788 -117 y Fo(2)1779 -62 y(2)1841 -83 y Fr(+)18 b(594368)p Fv(f)2217 -71 y Fo(2)2270 -83 y Fq(\000)g Fr(213546240)2731 -150 y Fn(\001)2778 -83 y Fv(e)2817 -117 y Fo(2)2817 -62 y Fk(t0)1121 125 y Fq(\000)p Fr(1745879040)1620 57 y Fn(\000)1653 125 y Fr(35)p Fv(f)1787 90 y Fo(2)1778 145 y(2)1841 125 y Fq(\000)g Fr(16576)p Fv(f)2175 137 y Fo(2)2228 125 y Fr(+)g(1577088)2605 57 y Fn(\001)2640 7 y(\023)2701 -17 y(\))706 350 y Fq(\002)785 283 y Fn(\000)823 350 y Fr(1944)p Fv(f)1041 316 y Fo(7)1032 371 y(2)1076 350 y Fv(n)1126 316 y Fo(4)1126 371 y(0)1163 283 y Fn(\001)1201 299 y FA(\000)p Fo(1)1308 350 y Fr(+)g Fq(O)r Fr(\()q(~)-43 b Fv(\017)1525 316 y Fo(4)1577 350 y Fr(\()p Fv(n)18 b Fq(\000)g Fv(n)1810 362 y Fo(0)1847 350 y Fr(\))1879 316 y Fo(5)1917 350 y Fr(\))c Fv(:)p Black 1909 w FB(\(209\))p Black -67 488 a(It)32 b(is)g(up)g(to)f(the)h (reader)e(to)i(truncate)f(this)h(to)g(some)f(required)f(order)h(in)h Fv(e)2366 500 y Fu(t)p Fo(0)2459 488 y FB(or)f(to)h(extend)g(it)g(in)g (orders)e(of)i Fr(~)-43 b Fv(\017)p FB(.)49 b(Ha)n(ving)31 b(found)-150 584 y(the)e(\020stationary\021)34 b Fv(n)503 554 y FA(\003)503 607 y Fo(1pN)655 584 y FB(from)29 b(Eq.)41 b(\()1049 584 y SDict begin H.S end 1049 584 a Black FB(199)p Black 1174 531 a SDict begin H.R end 1174 531 a 1174 584 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.198) cvn H.B /ANN pdfmark end 1174 584 a FB(\))29 b(and)g(from)g Fv(e)1635 554 y FA(\003)1635 604 y Fu(t)1673 584 y Fr(\()p Fv(n)1755 554 y FA(\003)1755 607 y Fo(1pN)1878 584 y Fr(\))p FB(,)g(one)g(can)g(obtain)f(the)i(asso)r(ciated)d Fv(t)3100 554 y FA(\003)3168 584 y FB(b)n(y)h(inserting)h(this)g(in)n (to)g(Eq.)-150 689 y(\()-118 689 y SDict begin H.S end -118 689 a Black FB(195)p Black 7 635 a SDict begin H.R end 7 635 a 7 689 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.195) cvn H.B /ANN pdfmark end 7 689 a FB(\).)36 b(This)25 b(in)g(turn)g(can)f(b)r(e)i(inserted)e(in)n(to)h(Eq.)36 b(\()1489 689 y SDict begin H.S end 1489 689 a Black FB(181)p Black 1614 635 a SDict begin H.R end 1614 635 a 1614 689 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.181) cvn H.B /ANN pdfmark end 1614 689 a FB(\))25 b(to)g(get)f(the)i(v)-5 b(alue)24 b(of)h(the)g(phase)g(at)f(the)h(stationary)f(p)r(oin)n(t)h Fv(t)3593 659 y FA(\003)3631 689 y FB(.)36 b(The)25 b(reader)-150 784 y(should)h(also)f(note)i(that)f(the)h(solution)f(to)g(Eq.)37 b(\()1386 784 y SDict begin H.S end 1386 784 a Black FB(186)p Black 1511 731 a SDict begin H.R end 1511 731 a 1511 784 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.186) cvn H.B /ANN pdfmark end 1511 784 a FB(\))27 b(will)f(in)n(tro)r (duce)g(new)g(1pN)h(correction)d(terms)j(to)f(the)h(m)n(ultip)r(ole)f (momen)n(ts)g(of)-150 880 y(Eqs.)37 b(\()75 880 y SDict begin H.S end 75 880 a Black FB(11a)p Black 200 826 a SDict begin H.R end 200 826 a 200 880 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.11a) cvn H.B /ANN pdfmark end 200 880 a FB(\))28 b(-)f(\()347 880 y SDict begin H.S end 347 880 a Black FB(11c)p Black 467 826 a SDict begin H.R end 467 826 a 467 880 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.11c) cvn H.B /ANN pdfmark end 467 880 a FB(\))h(.)148 1149 y SDict begin H.S end 148 1149 a 148 1149 a SDict begin 10.5 H.A end 148 1149 a 148 1149 a SDict begin [/View [/XYZ H.V]/Dest (section*.10) cvn /DEST pdfmark end 148 1149 a Fs(D.)88 b(Solution)32 b(to)e(the)f(SP)-7 b(A)29 b(condition)j(equation,)e Fe(j)c Fc(=)21 b(0)30 b Fs(and)g Fe(m)21 b Fd(6)p Fc(=)g(0)p Fs(.)39 b(The)30 b(pure)f(p)r(eriastron)j(phase)d(shift)-67 1362 y FB(The)f(stationary)e (phase)h(condition)g(for)g(the)h(pure)f(p)r(eriastron-dep)r(enden)n(t)g (terms)g(\(those)g(with)h Fv(j)g Fr(=)23 b(0)p FB(\))k(reads)1401 1511 y SDict begin H.S end 1401 1511 a 1401 1511 a SDict begin 11.5 H.A end 1401 1511 a 1401 1511 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.210) cvn /DEST pdfmark end 1401 1511 a 1419 1490 a Fr(_)1401 1511 y(\010)61 b(=)g(2)p Fv(\031)s(f)27 b Fq(\000)18 b Fv(m)p Fp(k)p Fv(n)1522 1690 y Fr(=)61 b(2)p Fv(\031)s(f)27 b Fq(\000)18 b Fv(mn)p Fr(\()p Fv(e)2085 1656 y FA(\003)2085 1710 y Fu(t)2123 1690 y Fr(\))c(3)2233 1634 y Fv(n)p Fr(\()p Fv(e)2354 1604 y FA(\003)2354 1654 y Fu(t)2392 1634 y Fr(\))2424 1604 y Fo(2)p Fu(=)p Fo(3)p 2220 1671 322 4 v 2220 1747 a Fr(1)k Fq(\000)g Fr(\()p Fv(e)2434 1718 y FA(\003)2434 1767 y Fu(t)2472 1747 y Fr(\))2504 1723 y Fo(2)2575 1690 y Fr(=)k(0)14 b Fv(;)p Black 1154 w FB(\(210\))p Black 1179 1931 a Fq(\))24 b Fv(g)s Fr(\()p Fv(e)1400 1943 y Fu(t)1428 1931 y Fr(\))50 b(:=)g(2)p Fv(\031)s(f)27 b Fq(\000)18 b Fv(mn)p Fr(\()p Fv(e)2085 1896 y FA(\003)2085 1951 y Fu(t)2123 1931 y Fr(\))c(3)2233 1875 y Fv(n)p Fr(\()p Fv(e)2354 1844 y FA(\003)2354 1895 y Fu(t)2392 1875 y Fr(\))2424 1844 y Fo(2)p Fu(=)p Fo(3)p 2220 1912 V 2220 1988 a Fr(1)k Fq(\000)g Fr(\()p Fv(e)2434 1959 y FA(\003)2434 2008 y Fu(t)2472 1988 y Fr(\))2504 1964 y Fo(2)2575 1931 y Fr(=)k(0)14 b Fv(;)p Black 1154 w FB(\(211\))p Black -67 2122 a(with)30 b Fv(n)p Fr(\()p Fv(e)245 2134 y Fu(t)274 2122 y Fr(\))g FB(tak)n(en)f(from)g(Eq.)43 b(\()960 2122 y SDict begin H.S end 960 2122 a Black FB(194)p Black 1085 2069 a SDict begin H.R end 1085 2069 a 1085 2122 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.194) cvn H.B /ANN pdfmark end 1085 2122 a FB(\).)f(Here,)30 b(w)n(e)f(pro)r(ceed)g(presen)n(ting)f(all)i(quan)n(tities)f(expressed) f(in)i(terms)f(of)g Fv(e)3582 2134 y Fu(t)3611 2122 y FB(.)43 b(T)-7 b(o)29 b(\034nd)h(the)-150 2217 y(solution)k(to)h(this)g (equation)f(analytically)-7 b(,)35 b(w)n(e)f(\034nd)h(it)g(con)n(v)n (enien)n(t)f(to)g(consider)g(the)h(p)r(erturbation)f(algorithm)f(from)i (Dan)n(b)n(y)f(&)-150 2313 y(Burkhardt)d([)285 2313 y SDict begin H.S end 285 2313 a FB(20)369 2313 y SDict begin 11.5 H.L end 369 2313 a 369 2313 a SDict begin [/Subtype /Link/Dest (cite.Danby:Burkhardt:1983) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 369 2313 a -1 w FB(])i(to)f(the)h(fourth)f(order.)50 b(W)-7 b(e)33 b(need)g(to)f(ha)n(v)n(e)f(a)h(nice)h(initial)f(guess)g(for)g Fv(e)2735 2325 y Fu(t)2764 2313 y FB(,)h(whic)n(h)g(w)n(e)f(tak)n(e)f (from)h(the)h(\034rst-order)-150 2408 y(expansion)26 b(of)i(Eq.)37 b(\()522 2408 y SDict begin H.S end 522 2408 a Black FB(210)p Black 647 2355 a SDict begin H.R end 647 2355 a 647 2408 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.210) cvn H.B /ANN pdfmark end 647 2408 a FB(\))28 b(in)g Fv(e)843 2420 y Fu(t)890 2408 y Fq(\000)18 b Fv(e)1012 2420 y Fu(t)p Fo(0)1074 2408 y FB(,)523 2630 y SDict begin H.S end 523 2630 a 523 2630 a SDict begin 11.5 H.A end 523 2630 a 523 2630 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.212) cvn /DEST pdfmark end 523 2630 a Fv(e)562 2587 y Fo([0])562 2651 y Fu(t)686 2630 y Fr(=)909 2520 y Fu(\031)r(f)p 820 2538 257 4 v 820 2589 a Fo(3)p Fu(mn)953 2597 y Fl(0)986 2573 y(5)p Fm(=)p Fl(3)1101 2490 y Fn(\000)1139 2557 y Fr(242)p Fv(e)1304 2527 y Fo(7)1304 2578 y Fu(t)p Fo(0)1383 2557 y Fr(+)g(124)p Fv(e)1631 2527 y Fo(5)1631 2578 y Fu(t)p Fo(0)1710 2557 y Fq(\000)g Fr(974)p Fv(e)1958 2527 y Fo(3)1958 2578 y Fu(t)p Fo(0)2037 2557 y Fr(+)g(608)p Fv(e)2285 2569 y Fu(t)p Fo(0)2345 2490 y Fn(\001)2402 2557 y Fr(+)g(178)p Fv(e)2650 2527 y Fo(5)2650 2578 y Fu(t)p Fo(0)2729 2557 y Fq(\000)g Fr(669)p Fv(e)2977 2527 y Fo(3)2977 2578 y Fu(t)p Fo(0)3056 2557 y Fq(\000)g Fr(784)p Fv(e)3304 2569 y Fu(t)p Fo(0)p 810 2611 2554 4 v 1659 2687 a Fr(3)c(\(19)p Fv(e)1870 2658 y Fo(4)1870 2709 y Fu(t)p Fo(0)1949 2687 y Fq(\000)k Fr(284)p Fv(e)2197 2658 y Fo(2)2197 2709 y Fu(t)p Fo(0)2276 2687 y Fq(\000)g Fr(160\))3388 2630 y Fv(;)p Black 484 w FB(\(212\))p Black -150 2818 a(noting)28 b(that)h(the)h(case)d Fv(m)e Fr(=)g(0)j FB(is)h(excluded.)40 b(This)29 b(can)f(b)r(e)i(inserted)e (in)n(to)h(an)f(iterativ)n(e)g(solution)g(algorithm,)g(whic)n(h)h(solv) n(es)e(for)h Fv(\016)-150 2913 y FB(in)g(the)g(expression)1836 3067 y SDict begin H.S end 1836 3067 a 1836 3067 a SDict begin 11.5 H.A end 1836 3067 a 1836 3067 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.213) cvn /DEST pdfmark end 1836 3067 a Fv(e)1875 3033 y FA(\003)1875 3088 y Fu(t)1963 3067 y Fr(=)49 b Fv(e)2116 3024 y Fo([0])2116 3088 y Fu(t)2209 3067 y Fr(+)18 b Fv(\016)f(;)p Black 1526 w FB(\(213\))p Black 1551 3208 a Fv(g)s Fr(\()p Fv(e)1665 3165 y Fo([0])1665 3229 y Fu(t)1758 3208 y Fr(+)h Fv(\016)s Fr(\))50 b(=)f(0)14 b Fv(:)p Black 1739 w FB(\(214\))p Black -150 3346 a(This)28 b Fv(\016)i FB(is)e(found)g(with)g(the)g (help)f(of)h(the)g(follo)n(wing)e(pro)r(cedure,)1207 3506 y SDict begin H.S end 1207 3506 a 1207 3506 a SDict begin 11.5 H.A end 1207 3506 a 1207 3506 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.215) cvn /DEST pdfmark end 1207 3506 a Fv(\016)1244 3518 y Fo(1)1342 3506 y Fr(=)61 b Fq(\000)1554 3449 y Fv(g)p 1543 3486 66 4 v 1543 3562 a(g)1586 3538 y FA(0)1632 3506 y Fv(;)p Black 2240 w FB(\(215\))p Black 1207 3704 a Fv(\016)1244 3716 y Fo(2)1342 3704 y Fr(=)g Fq(\000)1718 3648 y Fv(g)p 1543 3685 394 4 v 1543 3765 a(g)1586 3741 y FA(0)1627 3765 y Fr(+)1720 3732 y Fo(1)p 1720 3746 34 4 v 1720 3793 a(2)1763 3765 y Fv(\016)1800 3777 y Fo(1)1851 3765 y Fv(g)1894 3741 y FA(00)1960 3704 y Fv(;)p Black 1912 w FB(\(216\))p Black 1207 3918 a Fv(\016)1244 3930 y Fo(3)1342 3918 y Fr(=)g Fq(\000)1893 3862 y Fv(g)p 1543 3899 744 4 v 1543 3979 a(g)1586 3955 y FA(0)1627 3979 y Fr(+)1720 3947 y Fo(1)p 1720 3961 34 4 v 1720 4008 a(2)1763 3979 y Fv(\016)1800 3991 y Fo(2)1851 3979 y Fv(g)1894 3955 y FA(00)1954 3979 y Fr(+)2047 3947 y Fo(1)p 2047 3961 V 2047 4008 a(6)2090 3979 y Fv(\016)2130 3951 y Fo(2)2127 4002 y(2)2181 3979 y Fv(g)2224 3955 y FA(000)2309 3918 y Fv(;)p Black 1563 w FB(\(217\))p Black 1207 4133 a Fv(\016)1244 4145 y Fo(4)1342 4133 y Fr(=)g Fq(\000)2087 4077 y Fv(g)p 1543 4114 1132 4 v 1543 4194 a(g)1586 4170 y FA(0)1627 4194 y Fr(+)1720 4162 y Fo(1)p 1720 4176 34 4 v 1720 4223 a(2)1763 4194 y Fv(\016)1800 4206 y Fo(3)1851 4194 y Fv(g)1894 4170 y FA(00)1954 4194 y Fr(+)2047 4162 y Fo(1)p 2047 4176 V 2047 4223 a(6)2090 4194 y Fv(\016)2130 4166 y Fo(2)2127 4216 y(3)2181 4194 y Fv(g)2224 4170 y FA(000)2304 4194 y Fr(+)2414 4162 y Fo(1)p 2397 4176 67 4 v 2397 4223 a(24)2473 4194 y Fv(\016)2513 4166 y Fo(3)2510 4216 y(3)2550 4194 y Fv(g)2593 4170 y FA(0000)2698 4133 y Fv(;)p Black 1174 w FB(\(218\))p Black 1200 4370 a Fv(g)1243 4336 y Fu(p)1330 4370 y Fr(:=)1497 4314 y Fv(@)1546 4284 y Fu(p)p 1478 4351 126 4 v 1478 4427 a Fv(@)5 b(e)1566 4387 y Fu(p)1566 4447 y(t)1614 4370 y Fv(g)s Fr(\()p Fv(e)1728 4382 y Fu(t)1756 4370 y Fr(\))14 b Fv(:)p Black 2070 w FB(\(219\))p Black -150 4585 a(W)-7 b(e)27 b(ha)n(v)n(e)f Fv(e)222 4542 y Fo([4])222 4606 y Fu(t)320 4585 y Fr(=)c Fv(e)446 4542 y Fo([0])446 4606 y Fu(t)538 4585 y Fr(+)17 b Fv(\016)657 4597 y Fo(4)721 4585 y FB(as)26 b(the)h(fourth-order)e(solution)i(to)f(Eq.)38 b(\()2039 4585 y SDict begin H.S end 2039 4585 a Black FB(210)p Black 2163 4532 a SDict begin H.R end 2163 4532 a 2163 4585 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.210) cvn H.B /ANN pdfmark end 2163 4585 a FB(\))28 b(with)f(quin)n (tic)g(con)n(v)n(ergence,)e(and)h(again)g(extract)g Fv(n)p Fr(\()p Fv(e)3954 4542 y Fo([4])3954 4606 y Fu(t)4029 4585 y Fr(\))p FB(,)-150 4681 y Fv(t)18 b Fq(\000)g Fv(t)11 4693 y Fo(0)76 4681 y FB(and)28 b(so)f(on.)36 b(The)28 b(case)f Fv(m)22 b Fr(=)h(0)k FB(will)h(b)r(e)g(discussed)f(b)r(elo)n (w.)470 4950 y SDict begin H.S end 470 4950 a 470 4950 a SDict begin 10.5 H.A end 470 4950 a 470 4950 a SDict begin [/View [/XYZ H.V]/Dest (section*.11) cvn /DEST pdfmark end 470 4950 a Fs(E.)88 b(The)30 b(case)f Fe(j)d Fc(=)21 b(0)30 b Fs(and)f Fe(m)21 b Fc(=)g(0)p Fs(.)40 b(F)-7 b(ourier)30 b(transformation)h(of)f(a)g(slo)n(w-in-time)h(signal)-67 5163 y FB(In)d(Eq.)37 b(\()229 5163 y SDict begin H.S end 229 5163 a Black FB(177)p Black 354 5110 a SDict begin H.R end 354 5110 a 354 5163 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.177) cvn H.B /ANN pdfmark end 354 5163 a FB(\),)28 b(there)f(is)h(no)f(fast)g(oscillating)g(term)g(but)i(only) e(a)g(slo)n(w)f(v)-5 b(ariable)27 b(of)g(time)i(to)e(b)r(e)h(F)-7 b(ourier)26 b(transformed,)1424 5364 y SDict begin H.S end 1424 5364 a 1424 5364 a SDict begin 11.5 H.A end 1424 5364 a 1424 5364 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.220) cvn /DEST pdfmark end 1424 5364 a 1425 5342 a Fr(~)1424 5364 y Fv(h)p Fr(\()p Fv(f)9 b Fr(\))23 b(=)1696 5251 y Fn(Z)1779 5271 y Fo(+)p FA(1)1743 5439 y(\0001)1915 5364 y Fq(A)p Fr(\()p Fv(t)p Fr(\))14 b Fv(e)2128 5329 y Fu(i)p Fo(\()p FA(\000)p Fo(2)p Fu(\031)r(f)7 b(t)p Fo(\))2397 5364 y Fr(d)p Fv(t)14 b(:)1385 b FB(\(220\))p Black Black eop end end %%Page: 20 20 TeXDict begin HPSdict begin 20 19 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.20) cvn /DEST pdfmark end -158 -515 a Black 4001 -299 a FB(20)p Black -150 -83 a(The)34 b(term)g Fq(A)p Fr(\()p Fv(t)p Fr(\))i FB(dep)r(ends)e(on)g(time)h(only)f(due)g(to)g(RR.)h(Those)e(terms)h (are)f(non)n(trivially)g(dep)r(enden)n(t)i(on)f(time)h(and)f(ha)n(v)n (e)f(to)h(b)r(e)-150 12 y(treated)d(individually)h(when)g(they)g(are)f (requested)h(analytically)-7 b(.)48 b(In)32 b(principle,)h(one)f(w)n (ould)f(ha)n(v)n(e)g(to)g(express)g Fq(j)p Fv(E)5 b Fq(j)p FB(,)33 b Fv(n)f FB(and)g Fv(e)3949 24 y Fu(t)4010 12 y FB(as)-150 108 y(explicit)c(functions)g(of)f(time.)38 b(That)27 b(w)n(ould)g(include)h(in)n(v)n(ersion)e(of)h(App)r(ell)i (functions)f(or)e(p)r(erturbation)h(theory)-7 b(.)-67 203 y(Ho)n(w)n(ev)n(er,)22 b(for)g(the)h(case)f(of)g(inspiralling)g (compact)g(binaries,)h(they)g(will)g(not)f(b)r(e)h(able)g(to)f (signi\034can)n(tly)g(con)n(tribute)g(to)h(frequencies)-150 299 y(in)34 b(comparison)e(to)h(those)g(with)h(fast)g(oscillating)e (exp)r(onen)n(ts)h(as)g(w)n(e)g(compare)g(t)n(ypical)g(time)h(scales)e (for)h(one)g(orbit)g(and)h(for)f(the)-150 394 y(inspiral.)j(W)-7 b(e)28 b(will)g(therefore)e(imp)r(ose)i(the)g(follo)n(wing)e(relation.) 1491 570 y SDict begin H.S end 1491 570 a 1491 570 a SDict begin 11.5 H.A end 1491 570 a 1491 570 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.221) cvn /DEST pdfmark end 1491 570 a 1492 548 a Fr(~)1491 570 y Fv(h)p Fr(\()p Fv(f)9 b Fr(\))1653 582 y Fo(static)1840 570 y Fq(\034)1947 548 y Fr(~)1946 570 y Fv(h)o Fr(\()p Fv(f)g Fr(\))2107 582 y Fo(stationary)2420 570 y Fv(;)1452 b FB(\(221\))-150 760 y(where)94 738 y Fr(~)93 760 y Fv(h)141 772 y Fo(static)335 760 y FB(means)31 b(all)f(F)-7 b(ourier)29 b(in)n(tegrals)h(o)n(v)n(er) e(terms)j(where)f(the)h(mean)f(anomaly)g(\025)g(or)f(equiv)-5 b(alen)n(tly)e(,)32 b(the)f(time)g(\025)f(do)r(es)g(not)-150 855 y(app)r(ear)d(explicitly)-7 b(.)37 b(W)-7 b(e)28 b(state)f(that)h(the)g Fr(\()p Fv(j)g Fr(=)23 b(0)p Fv(;)14 b(m)22 b Fr(=)h(0\))k FB(F)-7 b(ourier)27 b(domain)g(terms)g(almost)g (v)-5 b(anish:)1632 1042 y SDict begin H.S end 1632 1042 a 1632 1042 a SDict begin 11.5 H.A end 1632 1042 a 1632 1042 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.222) cvn /DEST pdfmark end 1632 1042 a 1633 1020 a Fr(~)1632 1042 y Fv(h)p Fr(\()p Fv(f)9 b Fr(\))1794 1057 y Fo([)p Fu(j)s Fo(=0)p Fu(;m)p Fo(=0])2136 1042 y Fq(\031)23 b Fr(0)14 b Fv(;)1592 b FB(\(222\))-150 1218 y(for)21 b(the)h(frequency)f(domain)g(of)g(in)n(terest)g(for)g(the)h(regarded)e (detector.)34 b(LISA)22 b(for)f(example,)h(will)g(hardly)f(see)g(those) g(terms)g(op)r(erating)-150 1313 y(near)33 b Fv(n)91 1325 y Fo(0)163 1313 y Fq(\031)h Fr(0)p Fv(:)p Fr(001Hz)n FB(.)57 b(Let)35 b(us)f(giv)n(e)g(an)g(exemplary)f(n)n(um)n(b)r(er)h (to)g(supp)r(ort)g(this)h(statemen)n(t.)57 b(The)35 b(rate)e(of)h(c)n (hange)g(of)g(the)h(GW)-150 1409 y(frequency)27 b Fv(f)36 b FB(o)n(v)n(er)26 b(one)h(y)n(ear)f(will)i(b)r(e)g([)1106 1409 y SDict begin H.S end 1106 1409 a FB(7)1148 1409 y SDict begin 11.5 H.L end 1148 1409 a 1148 1409 a SDict begin [/Subtype /Link/Dest (cite.Tessmer:Gopakumar:2006) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1148 1409 a FB(])539 1667 y SDict begin H.S end 539 1667 a 539 1667 a SDict begin 11.5 H.A end 539 1667 a 539 1667 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.223) cvn /DEST pdfmark end 539 1667 a Fr(\001)p Fv(f)649 1679 y Fo(RR)772 1667 y Fq(\030)874 1611 y Fr(1)p Fv(:)p Fr(6)18 b Fq(\002)g Fr(10)1166 1581 y FA(\000)p Fo(9)p 870 1648 388 4 v 870 1726 a Fr(\(1)g Fq(\000)g Fv(e)1084 1697 y Fo(2)1084 1746 y Fu(t)1121 1726 y Fr(\))1153 1702 y Fo(7)p Fu(=)p Fo(2)1282 1550 y Fn(\022)1423 1611 y Fv(m)1496 1623 y Fu(t)p 1353 1648 243 4 v 1353 1724 a Fr(2)p Fv(:)p Fr(8)p Fv(M)1541 1736 y FA(\014)1605 1550 y Fn(\023)1676 1545 y Fl(5)p 1676 1554 29 3 v 1676 1587 a(3)1733 1575 y Fn(\020)1844 1611 y Fv(\021)p 1792 1648 148 4 v 1792 1724 a Fr(0)p Fv(:)p Fr(25)1950 1575 y Fn(\021)2013 1550 y(\022)2181 1611 y Fv(f)2222 1623 y Fu(r)p 2084 1648 272 4 v 2084 1724 a Fr(10)2168 1700 y FA(\000)p Fo(3)2256 1724 y Fr(Hz)2365 1550 y Fn(\023)2436 1545 y Fl(11)p 2436 1554 57 3 v 2451 1587 a(3)2521 1550 y Fn(\022)2582 1667 y Fr(1)g(+)2735 1611 y(73)p 2735 1648 84 4 v 2735 1724 a(24)2828 1667 y Fv(e)2867 1633 y Fo(2)2867 1688 y Fu(t)2922 1667 y Fr(+)3015 1611 y(37)p 3015 1648 V 3015 1724 a(96)3108 1667 y Fv(e)3147 1633 y Fo(4)3147 1688 y Fu(t)3184 1550 y Fn(\023)3259 1667 y Fr(Hz)c Fv(;)500 b FB(\(223\))-150 1903 y(where)33 b Fv(f)137 1915 y Fu(r)206 1903 y FB(is)g(the)h(radial)e(frequency)-7 b(,)34 b(giv)n(en)f(b)n(y)f Fv(f)1468 1915 y Fu(r)1537 1903 y Fr(=)g Fv(n)p Fr(\(2)p Fv(\031)s Fr(\))1840 1873 y FA(\000)p Fo(1)1930 1903 y FB(.)53 b(Let)34 b Fv(n)e Fr(=)g(10)2424 1873 y FA(\000)p Fo(3)2512 1903 y FB(,)j Fv(m)2643 1915 y Fo(1)2712 1903 y Fr(=)d Fv(m)2882 1915 y Fo(2)2952 1903 y Fr(=)g(1)p Fv(:)p Fr(4)p Fv(M)3237 1915 y FA(\014)3325 1903 y FB(and)h Fv(e)3531 1915 y Fu(t)3592 1903 y Fr(=)f(0)p Fv(:)p Fr(1)p FB(.)53 b(Then,)-150 1998 y Fr(\001)p Fv(f)-40 2010 y Fo(RR)83 1998 y Fq(\030)23 b Fr(2)p Fq(\002)o Fr(10)361 1968 y FA(\000)p Fo(12)482 1998 y Fr(Hz)28 b FB(and)g(the)g(scaled)e (energy)h(loss)g(is)g Fr(\001)p Fv(E)1795 2010 y Fo(RR)1919 1998 y Fq(\030)c Fr(3)18 b Fq(\002)g Fr(10)2234 1968 y FA(\000)p Fo(13)2355 1998 y FB(.)1114 2274 y SDict begin H.S end 1114 2274 a 1114 2274 a SDict begin 10.5 H.A end 1114 2274 a 1114 2274 a SDict begin [/View [/XYZ H.V]/Dest (section*.12) cvn /DEST pdfmark end 1114 2274 a Fs(F.)87 b(The)30 b(limit)h Fe(e)1710 2282 y Fb(t)1759 2274 y Fd(!)21 b Fc(0)p Fs(:)39 b(the)30 b(quasi-circular)j(case)-67 2487 y FB(The)c(quasi-circular)e(inspiral)h(has)h(b)r(een)h(discussed)e (extensiv)n(ely)h(in)g(the)h(literature,)e(esp)r(ecially)h(in)g([)3185 2487 y SDict begin H.S end 3185 2487 a FB(6)3227 2487 y SDict begin 11.5 H.L end 3227 2487 a 3227 2487 a SDict begin [/Subtype /Link/Dest (cite.Yunes:Arun:Berti:Will:2009) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 3227 2487 a FB(])g(whic)n(h)g(w)n(e)g(ha)n(v)n(e)f(oftenly)-150 2583 y(cited.)36 b(F)-7 b(or)21 b(further)i(information,)g(see)f(e.g.) 35 b([)1283 2583 y SDict begin H.S end 1283 2583 a FB(21)1367 2583 y SDict begin 11.5 H.L end 1367 2583 a 1367 2583 a SDict begin [/Subtype /Link/Dest (cite.Arun:Iyer:Sathyaprakash:Sundararajan:2005) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1367 2583 a -1 w FB(,)1411 2583 y SDict begin H.S end 1411 2583 a FB(22)1495 2583 y SDict begin 11.5 H.L end 1495 2583 a 1495 2583 a SDict begin [/Subtype /Link/Dest (cite.Damour:Iyer:Sathyaprakash:2000) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1495 2583 a -1 w FB(].)h(In)22 b(the)h(limit)h Fv(e)2045 2595 y Fu(t)2097 2583 y Fq(!)f Fr(0)p FB(,)g(all)f(elemen)n(ts)h(of)f (our)g(calculation)f(simplify)i(drastically)-7 b(.)-150 2678 y(The)28 b(follo)n(wing)e(equations,)1549 2854 y SDict begin H.S end 1549 2854 a 1549 2854 a SDict begin 11.5 H.A end 1549 2854 a 1549 2854 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.224) cvn /DEST pdfmark end 1549 2854 a Fp(M)50 b Fq(!)f Fv(u)14 b(;)p Black 1988 w FB(\(224\))p Black 1465 2975 a Fv(A)p Fr(\()p Fv(u)p Fr(\))51 b Fq(!)e Fr(1)14 b Fv(;)p Black 1994 w FB(\(225\))p Black 1403 3095 a Fv(f)1444 3107 y Fu(lm)1528 3095 y Fr(\()p Fv(u)p Fr(\))50 b Fq(!)f(F)1882 3107 y Fu(lm)1966 3095 y Fr(\()p Fv(E)5 b Fr(\))14 b Fv(;)p Black 1762 w FB(\(226\))p Black 1404 3215 a Fv(g)1444 3227 y Fu(lm)1528 3215 y Fr(\()p Fv(u)p Fr(\))50 b Fq(!)f(G)1871 3227 y Fu(lm)1956 3215 y 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y(quasi-circular)c(orbits.) 36 b(The)26 b(in\034nite)i(summation)e(series)f(in)i(Eq.)37 b(\()2019 3739 y SDict begin H.S end 2019 3739 a Black FB(187)p Black 2144 3686 a SDict begin H.R end 2144 3686 a 2144 3739 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.187) cvn H.B /ANN pdfmark end 2144 3739 a FB(\))26 b(shrink)g(to)h (one)f(single)g(term,)h(where)f(the)g(phase)g(term)h(has)-150 3834 y(to)g(b)r(e)g(replaced)f(b)n(y)h(the)g(one)f(in)h(Eq.)37 b(\()1086 3834 y SDict begin H.S end 1086 3834 a Black FB(229)p Black 1211 3781 a SDict begin H.R end 1211 3781 a 1211 3834 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.224) cvn H.B /ANN pdfmark end 1211 3834 a FB(\).)g(This)27 b(is)g(b)r(ecause)f(the)h Fr(sin)14 b Fv(u)27 b FB(terms)f(will)h(alw)n (a)n(ys)e(ha)n(v)n(e)h(a)g(factor)g Fv(e)3400 3846 y Fu(t)3456 3834 y FB(and)h(v)-5 b(anish)27 b(in)g(the)-150 3930 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b(;)p Black 1704 w FB(\(A5\))p Black 1519 1289 a Fq(\))260 b Fv(m)1935 1301 y Fo(1)2022 1289 y Fr(=)49 b(2)p Fv(k)21 b Fr(+)d Fv(j)h(;)p Black 1515 w FB(\(A6\))p Black 1519 1409 a Fq(\))260 b Fv(m)1935 1421 y Fo(2)2022 1409 y Fr(=)49 b(2)p Fv(k)21 b Fq(\000)d Fv(j)h(:)p Black 1515 w FB(\(A7\))p Black -150 1583 a(A)28 b(small)f(table)h(for)f(the)h Fr(cos)e FB(function)j(argumen)n(t)d(ma)n (y)h(help)h(\(note)f(that)h Fv(k)e Fq(\024)d Fv(m)k FB(alw)n(a)n(ys)f (holds\):)p Black 1594 1762 a Fy(m)e(:)35 b(k)p 1796 1788 4 88 v 36 w(0)59 b(1)g(2)g(3)h(4)f(5)p 1577 1792 780 4 v 1741 1853 a(1)p 1796 1879 4 88 v 37 w(1)34 b(-1)1741 1940 y(2)p 1796 1966 V 37 w(2)59 b(0)34 b(-2)1741 2027 y(3)p 1796 2053 V 37 w(3)59 b(1)34 b(-1)f(-3)1741 2114 y(4)p 1796 2140 V 37 w(4)59 b(2)g(0)34 b(-2)f(-4)1741 2201 y(5)p 1796 2228 V 37 w(5)59 b(3)g(1)34 b(-1)f(-3)g(-5)p Black -150 2400 a FB(The)f(zero)f(mo)r(de)h Fv(j)j Fr(=)30 b(0)i FB(alw)n(a)n(ys)e(app)r(ears)g(at)i(ev)n(en)g(n)n(um)n(b)r(ers)f Fv(m)p FB(,)i Fv(m)d Fr(=)g(2)p Fv(i)d Fr(with)h Fv(i)i Fr(=)g(\(1)p Fv(;)14 b Fr(2)p Fv(;)g(:::)p Fr(\))p FB(,)33 b(and)f Fv(m)21 b Fq(\000)g Fr(2)p Fv(k)33 b Fr(=)d(2)p Fv(i)20 b Fq(\000)h Fr(2)p Fv(k)33 b Fr(=)d(0)h FB(is)-150 2495 y(satis\034ed)c(b)n(y)g Fv(k)f Fr(=)d Fv(i)p FB(.)36 b(Th)n(us,)28 b(w)n(e)f(c)n(ho)r(ose)1685 2735 y SDict begin H.S end 1685 2735 a 1685 2735 a SDict begin 11.5 H.A end 1685 2735 a 1685 2735 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.8) cvn /DEST pdfmark end 1685 2735 a Fv(b)1721 2692 y Fo(\()p Fu(n)p Fo(\))1721 2757 y(0)1840 2735 y Fr(=)1955 2631 y FA(1)1928 2656 y Fn(X)1934 2833 y Fu(i)p Fo(=1)2062 2735 y Fv(\014)2113 2692 y Fo(\()p Fu(n)p Fo(\))2109 2758 y(2)p Fu(i;i)2226 2735 y Fv(:)1667 b FB(\(A8\))-150 2990 y(The)26 b(other)g(frequencies)g Fv(j)i(>)22 b Fr(0)p FB(,)27 b(app)r(earing)e(at)h Fv(m)d Fr(=)f Fv(j)f Fr(+)16 b(2)p Fv(i)25 b FB(with)i Fv(i)c Fr(=)f(\(0)p Fv(;)14 b Fr(1)p Fv(;)g Fr(2)p Fv(;)g(:::)p Fr(\))26 b FB(lead)g(to)g Fv(k)g Fr(=)c Fv(i)k 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Fv(e)c Fr(cos)e Fv(u)p Fr(\))1671 3758 y Fu(n)1749 3725 y Fr(=)23 b(1)18 b(+)g Fv(b)2016 3681 y Fo(\()p Fu(n)p Fo(\))2016 3747 y(0)2131 3725 y Fr(+)2241 3621 y FA(1)2214 3646 y Fn(X)2216 3823 y Fu(j)s Fo(=1)2348 3725 y Fv(b)2384 3681 y Fo(\()p Fu(n)p Fo(\))2384 3748 y Fu(j)2494 3725 y Fr(cos)13 b Fv(j)5 b(u)1168 b FB(\(A10\))-150 4015 y(with)24 b(the)g Fv(b)210 3972 y Fo(\()p Fu(n)p Fo(\))210 4039 y Fu(j)330 4015 y FB(de\034ned)g(in)g(Eqs.)36 b(\()929 4015 y SDict begin H.S end 929 4015 a Black FB(A8)p Black 1033 3958 a SDict begin H.R end 1033 3958 a 1033 4015 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.8) cvn H.B /ANN pdfmark end 1033 4015 a FB(\))24 b(and)f(\()1278 4015 y SDict begin H.S end 1278 4015 a Black FB(A9)p Black 1382 3958 a SDict begin H.R end 1382 3958 a 1382 4015 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.9) cvn H.B /ANN pdfmark end 1382 4015 a FB(\).)36 b(Ha)n(ving)22 b(this)i(at)g(hands,)g(it)g(is)f (easy)g(to)g(compute)h(the)g(asso)r(ciated)e(decomp)r(osition)-150 4111 y(of)592 4368 y SDict begin H.S end 592 4368 a 592 4368 a SDict begin 11.5 H.A end 592 4368 a 592 4368 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.11) cvn /DEST pdfmark end 592 4368 a 766 4311 a Fr(sin)13 b Fv(u)p 602 4349 492 4 v 602 4425 a Fr(\(1)18 b Fq(\000)g Fv(e)27 b Fr(cos)13 b Fv(u)p Fr(\))1048 4401 y Fu(n)1153 4368 y Fr(=)1267 4201 y Fn(0)1267 4350 y(@)1340 4368 y Fr(1)18 b(+)g Fv(b)1519 4325 y Fo(\()p Fu(n)p Fo(\))1519 4390 y(0)1634 4368 y Fr(+)1744 4264 y FA(1)1717 4289 y Fn(X)1720 4466 y Fu(j)s Fo(=1)1851 4368 y Fv(b)1887 4325 y Fo(\()p Fu(n)p Fo(\))1887 4391 y Fu(j)2011 4368 y Fr(cos)13 b Fv(j)5 b(u)2223 4201 y Fn(1)2223 4350 y(A)2323 4368 y Fr(sin)14 b Fv(u)1153 4667 y Fr(=)49 b(\(1)18 b(+)g Fv(b)1478 4624 y Fo(\()p Fu(n)p Fo(\))1478 4689 y(0)1575 4667 y Fr(\))c(sin)g Fv(u)k Fr(+)1896 4610 y(1)p 1896 4648 42 4 v 1896 4724 a(2)1988 4563 y FA(1)1961 4588 y Fn(X)1964 4765 y Fu(j)s Fo(=1)2095 4667 y Fv(b)2131 4624 y Fo(\()p Fu(n)p Fo(\))2131 4690 y Fu(j)2242 4667 y Fr(\(sin[\(1)g Fq(\000)g Fv(j)5 b Fr(\))p Fv(u)p Fr(])18 b(+)h(sin)o([\(1)g(+)f Fv(j)5 b Fr(\))p Fv(u)p Fr(]\))28 b Fv(:)p Black 531 w FB(\(A11\))p Black -150 4928 a(This)k(decomp)r(osition)f(demands)h (collecting)g(the)g(terms)g(ha)n(ving)f(the)h(same)g(frequency)f(as)h (w)n(ell.)50 b(Ha)n(ving)31 b Fv(m)h FB(times)g(u)g(in)h(the)f Fr(sin)-150 5023 y FB(argumen)n(t,)c(follo)n(wing)g Fv(j)34 b FB(will)29 b(lead)g(to)g Fr(sin\()p Fv(mu)p Fr(\))g FB(and)g Fq(\000)14 b Fr(sin)o(\()p Fv(mu)p Fr(\))29 b FB(for)f Fr(sin\(1)20 b Fq(\000)f Fv(j)5 b Fr(\))p Fv(u)28 b FB(in)i(the)f(\034rst)g(t)n(w)n(o)f(lines)h(and)g(for)f Fr(sin\(1)19 b(+)g Fv(j)5 b Fr(\))p Fv(u)29 b FB(in)-150 5119 y(the)f(third:)1473 5293 y SDict begin H.S end 1473 5293 a 1473 5293 a SDict begin 11.5 H.A end 1473 5293 a 1473 5293 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.12) cvn /DEST pdfmark end 1473 5293 a Fv(m)50 b Fr(=)23 b(\(1)18 b Fq(\000)g Fv(j)5 b Fr(\))23 b Fq(!)g Fv(j)28 b Fr(=)23 b(1)18 b Fq(\000)g Fv(m)c(;)p Black 1412 w FB(\(A12\))p Black 1409 5413 a Fq(\000)p Fv(m)49 b Fr(=)23 b(\(1)18 b Fq(\000)g Fv(j)5 b Fr(\))23 b Fq(!)g Fv(j)28 b Fr(=)23 b(1)18 b(+)g Fv(m)c(;)p Black 1412 w FB(\(A13\))p Black Black Black eop end end %%Page: 23 23 TeXDict begin HPSdict begin 23 22 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.23) cvn /DEST pdfmark end -158 -515 a Black 4001 -299 a FB(23)p Black Black 1772 -105 a Fy(j)p 1861 -79 4 88 v 86 w(1)47 b(2)59 b(3)g(4)p 1706 -76 522 4 v 1723 -15 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a SDict begin [/View [/XYZ H.V]/Dest (equation.A.16) cvn /DEST pdfmark end 1306 1482 a 1473 1426 a Fr(sin)14 b Fv(u)p 1316 1463 478 4 v 1316 1539 a Fr(\(1)k Fq(\000)g Fv(e)c Fr(cos)f Fv(u)p Fr(\))1749 1515 y Fu(n)1865 1482 y Fr(=)2018 1378 y FA(1)1991 1403 y Fn(X)1994 1580 y Fu(j)s Fo(=1)2125 1482 y Fv(S)2181 1439 y Fo(\()p Fu(n)p Fo(\))2176 1505 y Fu(j)2291 1482 y Fr(sin)h Fv(j)5 b(u)14 b(;)p Black 1343 w FB(\(A16\))p Black 1651 1743 a Fv(S)1707 1699 y Fo(\()p Fu(n)p Fo(\))1702 1765 y(1)1854 1743 y Fr(:=)49 b(1)18 b(+)g Fv(b)2170 1699 y Fo(\()p Fu(n)p Fo(\))2170 1765 y(0)2285 1743 y Fq(\000)2378 1686 y Fr(1)p 2378 1724 42 4 v 2378 1800 a(2)2430 1743 y Fv(b)2466 1699 y Fo(\()p Fu(n)p Fo(\))2466 1765 y(2)2576 1743 y Fv(;)p Black 1275 w FB(\(A17\))p Black 1633 1942 a Fv(S)1689 1899 y Fo(\()p Fu(n)p Fo(\))1684 1966 y Fu(j)s(>)p Fo(1)1854 1942 y Fr(:=)2001 1886 y(1)p 2001 1923 V 2001 1999 a(2)2066 1850 y Fn(\020)2116 1942 y Fv(b)2152 1899 y Fo(\()p Fu(n)p Fo(\))2152 1966 y Fu(j)s Fo(+1)2289 1942 y Fq(\000)g Fv(b)2408 1899 y Fo(\()p Fu(n)p Fo(\))2408 1966 y Fu(j)s FA(\000)p Fo(1)2528 1850 y Fn(\021)2605 1942 y Fv(:)p Black 1246 w FB(\(A18\))p Black 684 2248 a SDict begin H.S end 684 2248 a 684 2248 a SDict begin 10.5 H.A end 684 2248 a 684 2248 a SDict begin [/View [/XYZ H.V]/Dest (section*.17) cvn /DEST pdfmark end 684 2248 a Fs(App)r(endix)29 b(B:)g(F)-7 b(ourier)30 b(represen)n(tation)g(of)g(the)g Fc(sin)13 b Fe(mu)29 b Fs(and)h Fc(cos)13 b Fe(mu)30 b Fs(terms)-67 2462 y FB(In)k(the)h(previous)e(app)r(endix,)j(w)n(e)e (ha)n(v)n(e)f(pro)n(v)n(en)g(some)g(form)n(ulas)g(for)h(terms)g(app)r (earing)f(in)h(the)h(m)n(ultip)r(ole)f(expansion)g(of)g(the)-150 2557 y(far-zone)25 b(gra)n(vitational)e(\034eld.)37 b(No)n(w,)26 b(w)n(e)g(ha)n(v)n(e)f(to)h(express)f(simple)h(series)g(in)g Fr(sin)14 b Fv(mu)26 b FB(and)g Fr(cos)13 b Fv(mu)25 b FB(as)h(trigonometric)e(functions)j(in)-150 2653 y Fv(ml)r FB(.)36 b(Using)28 b(Eq.)37 b(\()435 2653 y SDict begin H.S end 435 2653 a Black FB(18)p Black 518 2600 a SDict begin H.R end 518 2600 a 518 2653 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.16) cvn H.B /ANN pdfmark end 518 2653 a FB(\),)28 b(w)n(e)f(ha)n(v)n(e)462 2935 y SDict begin H.S end 462 2935 a 462 2935 a SDict begin 11.5 H.A end 462 2935 a 462 2935 a SDict begin [/View [/XYZ H.V]/Dest (equation.B.1) cvn /DEST pdfmark end 462 2935 a 690 2879 a Fr(1)p 472 2916 478 4 v 472 2992 a(\(1)18 b Fq(\000)g Fv(e)c Fr(cos)f Fv(u)p Fr(\))905 2968 y Fu(n)1021 2935 y Fr(=)61 b(1)18 b(+)g Fv(b)1326 2892 y Fo(\()p Fu(n)p Fo(\))1326 2957 y(0)1441 2935 y Fr(+)1563 2831 y FA(1)1536 2856 y Fn(X)1524 3032 y Fu(m)p Fo(=1)1681 2935 y Fv(b)1717 2901 y Fo(\()p Fu(n)p Fo(\))1717 2956 y Fu(m)1828 2935 y Fr(cos)13 b Fv(mu)22 b Fr(=)h(1)18 b(+)g Fv(b)2363 2892 y Fo(\()p Fu(n)p Fo(\))2363 2957 y(0)2478 2935 y Fr(+)2599 2831 y FA(1)2572 2856 y Fn(X)2561 3032 y Fu(m)p Fo(=1)2718 2935 y Fv(b)2754 2901 y Fo(\()p Fu(n)p 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a 1501 1265 a SDict begin [/View [/XYZ H.V]/Dest (equation.D.2) cvn /DEST pdfmark end 1501 1265 a -34 x Fo([1])1576 1265 y Fv(P)1641 1222 y Fu(C)t(S;)p Fo([)p Fu(n)p Fo(])1629 1289 y Fu(j)s(>)p Fo(1)1858 1265 y Fr(=)1949 1158 y Fu(j)s FA(\000)p Fo(1)1947 1187 y Fn(X)1946 1365 y Fu(k)q Fo(=1)2081 1265 y Fq(A)2147 1222 y Fo(\()p Fu(n)p Fo(\))2147 1290 y Fu(k)2244 1265 y Fv(G)2309 1277 y Fu(j)s FA(\000)p Fu(k)3915 1265 y FB(\(D2\))-150 1524 y(P)n(art)i(2,)i Fq(\000)14 b Fr(sin)o([)p Fv(k)23 b Fq(\000)c Fv(m)p Fr(])p Fv(l)32 b FB(will)e(also)e(ha)n(v)n (e)h(p)r(ositiv)n(e)g(and)h(negativ)n(e)f(con)n(tributions,)g(for)h Fv(m)c Fr(=)h Fv(k)c Fq(\000)c Fv(j)35 b FB(\(implying)30 b Fv(k)g(>)c(j)5 b FB(\))31 b Fv(m)26 b Fr(=)h Fv(k)c Fr(+)c Fv(j)5 b FB(,)-150 1620 y(giving)1118 1919 y SDict begin H.S end 1118 1919 a 1118 1919 a SDict begin 11.5 H.A end 1118 1919 a 1118 1919 a SDict begin [/View [/XYZ H.V]/Dest (equation.D.3) cvn /DEST pdfmark end 1118 1919 a -34 x Fo([2])1193 1919 y Fv(P)1258 1876 y Fu(C)t(S;)p Fo([)p Fu(n)p Fo(])1246 1943 y Fu(j)s(>)p Fo(1)1476 1919 y Fr(=)22 b Fq(\000)1638 1863 y Fr(1)p 1638 1900 42 4 v 1638 1976 a(2)1730 1816 y FA(1)1704 1841 y Fn(X)1703 2019 y Fu(k)q Fo(=1)1838 1728 y Fn(0)1838 1874 y(B)1838 1927 y(@)1910 1919 y Fq(A)1976 1876 y Fo(\()p Fu(n)p Fo(\))1976 1944 y Fu(k)2074 1919 y Fv(G)2139 1931 y Fu(k)q FA(\000)p Fu(j)1910 1979 y Fn(|)p 1947 1979 102 10 v 102 w({z)p 2123 1979 V 102 w(})1973 2058 y Fo(for)27 b Fu(k)q(>j)2276 1919 y Fq(\000A)2407 1876 y Fo(\()p Fu(n)p Fo(\))2407 1944 y Fu(k)2504 1919 y Fv(G)2569 1931 y Fu(k)q Fo(+)p Fu(j)2692 1728 y Fn(1)2692 1874 y(C)2692 1927 y(A)2793 1919 y Fv(:)1099 b FB(\(D3\))-150 2224 y(Summation)28 b(of)f(part)g(1)g(and)h(2)f(yields)714 2598 y SDict begin H.S end 714 2598 a 714 2598 a SDict begin 11.5 H.A end 714 2598 a 714 2598 a SDict begin [/View [/XYZ H.V]/Dest (equation.D.4) cvn /DEST pdfmark end 714 2598 a Fv(P)779 2555 y Fu(C)t(S;)p Fo([)p Fu(n)p Fo(])767 2621 y Fu(j)996 2598 y Fr(:=)1117 2542 y(1)p 1117 2579 42 4 v 1117 2655 a(2)1182 2328 y Fn(8)1182 2403 y(>)1182 2428 y(>)1182 2453 y(>)1182 2478 y(>)1182 2502 y(<)1182 2652 y(>)1182 2677 y(>)1182 2702 y(>)1182 2727 y(>)1182 2752 y(:)1259 2490 y Fu(j)s FA(\000)p Fo(1)1256 2519 y Fn(X)1256 2698 y Fu(k)q Fo(=1)1391 2506 y Fn(\020)1440 2598 y Fq(A)1506 2555 y Fo(\()p Fu(n)p Fo(\))1506 2623 y Fu(k)1604 2598 y Fv(G)1669 2610 y Fu(j)s FA(\000)p Fu(k)1792 2506 y Fn(\021)1256 2741 y(|)p 1293 2741 219 10 v 219 w({z)p 1586 2741 V 219 w(})1438 2820 y Fo(for)g Fu(j)s(>)p Fo(1)1856 2598 y Fq(\000)1962 2494 y FA(1)1935 2519 y Fn(X)1935 2698 y Fu(k)q Fo(=1)2069 2406 y Fn(0)2069 2552 y(B)2069 2605 y(@)2142 2598 y Fq(A)2208 2555 y Fo(\()p Fu(n)p Fo(\))2208 2623 y Fu(k)2305 2598 y Fv(G)2370 2610 y Fu(k)q FA(\000)p Fu(j)2142 2658 y Fn(|)p 2179 2658 102 10 v 102 w({z)p 2355 2658 V 102 w(})2204 2737 y Fo(for)h Fu(k)q(>j)2508 2598 y Fq(\000A)2639 2555 y Fo(\()p Fu(n)p Fo(\))2639 2623 y Fu(k)2736 2598 y Fv(G)2801 2610 y Fu(k)q Fo(+)p 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1363 3302 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section*.6) cvn H.B /ANN pdfmark end 1363 3302 a 29 w FB(can)f(b)r(e)h(solv)n(ed)f(with)h (the)f(help)h(of)g(the)f(follo)n(wing)g(in)n(tegral)f(represen)n (tation)f(of)j(the)-150 3397 y(App)r(ellF1)e(function)g(\(see)g(h)n (ttp://dlmf.nist.go)n(v/)e(or)h([)1628 3397 y SDict begin H.S end 1628 3397 a FB(25)1712 3397 y SDict begin 11.5 H.L end 1712 3397 a 1712 3397 a SDict begin [/Subtype /Link/Dest (cite.Olver:Lozier:Boisvert:Clark:2010) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1712 3397 a -1 w FB(])g(for)g(further)h(information\).)642 3636 y SDict begin H.S end 642 3636 a 642 3636 a SDict begin 11.5 H.A end 642 3636 a 642 3636 a SDict begin [/View [/XYZ H.V]/Dest (equation.E.1) cvn /DEST pdfmark end 642 3636 a 656 3523 a Fn(Z)739 3544 y Fo(1)702 3712 y(0)790 3636 y Fr(d)p Fv(u)954 3580 y(u)1002 3550 y Fu(\013)p FA(\000)p Fo(1)1133 3580 y Fr(\(1)19 b Fq(\000)f Fv(u)p Fr(\))1389 3550 y Fu(\015)t FA(\000)p Fu(\013)p FA(\000)p Fo(1)p 907 3617 750 4 v 907 3693 a Fr(\(1)h Fq(\000)f Fv(ux)p Fr(\))1210 3669 y Fu(\014)1248 3677 y Fl(1)1284 3693 y Fr(\(1)g Fq(\000)g Fv(uy)s Fr(\))1583 3669 y Fu(\014)1621 3677 y Fl(2)1690 3636 y Fr(=)1788 3580 y(\000\()p Fv(\013)p Fr(\)\000\()p Fv(\015)24 b Fq(\000)18 b Fv(\013)p Fr(\))p 1788 3617 489 4 v 1950 3693 a(\000\()p Fv(\015)5 b Fr(\))2287 3636 y(App)r(ellF)2581 3656 y Fo(1)2632 3636 y Fr(\()p Fv(\013)p Fr(;)14 b Fv(\014)2801 3648 y Fo(1)2838 3636 y Fv(;)g(\014)2922 3648 y Fo(2)2960 3636 y Fr(;)g Fv(\015)5 b Fr(;)14 b Fv(x;)g(y)s Fr(\))27 b Fv(:)630 b FB(\(E1\))1255 3962 y SDict begin H.S end 1255 3962 a 1255 3962 a SDict begin 10.5 H.A end 1255 3962 a 1255 3962 a SDict begin [/View [/XYZ H.V]/Dest (section*.21) cvn /DEST pdfmark end 1255 3962 a Fs(App)r(endix)30 b(F:)f(A)n(ccuracy)g(of)h(\034nite)g (sums)-67 4176 y FB(In)36 b(Sections)376 4176 y SDict begin H.S end 376 4176 a Black FB(I)r(I)r(I)p Black 470 4119 a SDict begin H.R end 470 4119 a 470 4176 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section*.3) cvn H.B /ANN pdfmark end 470 4176 a 36 w FB(and)675 4176 y SDict begin H.S end 675 4176 a Black FB(IV)p Black 767 4119 a SDict begin H.R end 767 4119 a 767 4176 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section*.4) cvn H.B /ANN pdfmark end 767 4176 a 36 w FB(w)n(e)f(pro)n(vided)f(decomp)r(ositions)h(of)g(functions)h (of)f(some)g(\020elemen)n(tary\021)40 b(t)n(yp)r(e)c(in)g(terms)f(of)g Fv(u)g FB(whic)n(h)-150 4271 y(con)n(tain)c(in\034nite)i(summations.)49 b(Naturally)-7 b(,)33 b(for)e(n)n(umerics)g(it)i(is)f(imp)r(ortan)n(t)f (to)h(kno)n(w)f(ho)n(w)h(man)n(y)f(terms)h(are)f(needed)h(to)g(reac)n (h)-150 4367 y(some)j(desired)h(accuracy)-7 b(.)60 b(Considering)34 b(compact)i(binaries)f(with)h(small)g(eccen)n(tricities)f(only)-7 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/Link/Dest (equation.4.34) cvn H.B /ANN pdfmark end 3544 1111 a FB(\))g(and)g(\()3804 1111 y SDict begin H.S end 3804 1111 a Black FB(36)p Black 3887 1058 a SDict begin H.R end 3887 1058 a 3887 1111 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.34) cvn H.B /ANN pdfmark end 3887 1111 a FB(\))g(and)-150 1206 y(their)d(dep)r(endency)f(on)h(the)g (summation)f(index)h Fv(j)5 b FB(,)236 1451 y SDict begin H.S end 236 1451 a 236 1451 a SDict begin 11.5 H.A end 236 1451 a 236 1451 a SDict begin [/View [/XYZ H.V]/Dest (equation.F.6) cvn /DEST pdfmark end 236 1451 a 274 1395 a Fr(sin)14 b Fv(u)p 246 1432 220 4 v 246 1508 a(A)p Fr(\()p Fv(u)p Fr(\))420 1484 y Fu(n)525 1451 y Fr(=)666 1347 y FA(1)639 1372 y Fn(X)642 1549 y Fu(j)s Fo(=1)773 1451 y Fv(S)829 1408 y Fo(\()p Fu(n)p Fo(\))824 1474 y Fu(j)940 1451 y Fr(sin)f Fv(j)5 b(u)14 b(;)p Black 2745 w FB(\(F6\))p Black 322 1772 a Fv(S)378 1729 y Fo(\()p Fu(n)p Fo(\))373 1794 y(1)525 1772 y Fr(=)639 1680 y Fn(\020)689 1772 y Fr(1)k(+)g Fv(b)868 1729 y Fo(\()p Fu(n)p Fo(\))868 1794 y(0)965 1680 y Fn(\021)1033 1772 y Fq(\000)1126 1716 y Fr(1)p 1126 1753 42 4 v 1126 1829 a(2)1177 1772 y Fv(b)1213 1729 y Fo(\()p Fu(n)p Fo(\))1213 1794 y(2)1333 1772 y Fr(=)1421 1605 y Fn(0)1421 1754 y(@)1493 1772 y Fr(1)g(+)1636 1661 y FA(M)p Fu(=)p Fo(2)1649 1693 y Fn(X)1655 1870 y Fu(i)p Fo(=1)1795 1772 y Fv(\014)1846 1729 y Fo(\()p Fu(n)p Fo(\))1842 1795 y(2)p Fu(i;i)1946 1605 y Fn(1)1946 1754 y(A)2037 1772 y Fq(\000)2130 1716 y Fr(1)p 2130 1753 V 2130 1829 a(2)2195 1661 y Fo(\()p FA(M\000)p Fo(2\))p Fu(=)p Fo(2)2276 1693 y Fn(X)2283 1870 y Fu(i)p Fo(=0)2491 1680 y Fn(\020)2541 1772 y Fv(\014)2592 1729 y Fo(\()p Fu(n)p Fo(\))2588 1795 y(2+2)p Fu(i;i)2794 1772 y Fr(+)g Fv(\014)2928 1729 y Fo(\()p Fu(n)p Fo(\))2924 1795 y(2+2)p Fu(i;)p Fo(2+)p Fu(i)3196 1680 y Fn(\021)3264 1772 y Fr(+)g Fq(O)r Fr(\()p Fv(e)3486 1737 y FA(M)p Fo(+1)3653 1772 y Fr(\))c Fv(;)p Black 202 w FB(\(F7\))p Black 304 2104 a Fv(S)360 2061 y Fo(\()p Fu(n)p Fo(\))355 2127 y Fu(j)s(>)p Fo(1)525 2104 y Fr(=)649 2048 y(1)p 649 2085 V 649 2161 a(2)715 1937 y Fn(0)715 2086 y(@)787 1993 y Fo(\()p FA(M\000)p Fu(j)s Fo(+1\))p Fu(=)p Fo(2)909 2025 y Fn(X)915 2202 y Fu(i)p Fo(=0)1165 2012 y Fn(\020)1214 2104 y Fv(\014)1265 2061 y Fo(\()p Fu(n)p Fo(\))1261 2127 y Fu(j)s FA(\000)p Fo(1+2)p Fu(i;i)1551 2104 y Fr(+)k Fv(\014)1685 2061 y Fo(\()p Fu(n)p Fo(\))1681 2127 y Fu(j)s FA(\000)p Fo(1+2)p Fu(i;j)s FA(\000)p Fo(1+)p Fu(i)2118 2012 y Fn(\021)2186 2104 y Fq(\000)2269 1993 y Fo(\()p FA(M\000)p Fu(j)s FA(\000)p Fo(1\))p Fu(=)p Fo(2)2392 2025 y Fn(X)2398 2202 y Fu(i)p Fo(=0)2648 2012 y Fn(\020)2697 2104 y Fv(\014)2748 2061 y Fo(\()p Fu(n)p Fo(\))2744 2127 y Fu(j)s Fo(+1+2)p Fu(i;i)3033 2104 y Fr(+)g Fv(\014)3167 2061 y Fo(\()p Fu(n)p Fo(\))3163 2127 y Fu(j)s Fo(+1+2)p Fu(i;j)s Fo(+1+)p Fu(i)3598 2012 y Fn(\021)3648 1937 y(1)3648 2086 y(A)639 2340 y Fr(+)p Fq(O)r Fr(\()p Fv(e)843 2305 y FA(M)p Fo(+1)1009 2340 y Fr(\))c Fv(;)p Black 2846 w FB(\(F8\))p Black 198 2480 a Fq(\))23 b Fv(S)360 2437 y Fo(\()p Fu(n)p Fo(\))355 2504 y Fu(j)s(>)p Fo(1)525 2480 y Fr(=)49 b Fq(O)r Fr(\()p Fv(e)778 2446 y Fu(j)s FA(\000)p Fo(1)899 2480 y Fr(\))18 b(+)g Fq(O)r Fr(\()p Fv(e)1171 2446 y Fu(j)s Fo(+1)1291 2480 y Fr(\))23 b(=)g Fq(O)r Fr(\()p Fv(e)1573 2446 y Fu(j)s FA(\000)p Fo(1)1693 2480 y Fr(\))14 b Fv(;)p Black 2162 w FB(\(F9\))p Black 274 2644 a Fr(sin)g Fv(u)p 246 2681 220 4 v 246 2757 a(A)p Fr(\()p Fv(u)p Fr(\))420 2733 y Fu(n)525 2700 y Fr(=)639 2596 y FA(M)p Fo(+1)660 2621 y Fn(X)663 2798 y Fu(j)s Fo(=1)815 2700 y Fv(S)871 2657 y Fo(\()p Fu(n)p Fo(\))866 2723 y Fu(j)982 2700 y Fr(sin)g Fv(j)5 b(u)18 b Fr(+)g Fq(O)r Fr(\()p Fv(e)1425 2666 y FA(M)p Fo(+1)1591 2700 y Fr(\))c Fv(:)p Black 2222 w FB(\(F10\))p Black -150 2971 a(F)-7 b(or)27 b(the)h(F)-7 b(ourier)26 b(represen)n(tation,)g(w)n(e)h(remem)n(b)r(er)g(Eq.)37 b(\()1697 2971 y SDict begin H.S end 1697 2971 a Black FB(37)p Black 1781 2917 a SDict begin H.R end 1781 2917 a 1781 2971 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.37) cvn H.B /ANN pdfmark end 1781 2971 a FB(\))28 b(and)f(tak)n(e)g (the)h(expansion)e(of)i(the)g(Bessel)f(co)r(e\036cien)n(ts)g([)3640 2971 y SDict begin H.S end 3640 2971 a FB(12)3724 2971 y SDict begin 11.5 H.L end 3724 2971 a 3724 2971 a SDict begin [/Subtype /Link/Dest (cite.Colwell:1993) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 3724 2971 a -1 w FB(],)1367 3220 y SDict begin H.S end 1367 3220 a 1367 3220 a SDict begin 11.5 H.A end 1367 3220 a 1367 3220 a SDict begin [/View [/XYZ H.V]/Dest (equation.F.11) cvn /DEST pdfmark end 1367 3220 a Fv(J)1413 3232 y Fu(n)1458 3220 y Fr(\()p Fv(x)p Fr(\))d(=)f Fv(x)1728 3185 y Fu(n)1828 3116 y FA(1)1801 3141 y Fn(X)1801 3319 y Fu(k)q Fo(=0)2060 3163 y Fr(\()p Fq(\000)p Fr(1\))2231 3133 y Fu(k)2285 3163 y Fv(x)2332 3133 y Fo(2)p Fu(k)p 1946 3201 575 4 v 1946 3277 a Fr(2)1988 3253 y Fo(2)p Fu(k)q Fo(+)p Fu(n)2153 3277 y Fv(k)s Fr(!)14 b(\()p Fv(k)21 b Fr(+)d Fv(n)p Fr(\)!)2544 3220 y Fv(;)1315 b FB(\(F11\))-150 3476 y(for)27 b(the)h(determination)f(of)h(the)g (limit)g(for)f(our)g(\034nite)h(sums,)1208 3675 y SDict begin H.S end 1208 3675 a 1208 3675 a SDict begin 11.5 H.A end 1208 3675 a 1208 3675 a SDict begin [/View [/XYZ H.V]/Dest (equation.F.12) cvn /DEST pdfmark end 1208 3675 a 3 w Fr(\026)-45 b Fv(\015)1256 3641 y Fu(m)1251 3696 y(j)1369 3675 y Fr(=)1493 3619 y Fv(m)p 1493 3656 73 4 v 1510 3732 a(j)1604 3675 y Fr(\()p Fv(J)1682 3687 y Fu(j)s FA(\000)p Fu(m)1828 3675 y Fr(\()p Fv(j)5 b(e)p Fr(\))p Fq(\000)o Fv(J)2080 3687 y Fu(j)s Fo(+)p Fu(m)2225 3675 y Fr(\()p Fv(j)g(e)p Fr(\)\))24 b(=)f Fq(O)r Fr(\()p Fv(e)2650 3641 y Fu(j)s FA(\000)p Fu(m)2796 3675 y Fr(\))14 b Fv(;)p Black 1017 w FB(\(F12\))p Black 967 3933 a Fq(\))24 b Fr(cos)13 b Fv(mu)49 b Fr(=)1483 3829 y FA(M)p Fo(+)p Fu(m)1517 3854 y Fn(X)1520 4031 y Fu(j)s Fo(=1)1688 3933 y Fr(\026)-45 b Fv(\015)1733 3899 y Fu(m)1728 3954 y(j)1809 3933 y Fr(cos)13 b Fv(j)5 b Fp(M)18 b Fq(\000)g(O)r Fr(\()p Fv(e)2304 3899 y FA(M)p Fo(+1)2471 3933 y Fr(\))c Fv(;)p Black 1342 w FB(\(F13\))p Black 1156 4236 a Fq(A)1222 4193 y Fo(\()p Fu(n)p Fo(\))1222 4259 y Fu(j)1369 4236 y Fr(=)1516 4132 y FA(M)1495 4157 y Fn(X)1483 4333 y Fu(m)p Fo(=1)1640 4236 y Fv(\015)1688 4201 y Fu(m)1683 4256 y(j)1751 4236 y Fv(b)1787 4201 y Fo(\()p Fu(n)p Fo(\))1787 4256 y Fu(m)1906 4236 y Fr(=)23 b Fq(O)r Fr(\()p Fv(e)2133 4201 y Fu(j)s FA(\000)p Fu(m)2279 4236 y Fr(\))14 b Fq(O)r Fr(\()p Fv(e)2464 4201 y Fu(m)2528 4236 y Fr(\))23 b(=)g Fq(O)r Fr(\()p Fv(e)2810 4201 y Fu(j)2845 4236 y Fr(\))14 b Fv(;)p Black 968 w FB(\(F14\))p Black 1048 4526 a Fv(A)p Fr(\()p Fv(u)p Fr(\))1222 4492 y FA(\000)p Fu(n)1369 4526 y Fr(=)49 b(\(1)18 b(+)g Fv(b)1694 4483 y Fo(\()p Fu(n)p Fo(\))1694 4548 y(0)1791 4526 y Fr(\))h(+)1946 4422 y FA(M)1925 4447 y Fn(X)1927 4624 y Fu(j)s Fo(=1)2059 4526 y Fq(A)2125 4483 y Fo(\()p Fu(n)p Fo(\))2125 4549 y Fu(j)2236 4526 y Fr(cos)13 b Fv(j)5 b Fp(M)18 b Fr(+)g Fq(O)r Fr(\()p Fv(e)2731 4492 y FA(M)p Fo(+1)2897 4526 y Fr(\))c Fv(:)p Black 916 w FB(\(F15\))p Black -150 4797 a(In)29 b(the)h(third)f(line)g(ab)r(o)n(v)n(e)f(w)n(e)h(used)g (Eq.)42 b(\()1216 4797 y SDict begin H.S end 1216 4797 a Black FB(F5)p Black 1312 4740 a SDict begin H.R end 1312 4740 a 1312 4797 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.F.1) cvn H.B /ANN pdfmark end 1312 4797 a FB(\))29 b(to)g(truncate)g(the)h(n)n(um)n(b)r(er)f Fv(j)34 b FB(in)29 b Fr(cos)13 b Fv(j)5 b(u)p FB(.)41 b(F)-7 b(or)29 b(the)g(other)g (relev)-5 b(an)n(t)28 b(term)h(with)h Fr(sin)14 b Fv(u)-150 4892 y FB(w)n(e)27 b(consider)1114 5071 y SDict begin H.S end 1114 5071 a 1114 5071 a SDict begin 11.5 H.A end 1114 5071 a 1114 5071 a SDict begin [/View [/XYZ H.V]/Dest (equation.F.16) cvn /DEST pdfmark end 1114 5071 a 4 w Fr(\026)-46 b Fv(\033)1164 5036 y Fu(m)1161 5091 y(j)1277 5071 y Fr(=)1402 5014 y Fv(m)p 1402 5051 73 4 v 1419 5128 a(j)1499 5071 y Fr(\()p Fv(J)1577 5083 y Fu(j)s FA(\000)p Fu(m)1723 5071 y Fr(\()p Fv(j)5 b(e)p Fr(\)+)o Fv(J)1975 5083 y Fu(j)s Fo(+)p Fu(m)2120 5071 y Fr(\()p Fv(j)g(e)p Fr(\)\))24 b(=)e Fq(O)r Fr(\()p Fv(e)2544 5036 y Fu(j)s FA(\000)p Fu(m)2691 5071 y Fr(\))14 b Fv(;)p Black 1122 w FB(\(F16\))p Black 885 5329 a Fq(\))23 b Fr(sin)14 b Fv(mu)49 b Fr(=)1392 5225 y FA(M)p Fo(+)p Fu(m)1426 5250 y Fn(X)1428 5427 y Fu(j)s Fo(=1)1594 5329 y Fq(S)1650 5286 y Fo(\()p Fu(m)p Fo(\))1644 5352 y Fu(j)1779 5329 y Fr(sin)14 b Fv(j)5 b Fp(M)18 b Fr(+)g Fq(O)r Fr(\()p Fv(e)2265 5294 y FA(M)p Fo(+1)2431 5329 y Fr(\))c Fv(;)p Black 1382 w FB(\(F17\))p Black Black Black eop end end %%Page: 27 27 TeXDict begin HPSdict begin 27 26 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -158 -515 a SDict begin H.S end -158 -515 a Black Black -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.27) cvn /DEST pdfmark end -158 -515 a Black 4001 -299 a FB(27)p Black 1074 -14 a Fq(S)1130 -57 y Fo(\()p Fu(n)p Fo(\))1124 9 y Fu(j)1277 -14 y Fr(=)1392 -118 y FA(M)p Fo(+1)1413 -93 y Fn(X)1401 83 y Fu(m)p Fo(=1)1572 -14 y Fr(\026)-46 b Fv(\033)1618 -49 y Fu(m)1615 6 y(j)1681 -14 y Fv(S)1737 -49 y Fo(\()p Fu(n)p Fo(\))1732 6 y Fu(m)1857 -14 y Fr(=)23 b Fq(O)r Fr(\()p Fv(e)2084 -49 y Fu(j)s FA(\000)p Fu(m)2230 -14 y Fr(\))14 b Fq(O)r Fr(\()p Fv(e)2415 -49 y Fu(m)p FA(\000)p Fo(1)2563 -14 y Fr(\))24 b(=)e Fq(O)r Fr(\()p Fv(e)2845 -49 y Fu(j)s FA(\000)p Fo(1)2966 -14 y Fr(\))14 b Fv(;)p Black 847 w FB(\(F18\))p Black 1026 220 a Fr(sin)g Fv(u)p 998 257 220 4 v 998 333 a(A)p Fr(\()p Fv(u)p Fr(\))1172 309 y Fu(n)1277 276 y Fr(=)1392 172 y FA(M)p Fo(+1)1413 197 y Fn(X)1415 374 y Fu(j)s Fo(=1)1568 276 y Fq(S)1624 233 y Fo(\()p Fu(n)p Fo(\))1618 299 y Fu(j)1735 276 y Fr(sin)g Fv(j)5 b Fp(M)18 b Fr(+)g Fq(O)r Fr(\()p Fv(e)2221 242 y FA(M)p Fo(+1)2388 276 y Fr(\))c Fv(:)p Black 1425 w FB(\(F19\))p Black -150 547 a(In)32 b(the)f(third)h(line)f(again,)g (w)n(e)g(used)h(\()1077 547 y SDict begin H.S end 1077 547 a Black FB(F9)p Black 1173 490 a SDict begin H.R end 1173 490 a 1173 547 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.F.6) cvn H.B /ANN pdfmark end 1173 547 a FB(\))f(for)g(the)h (index)f Fv(j)37 b FB(in)31 b Fr(sin)14 b Fv(j)5 b(u)31 b FB(and)g(in)h(the)f(last)h(Eq.)48 b(\()2923 547 y SDict begin H.S end 2923 547 a Black FB(F10)p Black 3061 490 a SDict begin H.R end 3061 490 a 3061 547 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.F.6) cvn H.B /ANN pdfmark end 3061 547 a FB(\).)g(W)-7 b(e)32 b(will)f(also)g(consider)f(the) -150 643 y(expansion)c(co)r(e\036cien)n(ts)i(of)f Fv(v)k FB(from)c(Eq.)37 b(\()1209 643 y SDict begin H.S end 1209 643 a Black FB(23)p Black 1293 589 a SDict begin H.R end 1293 589 a 1293 643 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.23) cvn H.B /ANN pdfmark end 1293 643 a FB(\).)g(With)28 b(the)g(regular)e(solution)h(to)h(Eq.)37 b(\()2632 643 y SDict begin H.S end 2632 643 a Black FB(24)p Black 2715 589 a SDict begin H.R end 2715 589 a 2715 643 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.24) cvn H.B /ANN pdfmark end 2715 643 a FB(\))28 b(at)f Fv(e)c Fr(=)g(0)p FB(,)1481 887 y SDict begin H.S end 1481 887 a 1481 887 a SDict begin 11.5 H.A end 1481 887 a 1481 887 a SDict begin [/View [/XYZ H.V]/Dest (equation.F.20) cvn /DEST pdfmark end 1481 887 a Fv(\013)h Fr(=)1655 831 y(1)18 b Fq(\000)1798 761 y(p)p 1868 761 219 4 v 1868 831 a Fr(1)f Fq(\000)i Fv(e)2050 807 y Fo(2)p 1655 868 432 4 v 1852 944 a Fv(e)2119 887 y Fr(=)k Fq(O)r Fr(\()p Fv(e)2346 852 y Fo(1)2384 887 y Fr(\))14 b Fv(;)1429 b FB(\(F20\))-150 1092 y(their)28 b(order)e(is)h(calculated)g(to)h(b)r(e)962 1321 y SDict begin H.S end 962 1321 a 962 1321 a SDict begin 11.5 H.A end 962 1321 a 962 1321 a SDict begin [/View [/XYZ H.V]/Dest (equation.F.21) cvn /DEST pdfmark end 962 1321 a Fv(G)1027 1333 y Fu(m)1091 1321 y Fr(\()p Fv(e)p Fr(\))50 b(=)1384 1264 y(2)p 1368 1301 73 4 v 1368 1378 a Fv(m)1451 1321 y(J)1497 1333 y Fu(m)1560 1321 y Fr(\()p Fv(me)p Fr(\))19 b(+)1865 1217 y FA(1)1838 1242 y Fn(X)1840 1418 y Fu(s)p Fo(=1)1972 1321 y Fv(\013)2025 1286 y Fu(s)2074 1321 y Fr([)p Fv(J)2143 1333 y Fu(m)p FA(\000)p Fu(s)2290 1321 y Fr(\()p Fv(me)p Fr(\))f Fq(\000)g Fv(J)2613 1333 y Fu(m)p Fo(+)p Fu(s)2759 1321 y Fr(\()p Fv(me)p Fr(\)])1244 1596 y(=)49 b Fq(O)r Fr(\()p Fv(e)1497 1562 y Fu(m)1561 1596 y Fr(\))18 b(+)1721 1492 y FA(1)1694 1517 y Fn(X)1697 1693 y Fu(s)p Fo(=1)1828 1596 y Fq(O)r Fr(\()p Fv(e)1967 1562 y Fu(s)2003 1596 y Fr(\))2049 1529 y Fn(\002)2084 1596 y Fq(O)r Fr(\()p Fv(e)2223 1562 y Fu(m)p FA(\000)p Fu(s)2369 1596 y Fr(\))h Fq(\000)f(O)r Fr(\()p Fv(e)2642 1562 y Fu(m)p Fo(+)p Fu(s)2788 1596 y Fr(\))2820 1529 y Fn(\003)1244 1797 y Fr(=)49 b Fq(O)r Fr(\()p Fv(e)1497 1763 y Fu(m)1561 1797 y Fr(\))14 b Fv(:)p Black 2252 w FB(\(F21\))p Black 1268 2073 a SDict begin H.S end 1268 2073 a 1268 2073 a SDict begin 10.5 H.A end 1268 2073 a 1268 2073 a SDict begin [/View [/XYZ H.V]/Dest (section*.23) cvn /DEST pdfmark end 1268 2073 a Fs(2.)88 b(A)n(ccuracy)30 b(of)g(\034nite)g(sum)f(pro)r(ducts)-150 2287 y FB(The)h(pro)r(duct)f (of)g(t)n(w)n(o)g(sin)h(series)e(can)h(b)r(e)h(truncated)g(taking)e(in) n(to)i(accoun)n(t)e(the)i(individual)g(series')e(terms.)43 b(In)29 b(the)h(follo)n(wing)f(w)n(e)-150 2382 y(will)d(use)g(the)h (fact)f(that)g(those)g(co)r(e\036cien)n(ts)f(could)h(already)e(b)r(e)j (truncated)f(to)g(some)f(\034nite)i(order)d Fq(M)i FB(in)g(the)h (previous)e(subsection,)-150 2478 y(see)i(Eqs.)37 b(\()209 2478 y SDict begin H.S end 209 2478 a Black FB(F14)p Black 347 2420 a SDict begin H.R end 347 2420 a 347 2478 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.F.12) cvn H.B /ANN pdfmark end 347 2478 a FB(\))28 b(and)f(\()600 2478 y SDict begin H.S end 600 2478 a Black FB(F18)p Black 737 2420 a SDict begin H.R end 737 2420 a 737 2478 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.F.16) cvn H.B /ANN pdfmark end 737 2478 a FB(\),)h(and)g(that)g(they)f(are)g(at)g (least)h(of)f(some)g(order)f(of)i Fv(e)f FB(themselv)n(es,)828 2735 y SDict begin H.S end 828 2735 a 828 2735 a SDict begin 11.5 H.A end 828 2735 a 828 2735 a SDict begin [/View [/XYZ H.V]/Dest (equation.F.22) cvn /DEST pdfmark end 828 2735 a 842 2593 a Fn( )935 2631 y FA(1)908 2656 y Fn(X)908 2835 y Fu(k)q Fo(=1)1042 2735 y Fq(S)1098 2692 y Fo(\()p Fu(n)p Fo(\))1092 2760 y Fu(k)1210 2735 y Fr(sin)13 b Fv(k)s Fp(M)1462 2593 y Fn(!)g( )1646 2631 y FA(1)1619 2656 y Fn(X)1607 2832 y Fu(m)p Fo(=1)1764 2735 y Fv(G)1829 2747 y Fu(m)1906 2735 y Fr(sin)h Fv(m)p Fp(M)2186 2593 y Fn(!)2301 2735 y Fr(=)2442 2631 y FA(1)2415 2656 y Fn(X)2418 2833 y Fu(j)s Fo(=0)2549 2735 y Fv(P)2614 2692 y Fu(S)s(S;)p Fo([)p Fu(n)p Fo(])2602 2758 y Fu(j)2815 2735 y Fr(cos)f Fv(j)5 b Fp(M)14 b Fv(:)p Black 775 w FB(\(F22\))p Black -150 3001 a(F)-7 b(or)25 b(clarit)n(y)-7 b(,)26 b(w)n(e)g(again)f(write)g(do)n(wn)h(the)g(de\034nitions)h(and)f (mak)n(e)f(use)h(of)g(the)g(elab)r(orated)f(orders,)g(where)h(some)f (abuse)h(of)g(notation)-150 3096 y(is)h(made,)294 3341 y SDict begin H.S end 294 3341 a 294 3341 a SDict begin 11.5 H.A end 294 3341 a 294 3341 a SDict begin [/View [/XYZ H.V]/Dest (equation.F.23) cvn /DEST pdfmark end 294 3341 a Fv(P)359 3298 y Fu(S)s(S;)p Fo([)p Fu(n)p Fo(])347 3363 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3630 a Black -40 3758 a FB(1.)p Black 33 w(eac)n(h)26 b Fq(A)310 3715 y Fo(\()p Fu(n)p Fo(\))310 3781 y Fu(j)435 3758 y FB(term)i(at)f Fv(j)769 3770 y FA(M)875 3758 y Fr(=)22 b Fq(M)p FB(,)-150 3826 y SDict begin H.S end -150 3826 a -150 3826 a SDict begin 11.5 H.A end -150 3826 a -150 3826 a SDict begin [/View [/XYZ H.V]/Dest (Item.6) cvn /DEST pdfmark end -150 3826 a Black -40 3954 a FB(2.)p Black 33 w(eac)n(h)k Fq(S)300 3911 y Fo(\()p Fu(n)p Fo(\))294 3977 y Fu(j)426 3954 y FB(term)h(at)h Fv(j)760 3966 y FA(M)865 3954 y Fr(=)23 b Fq(M)18 b Fr(+)g(1)p FB(,)-150 4022 y SDict begin H.S end -150 4022 a -150 4022 a SDict begin 11.5 H.A end -150 4022 a -150 4022 a SDict begin [/View [/XYZ H.V]/Dest (Item.7) cvn /DEST pdfmark end -150 4022 a Black -40 4150 a FB(3.)p Black 33 w(eac)n(h)26 b Fv(P)309 4107 y Fu(S)s(S;)p Fo([)p Fu(n)p Fo(])297 4173 y Fu(j)524 4150 y FB(term)h(at)h Fv(j)858 4162 y FA(M)963 4150 y Fr(=)23 b Fq(M)18 b Fr(+)g(1)p FB(,)27 b(and)-150 4217 y SDict begin H.S end -150 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