CINXE.COM

<!DOCTYPE html><html lang="en"> <head> <meta http-equiv="content-type" content="text/html; charset=UTF-8"> <title>DLMF: Index</title> <link rel="shortcut icon" href=".././style/DLMF-16.png" type="image/png"> <script type="text/javascript"></script> <script type="text/javascript" src=".././style/jquery.js"></script> <script type="text/javascript" src=".././style/jquery.leaveNotice.js"></script> <script type="text/javascript" src=".././style/DLMF.js"></script> <script async="" type="text/javascript" id="_fed_an_ua_tag" src="https://dap.digitalgov.gov/Universal-Federated-Analytics-Min.js?agency=DOC&subagency=NIST&pua=UA-37115410-44&yt=true&exts=ppsx,pps,f90,sch,rtf,wrl,txz,m1v,xlsm,msi,xsd,f,tif,eps,mpg,xml,pl,xlt,c"></script><script async="" src="https://www.googletagmanager.com/gtag/js?id=G-BZB64W4M0L"></script><script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-BZB64W4M0L'); </script> <link rel="stylesheet" type="text/css" href=".././style/LaTeXML.css"> <link rel="stylesheet" type="text/css" href=".././style/DLMF.css"> <link rel="stylesheet" type="text/css" href=".././style/print.css" media="print"> <meta name="viewport" content="width=device-width, initial-scale=1"> <link rel="top" href=".././" title="DLMF"> <link rel="copyright" href=".././about/notices" title="Copyright"> <link rel="search" href=".././search/search" title="Search"> <link rel="help" class="ltx_help" href=".././help/" title="Help"> <link rel="start" href=".././" title=""> <link rel="prev" href=".././bib/Z" title="Bibliography Z ‣ Bibliography"> <link rel="next" href=".././idx/B" title="Index B ‣ Index"> <link rel="document" href=".././front/foreword" title="Foreword"> <link rel="document" href=".././front/preface" title="Preface"> <link rel="document" href=".././front/introduction" title="Mathematical Introduction"> <link rel="chapter" href=".././1" title="Chapter 1 Algebraic and Analytic Methods"> <link rel="chapter" href=".././2" title="Chapter 2 Asymptotic Approximations"> <link rel="chapter" href=".././3" title="Chapter 3 Numerical Methods"> <link rel="chapter" href=".././4" title="Chapter 4 Elementary Functions"> <link rel="chapter" href=".././5" title="Chapter 5 Gamma Function"> <link rel="chapter" href=".././6" title="Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals"> <link rel="chapter" href=".././7" title="Chapter 7 Error Functions, Dawson’s and Fresnel Integrals"> <link rel="chapter" href=".././8" title="Chapter 8 Incomplete Gamma and Related Functions"> <link rel="chapter" href=".././9" title="Chapter 9 Airy and Related Functions"> <link rel="chapter" href=".././10" title="Chapter 10 Bessel Functions"> <link rel="chapter" href=".././11" title="Chapter 11 Struve and Related Functions"> <link rel="chapter" href=".././12" title="Chapter 12 Parabolic Cylinder Functions"> <link rel="chapter" href=".././13" title="Chapter 13 Confluent Hypergeometric Functions"> <link rel="chapter" href=".././14" title="Chapter 14 Legendre and Related Functions"> <link rel="chapter" href=".././15" title="Chapter 15 Hypergeometric Function"> <link rel="chapter" href=".././16" title="Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function"> <link rel="chapter" href=".././17" title="Chapter 17 𝑞-Hypergeometric and Related Functions"> <link rel="chapter" href=".././18" title="Chapter 18 Orthogonal Polynomials"> <link rel="chapter" href=".././19" title="Chapter 19 Elliptic Integrals"> <link rel="chapter" href=".././20" title="Chapter 20 Theta Functions"> <link rel="chapter" href=".././21" title="Chapter 21 Multidimensional Theta Functions"> <link rel="chapter" href=".././22" title="Chapter 22 Jacobian Elliptic Functions"> <link rel="chapter" href=".././23" title="Chapter 23 Weierstrass Elliptic and Modular Functions"> <link rel="chapter" href=".././24" title="Chapter 24 Bernoulli and Euler Polynomials"> <link rel="chapter" href=".././25" title="Chapter 25 Zeta and Related Functions"> <link rel="chapter" href=".././26" title="Chapter 26 Combinatorial Analysis"> <link rel="chapter" href=".././27" title="Chapter 27 Functions of Number Theory"> <link rel="chapter" href=".././28" title="Chapter 28 Mathieu Functions and Hill’s Equation"> <link rel="chapter" href=".././29" title="Chapter 29 Lamé Functions"> <link rel="chapter" href=".././30" title="Chapter 30 Spheroidal Wave Functions"> <link rel="chapter" href=".././31" title="Chapter 31 Heun Functions"> <link rel="chapter" href=".././32" title="Chapter 32 Painlevé Transcendents"> <link rel="chapter" href=".././33" title="Chapter 33 Coulomb Functions"> <link rel="chapter" href=".././34" title="Chapter 34 3⁢𝑗,6⁢𝑗,9⁢𝑗 Symbols"> <link rel="chapter" href=".././35" title="Chapter 35 Functions of Matrix Argument"> <link rel="chapter" href=".././36" title="Chapter 36 Integrals with Coalescing Saddles"> <link rel="bibliography" href=".././bib/" title="Bibliography"> <link rel="glossary" href=".././not/" title="Notations"> <link rel="document" href=".././lof/" title="List of Figures"> <link rel="document" href=".././lot/" title="List of Tables"> <link rel="document" href=".././software/" title="Software Index"> <link rel="document" href=".././errata/" title="Errata"> <link rel="document" href=".././help/" title="Need Help?"> <link rel="document" href=".././search/" title="Advanced Search"> <link rel="document" href=".././about/" title="About the Project"> <link rel="index" href=".././idx/B" title="Index B ‣ Index"> <link rel="index" href=".././idx/C" title="Index C ‣ Index"> <link rel="index" href=".././idx/D" title="Index D ‣ Index"> <link rel="index" href=".././idx/E" title="Index E ‣ Index"> <link rel="index" href=".././idx/F" title="Index F ‣ Index"> <link rel="index" href=".././idx/G" title="Index G ‣ Index"> <link rel="index" href=".././idx/H" title="Index H ‣ Index"> <link rel="index" href=".././idx/I" title="Index I ‣ Index"> <link rel="index" href=".././idx/J" title="Index J ‣ Index"> <link rel="index" href=".././idx/K" title="Index K ‣ Index"> <link rel="index" href=".././idx/L" title="Index L ‣ Index"> <link rel="index" href=".././idx/M" title="Index M ‣ Index"> <link rel="index" href=".././idx/N" title="Index N ‣ Index"> <link rel="index" href=".././idx/O" title="Index O ‣ Index"> <link rel="index" href=".././idx/P" title="Index P ‣ Index"> <link rel="index" href=".././idx/Q" title="Index Q ‣ Index"> <link rel="index" href=".././idx/R" title="Index R ‣ Index"> <link rel="index" href=".././idx/S" title="Index S ‣ Index"> <link rel="index" href=".././idx/T" title="Index T ‣ Index"> <link rel="index" href=".././idx/U" title="Index U ‣ Index"> <link rel="index" href=".././idx/V" title="Index V ‣ Index"> <link rel="index" href=".././idx/W" title="Index W ‣ Index"> <link rel="index" href=".././idx/Z" title="Index Z ‣ Index"> <meta name="keywords" lang="en" content=" symbols, -difference equation, Abel means, Abel summability, Abelian functions, Abel–Plana formula, Absolutely continuous integration measure, Airy functions, Airy transform, Airy’s equation, Aitken’s , Al-Salam–Chihara polynomials, Anderson localization, Anger function, Anger–Weber functions, Appell functions, Askey polynomials, Askey scheme for orthogonal polynomials, Askey–Gasper inequality, Askey–Wilson class orthogonal polynomials, Askey–Wilson polynomials, Bessel functions, Bleistein’s method, Chebyshev-series expansions, Chester–Friedman–Ursell method, Coulomb functions, Darboux’s method, Dedekind modular function, Dedekind sum, Dirac delta, Euler–Maclaurin formula, Euler’s pentagonal number theorem, Fabry’s transformation, Fourier integrals, Fourier series, Fresnel integrals, Gegenbauer function, Goldbach conjecture, Haar’s method, Hankel functions, Heine’s formula, Hermite, Heun functions and Heun’s equation, Jacobi, Jacobi function, Jacobian elliptic functions, Jacobi’s identities, Kelvin functions, Lagrange’s formula, Laguerre, Lamé functions, Laplace transforms, Laplace’s method, Legendre, Legendre polynomials, Legendre’s elliptic integrals, Liouville transformation, Liouville–Green (or WKBJ) approximations, Liouville–Green (or WKBJ) type approximations, Liouville–Green approximation theorem, Lommel functions, Maclaurin series, Meixner–Pollaczek, Mellin transform, Mellin transform methods, Mellin transforms, Mellin–Barnes integrals, Olver’s, Padé, Painlevé equations, Poincaré type, Ramanujan’s identity, Ramanujan’s tau function, Riccati form, Riemann surface, Rodrigues-type formulas, Stieltjes transforms, Stokes phenomenon, Struve functions, Waring’s problem, Watson’s lemma, Whipple’s formula, Wronskian, Wronskians, absolute error, acceleration of convergence, accumulation point, acoustics, addition theorems, additive number theory, aerodynamics, affine Weyl groups, algebraic Lamé functions, algebraic curves, algebraic equations, algebraic operations, almost Mathiew equation, alternant, amplitude (, analytic continuation, analytic continuation of matrix elements of the resolvent onto higher Riemann sheets, analytic continuation onto higher Riemann sheets, analytic function, analytic properties, and Zhedanov algebra, and corecursive OP’s, angle between, angle between arcs, angular momenta, angular momentum, angular momentum coupling coefficients, angular momentum operator, annulus, antenna research, applications, approximation techniques, approximations, arc length, arc(s), area of triangle, argument principle, arithmetic Fourier transform, arithmetic mean, arithmetic progression, arithmetic-geometric mean, arithmetics, as eigenfunctions of a , associated Anger–Weber function, associated Hermite polynomials, associated Laguerre functions, associated Laguerre polynomials, associated Legendre equation, associated Legendre functions, associated orthogonal polynomials, astrophysics, asymptotic and order symbols, asymptotic approximations, asymptotic approximations and expansions, asymptotic approximations of integrals, asymptotic approximations of sums and sequences, asymptotic expansions, asymptotic scale or sequence, asymptotic solutions of difference equations, asymptotic solutions of differential equations, asymptotic solutions of transcendental equations, at infinity, atomic photo-ionization, atomic physics, atomic spectra, atomic spectroscopy, attractive potentials, auxiliary functions for Fresnel integrals, auxiliary functions for sine and cosine integrals, axially symmetric potential theory, behavior at singularities, by reflection, canonical integrals, cases of failure, characteristic equation, classification of cases, coalescing critical points, coalescing peak and endpoint, coalescing saddle points, coalescing transition points, coincident characteristic values, complex, complex variables, computation, confluent hypergeometric functions, connection formulas, connection formulas across transition points, continued fractions, cross-products, definite, definition, definitions, degree, derivatives, determinant, differential equation, differentiation, dilatation transformations, discriminant function, distributional methods, double asymptotic properties, duality, elliptic integrals, entire functions, envelope functions, error bounds, error functions, error functions and Voigt functions, error-control function, exact rational, expansions in Chebyshev series, expansions in series of, exponent pairs, exponentially-improved, exponentially-improved expansions, extensions, floating-point, for products, for sequences, for series, generalized, generalized hypergeometric functions, generating functions, graphics, graphs, hyperasymptotic expansions, hypergeometric function, hypergeometric representations, identities, improved accuracy via numerical transformations, in a domain, in terms of Airy functions, in terms of Bessel functions of fixed order, in terms of Bessel functions of variable order, in terms of elementary functions, in the complex plane, incomplete, indefinite, initial values, integer degree and order, integer order, integral identities, integral representation, integral representations, integrals, integration, integration by parts, interrelations, interrelations with other orthogonal polynomials, interval, inverse Laplace transform, inverse Laplace transforms, irregular singularities of rank 1, iterated, large argument, large degree, large order, least squares, level-index, limit relations, limit-preserving, logarithms of, mathematical, matrix elements of the resolvent, method of stationary phase, method of steepest descents, minimax polynomials, minimax rational functions, modified Bessel functions, modulus and phase, monic, monotonicity, multidimensional integrals, nonsymmetric, notation, null, numerical use of, numerically satisfactory solutions, of products, of the first kind, of the second kind, order, orthogonality, orthogonality properties, parabolic cylinder functions, parametrization via Jacobian elliptic functions, partial differential equations, partition function, physical, power series, powers of, principal values, principal values (or branches), products, products of Bessel functions, quadratic, quasi-periodicity, random potentials, re-expansion of remainder terms, real variables, recurrence relation, recurrence relations, reduction formulas, relation to , relation to Bessel functions, relation to Gudermannian function, relation to Legendre’s elliptic integrals, relation to confluent hypergeometric functions, relation to elliptic integrals, relation to modulus and phase, relation to symmetric elliptic integrals, relation to umbilics, relation to zeros, relations to hypergeometric functions, relations to other functions, repeated, representation as , representation by squares, resurgence, reversion of, series expansions, ship waves, singular continuous spectra, singularities, small , small , special values, spherical coordinates, spherical trigonometry, splines, standard solutions, substitution of, summation by parts, sums, symmetric elliptic integrals, symmetry, tables, transformations of variables, transition points, turning points, type 2 Pollaczek, ultraspherical, uniform, uniform asymptotic approximations, uniformization, uniqueness, unrestricted, values on the cut, via connection formulas, with a parameter, with respect to degree or order, zeros"> </head> <body class="color_default textfont_default titlefont_default fontsize_default navbar_default"> <div class="ltx_page_navbar"> <div class="ltx_page_navlogo"><a href=".././" title="Digital Library of Mathematical Functions">DLMF</a></div> <div class="ltx_page_navitems"> <form method="get" action=".././search/search"> <ul> <li><a href=".././idx/">Index</a></li> <li><a href=".././not/">Notations</a></li> <li><input type="text" name="q" value="" size="6" class="ltx_page_navitem_search"><button type="submit">Search</button></li> <li><a href=".././help/" class="ltx_help">Help?</a></li> <li><a href=".././help/cite">Citing</a></li> <li><a href=".././help/customize" class="ltx_customize">Customize</a></li> </ul> </form> </div> <div class="ltx_page_navsponsors"> <div><a href=".././about/" class="ltx_page_navabout">About the Project</a></div> </div> </div> <div class="ltx_page_main"> <div class="ltx_page_header"> <a href=".././bib/Z" title="In Bibliography" class="ltx_ref" rel="prev">Bibliography Z</a><a href=".././idx/B" title="In Index" class="ltx_ref" rel="next">Index B</a> </div> <div class="ltx_page_content"> <section class="ltx_index"> <h1 class="ltx_title ltx_title_index">Index</h1> <nav class="ltx_TOC"> <div class="ltx_toclist ltx_toc_verycompact">♦A♦<a href=".././idx/B" title="Index B ‣ Index" class="ltx_ref">B</a>♦<a href=".././idx/C" title="Index C ‣ Index" class="ltx_ref">C</a>♦<a href=".././idx/D" title="Index D ‣ Index" class="ltx_ref">D</a>♦<a href=".././idx/E" title="Index E ‣ Index" class="ltx_ref">E</a>♦<a href=".././idx/F" title="Index F ‣ Index" class="ltx_ref">F</a>♦<a href=".././idx/G" title="Index G ‣ Index" class="ltx_ref">G</a>♦<a href=".././idx/H" title="Index H ‣ Index" class="ltx_ref">H</a>♦<a href=".././idx/I" title="Index I ‣ Index" class="ltx_ref">I</a>♦<a href=".././idx/J" title="Index J ‣ Index" class="ltx_ref">J</a>♦<a href=".././idx/K" title="Index K ‣ Index" class="ltx_ref">K</a>♦<a href=".././idx/L" title="Index L ‣ Index" class="ltx_ref">L</a>♦<a href=".././idx/M" title="Index M ‣ Index" class="ltx_ref">M</a>♦<a href=".././idx/N" title="Index N ‣ Index" class="ltx_ref">N</a>♦<a href=".././idx/O" title="Index O ‣ Index" class="ltx_ref">O</a>♦<a href=".././idx/P" title="Index P ‣ Index" class="ltx_ref">P</a>♦<a href=".././idx/Q" title="Index Q ‣ Index" class="ltx_ref">Q</a>♦<a href=".././idx/R" title="Index R ‣ Index" class="ltx_ref">R</a>♦<a href=".././idx/S" title="Index S ‣ Index" class="ltx_ref">S</a>♦<a href=".././idx/T" title="Index T ‣ Index" class="ltx_ref">T</a>♦<a href=".././idx/U" title="Index U ‣ Index" class="ltx_ref">U</a>♦<a href=".././idx/V" title="Index V ‣ Index" class="ltx_ref">V</a>♦<a href=".././idx/W" title="Index W ‣ Index" class="ltx_ref">W</a>♦<a href=".././idx/Z" title="Index Z ‣ Index" class="ltx_ref">Z</a>♦</div></nav> <div class="ltx_page_columns"> <div class="ltx_page_column1"> <ul class="ltx_indexlist"> <li id="Abelmeans" class="ltx_indexentry"> Abel means <a href=".././1.15#Px7" title="Abel Means ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.15(iii)</a> </li> <li id="Abelsummability" class="ltx_indexentry"> Abel summability <a href=".././1.15#Px1" title="Abel Summability ‣ §1.15(i) Definitions for Series ‣ §1.15 Summability Methods ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.15(i)</a>, <a href=".././1.15#Px10" title="Abel Summability ‣ §1.15(iv) Definitions for Integrals ‣ §1.15 Summability Methods ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.15(iv)</a> </li> <li id="AbelPlanaformula" class="ltx_indexentry"> Abel–Plana formula <a href=".././2.10#i.p2" title="§2.10(i) Euler–Maclaurin Formula ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.10(i)</a> </li> <li id="Abelianfunctions" class="ltx_indexentry"> Abelian functions <a href=".././21.8" title="§21.8 Abelian Functions ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref">§21.8</a> </li> <li id="absoluteerror" class="ltx_indexentry"> absolute error <a href=".././3.1#v.p1" title="§3.1(v) Error Measures ‣ §3.1 Arithmetics and Error Measures ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.1(v)</a> </li> <li id="Absolutelycontinuousintegrationmeasure" class="ltx_indexentry"> Absolutely continuous integration measure <a href=".././1.4#Px12" title="Absolutely Continuous Stieltjes Measure ‣ §1.4(v) Definite Integrals ‣ §1.4 Calculus of One Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.4(v)</a> </li> <li id="accelerationofconvergence" class="ltx_indexentry"> acceleration of convergence <ul class="ltx_indexlist"> <li id="accelerationofconvergence.definition" class="ltx_indexentry"> definition <a href=".././3.9#i.p3" title="§3.9(i) Sequence Transformations ‣ §3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.9(i)</a> </li> <li id="accelerationofconvergence.forsequences" class="ltx_indexentry"> for sequences <a href=".././3.9" title="§3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.9</a>—<a href=".././3.9#vi" title="§3.9(vi) Applications and Further Transformations ‣ §3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.9(vi)</a> </li> <li id="accelerationofconvergence.forseries" class="ltx_indexentry"> for series <a href=".././3.9" title="§3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.9</a>—<a href=".././3.9#vi" title="§3.9(vi) Applications and Further Transformations ‣ §3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.9(vi)</a> </li> <li id="accelerationofconvergence.limitpreserving" class="ltx_indexentry"> limit-preserving <a href=".././3.9#i.p2" title="§3.9(i) Sequence Transformations ‣ §3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.9(i)</a> </li> </ul> </li> <li id="accumulationpoint" class="ltx_indexentry"> accumulation point <a href=".././1.9#Px10.p2" title="Point Sets in ℂ ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.9(ii)</a> </li> <li id="acoustics" class="ltx_indexentry"> acoustics <ul class="ltx_indexlist"> <li id="acoustics.canonicalintegrals" class="ltx_indexentry"> canonical integrals <a href=".././36.14#iv" title="§36.14(iv) Acoustics ‣ §36.14 Other Physical Applications ‣ Applications ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref">§36.14(iv)</a> </li> </ul> </li> <li id="additivenumbertheory" class="ltx_indexentry"> additive number theory <a href=".././27#PT3" title="Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">Ch.27</a>—<a href=".././27.14#vi.p3" title="§27.14(vi) Ramanujan’s Tau Function ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.14(vi)</a> <ul class="ltx_indexlist"> <li id="additivenumbertheory.Dedekindmodularfunction" class="ltx_indexentry"> Dedekind modular function <a href=".././27.14#iv" title="§27.14(iv) Relation to Modular Functions ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.14(iv)</a> </li> <li id="additivenumbertheory.Dedekindsum" class="ltx_indexentry"> Dedekind sum <a href=".././27.14#iii.p1" title="§27.14(iii) Asymptotic Formulas ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.14(iii)</a> </li> <li id="additivenumbertheory.discriminantfunction" class="ltx_indexentry"> discriminant function <a href=".././27.14#vi" title="§27.14(vi) Ramanujan’s Tau Function ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.14(vi)</a> </li> <li id="additivenumbertheory.Eulerspentagonalnumbertheorem" class="ltx_indexentry"> Euler’s pentagonal number theorem <a href=".././27.14#ii.p1" title="§27.14(ii) Generating Functions and Recursions ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.14(ii)</a> </li> <li id="additivenumbertheory.Goldbachconjecture" class="ltx_indexentry"> Goldbach conjecture <a href=".././27.13#ii" title="§27.13(ii) Goldbach Conjecture ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.13(ii)</a> </li> <li id="additivenumbertheory.Jacobisidentities" class="ltx_indexentry"> Jacobi’s identities <a href=".././27.13#iv" title="§27.13(iv) Representation by Squares ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.13(iv)</a> </li> <li id="additivenumbertheory.notation" class="ltx_indexentry"> notation <a href=".././27.1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.1</a> </li> <li id="additivenumbertheory.partitionfunction" class="ltx_indexentry"> partition function <a href=".././27.13#i" title="§27.13(i) Introduction ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.13(i)</a> <ul class="ltx_indexlist"> <li id="additivenumbertheory.partitionfunction.unrestricted" class="ltx_indexentry"> unrestricted <a href=".././27.14#i.p1" title="§27.14(i) Partition Functions ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.14(i)</a> </li> </ul> </li> <li id="additivenumbertheory.Ramanujansidentity" class="ltx_indexentry"> Ramanujan’s identity <a href=".././27.14#v" title="§27.14(v) Divisibility Properties ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.14(v)</a> </li> <li id="additivenumbertheory.Ramanujanstaufunction" class="ltx_indexentry"> Ramanujan’s tau function <a href=".././27.14#vi" title="§27.14(vi) Ramanujan’s Tau Function ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.14(vi)</a> </li> <li id="additivenumbertheory.representationbysquares" class="ltx_indexentry"> representation by squares <a href=".././27.13#iv" title="§27.13(iv) Representation by Squares ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.13(iv)</a> </li> <li id="additivenumbertheory.Waringsproblem" class="ltx_indexentry"> Waring’s problem <a href=".././27.13#iii" title="§27.13(iii) Waring’s Problem ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.13(iii)</a> </li> </ul> </li> <li id="aerodynamics" class="ltx_indexentry"> aerodynamics <ul class="ltx_indexlist"> <li id="aerodynamics.Struvefunctions" class="ltx_indexentry"> Struve functions <a href=".././11.12" title="§11.12 Physical Applications ‣ Applications ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.12</a> </li> </ul> </li> <li id="affineWeylgroups" class="ltx_indexentry"> affine Weyl groups <ul class="ltx_indexlist"> <li id="affineWeylgroups.Painleveequations" class="ltx_indexentry"> Painlevé equations <a href=".././32.7#viii" title="§32.7(viii) Affine Weyl Groups ‣ §32.7 Bäcklund Transformations ‣ Properties ‣ Chapter 32 Painlevé Transcendents" class="ltx_ref">§32.7(viii)</a> </li> </ul> </li> <li id="Airyfunctions" class="ltx_indexentry"> Airy functions <a href=".././9.1" title="§9.1 Special Notation ‣ Notation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.1</a> <ul class="ltx_indexlist"> <li id="Airyfunctions.analyticproperties" class="ltx_indexentry"> analytic properties <a href=".././9.2#i.p1" title="§9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.2(i)</a> </li> <li id="Airyfunctions.applications" class="ltx_indexentry"> applications <ul class="ltx_indexlist"> <li id="Airyfunctions.applications.mathematical" class="ltx_indexentry"> mathematical <a href=".././9.15" title="§9.15 Mathematical Applications ‣ Applications ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.15</a> </li> <li id="Airyfunctions.applications.physical" class="ltx_indexentry"> physical <a href=".././9.16" title="§9.16 Physical Applications ‣ Applications ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.16</a>—<a href=".././9.16#p6" title="§9.16 Physical Applications ‣ Applications ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.16</a> </li> <li id="Airyfunctions.applications.shipwaves" class="ltx_indexentry"> ship waves <a href=".././36.13" title="§36.13 Kelvin’s Ship-Wave Pattern ‣ Applications ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref">§36.13</a> </li> </ul> </li> <li id="Airyfunctions.approximations" class="ltx_indexentry"> approximations <ul class="ltx_indexlist"> <li id="Airyfunctions.approximations.expansionsinChebyshevseries" class="ltx_indexentry"> expansions in Chebyshev series <a href=".././9.19#ii" title="§9.19(ii) Expansions in Chebyshev Series ‣ §9.19 Approximations ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.19(ii)</a> </li> <li id="Airyfunctions.approximations.intermsofelementaryfunctions" class="ltx_indexentry"> in terms of elementary functions <a href=".././9.19#i" title="§9.19(i) Approximations in Terms of Elementary Functions ‣ §9.19 Approximations ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.19(i)</a> </li> <li id="Airyfunctions.approximations.inthecomplexplane" class="ltx_indexentry"> in the complex plane <a href=".././9.19#iii" title="§9.19(iii) Approximations in the Complex Plane ‣ §9.19 Approximations ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.19(iii)</a> </li> </ul> </li> <li id="Airyfunctions.asymptoticexpansions" class="ltx_indexentry"> asymptotic expansions <a href=".././9.7" title="§9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.7</a>—<a href=".././9.7#v.p3" title="§9.7(v) Exponentially-Improved Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.7(v)</a> <ul class="ltx_indexlist"> <li id="Airyfunctions.asymptoticexpansions.errorbounds" class="ltx_indexentry"> error bounds <a href=".././9.7#iii" title="§9.7(iii) Error Bounds for Real Variables ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.7(iii)</a>, <a href=".././9.7#iv" title="§9.7(iv) Error Bounds for Complex Variables ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.7(iv)</a> </li> <li id="Airyfunctions.asymptoticexpansions.exponentiallyimproved" class="ltx_indexentry"> exponentially-improved <a href=".././9.7#v" title="§9.7(v) Exponentially-Improved Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.7(v)</a> </li> </ul> </li> <li id="Airyfunctions.computation" class="ltx_indexentry"> computation <a href=".././9.17" title="§9.17 Methods of Computation ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.17</a>—<a href=".././9.17#v" title="§9.17(v) Zeros ‣ §9.17 Methods of Computation ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.17(v)</a> </li> <li id="Airyfunctions.connectionformulas" class="ltx_indexentry"> connection formulas <a href=".././9.2#v" title="§9.2(v) Connection Formulas ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.2(v)</a> </li> <li id="Airyfunctions.definitions" class="ltx_indexentry"> definitions <a href=".././9.2#i" title="§9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.2(i)</a> </li> <li id="Airyfunctions.differentialequation" class="ltx_indexentry"> differential equation <a href=".././9.2#i" title="§9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.2(i)</a> <ul class="ltx_indexlist"> <li id="Airyfunctions.differentialequation.forproducts" class="ltx_indexentry"> for products <a href=".././9.11#i" title="§9.11(i) Differential Equation ‣ §9.11 Products ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.11(i)</a> </li> <li id="Airyfunctions.differentialequation.initialvalues" class="ltx_indexentry"> initial values <a href=".././9.2#ii" title="§9.2(ii) Initial Values ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.2(ii)</a> </li> <li id="Airyfunctions.differentialequation.numericallysatisfactorysolutions" class="ltx_indexentry"> numerically satisfactory solutions <a href=".././9.2#iii" title="§9.2(iii) Numerically Satisfactory Pairs of Solutions ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.2(iii)</a> </li> <li id="Airyfunctions.differentialequation.Riccatiform" class="ltx_indexentry"> Riccati form <a href=".././9.2#vi" title="§9.2(vi) Riccati Form of Differential Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.2(vi)</a> </li> </ul> </li> <li id="Airyfunctions.Diracdelta" class="ltx_indexentry"> Dirac delta <a href=".././1.17#Px4" title="Airy Functions (§9.2) ‣ §1.17(ii) Integral Representations ‣ §1.17 Integral and Series Representations of the Dirac Delta ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.17(ii)</a> </li> <li id="Airyfunctions.envelopefunctions" class="ltx_indexentry"> envelope functions <a href=".././2.8#iii.p4" title="§2.8(iii) Case II: Simple Turning Point ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(iii)</a> </li> <li id="Airyfunctions.generalized" class="ltx_indexentry"> generalized, see <a href=".././idx/G#generalizedAiryfunctions" title="Index G ‣ Index" class="ltx_ref">generalized Airy functions</a>. </li> <li id="Airyfunctions.graphics" class="ltx_indexentry"> graphics <a href=".././9.3" title="§9.3 Graphics ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.3</a>—<a href=".././9.3#ii" title="§9.3(ii) Complex Variable ‣ §9.3 Graphics ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.3(ii)</a> </li> <li id="Airyfunctions.incomplete" class="ltx_indexentry"> incomplete <a href=".././9.14" title="§9.14 Incomplete Airy Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.14</a> </li> <li id="Airyfunctions.integralidentities" class="ltx_indexentry"> integral identities <a href=".././36.9" title="§36.9 Integral Identities ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref">§36.9</a> </li> <li id="Airyfunctions.integralrepresentations" class="ltx_indexentry"> integral representations <a href=".././9.11#iii" title="§9.11(iii) Integral Representations ‣ §9.11 Products ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.11(iii)</a>, <a href=".././9.5#i" title="§9.5(i) Real Variable ‣ §9.5 Integral Representations ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.5(i)</a>, <a href=".././9.5#ii" title="§9.5(ii) Complex Variable ‣ §9.5 Integral Representations ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.5(ii)</a> </li> <li id="Airyfunctions.integrals" class="ltx_indexentry"> integrals <ul class="ltx_indexlist"> <li id="Airyfunctions.integrals.approximations" class="ltx_indexentry"> approximations <a href=".././9.19#i.p1" title="§9.19(i) Approximations in Terms of Elementary Functions ‣ §9.19 Approximations ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.19(i)</a>, <a href=".././9.19#I1.i1" title="In §9.19(i) Approximations in Terms of Elementary Functions ‣ §9.19 Approximations ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">1st item</a>, <a href=".././9.19#I2.i1" title="In §9.19(ii) Expansions in Chebyshev Series ‣ §9.19 Approximations ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">1st item</a>, <a href=".././9.19#I2.i2" title="In §9.19(ii) Expansions in Chebyshev Series ‣ §9.19 Approximations ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">2nd item</a>, <a href=".././9.19#ii.p1" title="§9.19(ii) Expansions in Chebyshev Series ‣ §9.19 Approximations ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.19(ii)</a>, <a href=".././9.19#iii.p1" title="§9.19(iii) Approximations in the Complex Plane ‣ §9.19 Approximations ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.19(iii)</a> </li> <li id="Airyfunctions.integrals.asymptoticapproximations" class="ltx_indexentry"> asymptotic approximations <a href=".././9.10#ii" title="§9.10(ii) Asymptotic Approximations ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.10(ii)</a> </li> <li id="Airyfunctions.integrals.definite" class="ltx_indexentry"> definite <a href=".././9.10#iv" title="§9.10(iv) Definite Integrals ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.10(iv)</a> </li> <li id="Airyfunctions.integrals.indefinite" class="ltx_indexentry"> indefinite <a href=".././9.10#i" title="§9.10(i) Indefinite Integrals ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.10(i)</a> </li> <li id="Airyfunctions.integrals.ofproducts" class="ltx_indexentry"> of products <a href=".././9.11#iv" title="§9.11(iv) Indefinite Integrals ‣ §9.11 Products ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.11(iv)</a>, <a href=".././9.11#v" title="§9.11(v) Definite Integrals ‣ §9.11 Products ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.11(v)</a> </li> <li id="Airyfunctions.integrals.repeated" class="ltx_indexentry"> repeated <a href=".././9.10#viii" title="§9.10(viii) Repeated Integrals ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.10(viii)</a> </li> <li id="Airyfunctions.integrals.tables" class="ltx_indexentry"> tables <a href=".././9.18#v" title="§9.18(v) Integrals ‣ §9.18 Tables ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.18(v)</a> </li> </ul> </li> <li id="Airyfunctions.Laplacetransforms" class="ltx_indexentry"> Laplace transforms <a href=".././9.10#v" title="§9.10(v) Laplace Transforms ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.10(v)</a> </li> <li id="Airyfunctions.Maclaurinseries" class="ltx_indexentry"> Maclaurin series <a href=".././9.4" title="§9.4 Maclaurin Series ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.4</a> </li> <li id="Airyfunctions.Mellintransform" class="ltx_indexentry"> Mellin transform <a href=".././9.10#vi" title="§9.10(vi) Mellin Transform ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.10(vi)</a> </li> <li id="Airyfunctions.modulusandphase" class="ltx_indexentry"> modulus and phase <ul class="ltx_indexlist"> <li id="Airyfunctions.modulusandphase.asymptoticexpansions" class="ltx_indexentry"> asymptotic expansions <a href=".././9.8#iv" title="§9.8(iv) Asymptotic Expansions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.8(iv)</a>—<a href=".././9.8#iv.p3" title="§9.8(iv) Asymptotic Expansions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.8(iv)</a> </li> <li id="Airyfunctions.modulusandphase.definitions" class="ltx_indexentry"> definitions <a href=".././9.8#i" title="§9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.8(i)</a> </li> <li id="Airyfunctions.modulusandphase.graphs" class="ltx_indexentry"> graphs <a href=".././9.3#i" title="§9.3(i) Real Variable ‣ §9.3 Graphics ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.3(i)</a> </li> <li id="Airyfunctions.modulusandphase.identities" class="ltx_indexentry"> identities <a href=".././9.8#ii" title="§9.8(ii) Identities ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.8(ii)</a> </li> <li id="Airyfunctions.modulusandphase.monotonicity" class="ltx_indexentry"> monotonicity <a href=".././9.8#iii" title="§9.8(iii) Monotonicity ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.8(iii)</a> </li> <li id="Airyfunctions.modulusandphase.relationtoBesselfunctions" class="ltx_indexentry"> relation to Bessel functions <a href=".././9.8#i" title="§9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.8(i)</a> </li> <li id="Airyfunctions.modulusandphase.relationtozeros" class="ltx_indexentry"> relation to zeros <a href=".././9.9#ii" title="§9.9(ii) Relation to Modulus and Phase ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.9(ii)</a> </li> </ul> </li> <li id="Airyfunctions.notation" class="ltx_indexentry"> notation <a href=".././9.1" title="§9.1 Special Notation ‣ Notation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.1</a> </li> <li id="Airyfunctions.products" class="ltx_indexentry"> products <ul class="ltx_indexlist"> <li id="Airyfunctions.products.differentialequation" class="ltx_indexentry"> differential equation <a href=".././9.11#i" title="§9.11(i) Differential Equation ‣ §9.11 Products ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.11(i)</a> </li> <li id="Airyfunctions.products.integralrepresentations" class="ltx_indexentry"> integral representations <a href=".././9.11#iii" title="§9.11(iii) Integral Representations ‣ §9.11 Products ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.11(iii)</a> </li> <li id="Airyfunctions.products.integrals" class="ltx_indexentry"> integrals <a href=".././9.11#iv" title="§9.11(iv) Indefinite Integrals ‣ §9.11 Products ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.11(iv)</a>, <a href=".././9.11#v" title="§9.11(v) Definite Integrals ‣ §9.11 Products ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.11(v)</a> </li> <li id="Airyfunctions.products.Wronskian" class="ltx_indexentry"> Wronskian <a href=".././9.11#ii" title="§9.11(ii) Wronskian ‣ §9.11 Products ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.11(ii)</a> </li> </ul> </li> <li id="Airyfunctions.relationtoumbilics" class="ltx_indexentry"> relation to umbilics <a href=".././36.2#ii.p4" title="§36.2(ii) Special Cases ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref">§36.2(ii)</a> </li> <li id="Airyfunctions.relationstootherfunctions" class="ltx_indexentry"> relations to other functions <ul class="ltx_indexlist"> <li id="Airyfunctions.relationstootherfunctions.Besselfunctions" class="ltx_indexentry"> Bessel functions <a href=".././9.6#i" title="§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.6(i)</a>—<a href=".././9.6#ii.p1" title="§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.6(ii)</a> </li> <li id="Airyfunctions.relationstootherfunctions.confluenthypergeometricfunctions" class="ltx_indexentry"> confluent hypergeometric functions <a href=".././13.18#iii" title="§13.18(iii) Modified Bessel Functions ‣ §13.18 Relations to Other Functions ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref">§13.18(iii)</a>, <a href=".././13.6#iii" title="§13.6(iii) Modified Bessel Functions ‣ §13.6 Relations to Other Functions ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref">§13.6(iii)</a>, <a href=".././9.6#iii" title="§9.6(iii) Airy Functions as Confluent Hypergeometric Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.6(iii)</a> </li> <li id="Airyfunctions.relationstootherfunctions.Hankelfunctions" class="ltx_indexentry"> Hankel functions <a href=".././9.6#i" title="§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.6(i)</a>—<a href=".././9.6#ii.p1" title="§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.6(ii)</a> </li> <li id="Airyfunctions.relationstootherfunctions.modifiedBesselfunctions" class="ltx_indexentry"> modified Bessel functions <a href=".././9.6#i" title="§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.6(i)</a>—<a href=".././9.6#ii.p1" title="§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.6(ii)</a> </li> </ul> </li> <li id="Airyfunctions.Stieltjestransforms" class="ltx_indexentry"> Stieltjes transforms <a href=".././9.10#vii" title="§9.10(vii) Stieltjes Transforms ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.10(vii)</a> </li> <li id="Airyfunctions.tables" class="ltx_indexentry"> tables <ul class="ltx_indexlist"> <li id="Airyfunctions.tables.complexvariables" class="ltx_indexentry"> complex variables <a href=".././9.18#iii" title="§9.18(iii) Complex Variables ‣ §9.18 Tables ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.18(iii)</a> </li> <li id="Airyfunctions.tables.integrals" class="ltx_indexentry"> integrals <a href=".././9.18#v" title="§9.18(v) Integrals ‣ §9.18 Tables ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.18(v)</a> </li> <li id="Airyfunctions.tables.realvariables" class="ltx_indexentry"> real variables <a href=".././9.18#ii" title="§9.18(ii) Real Variables ‣ §9.18 Tables ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.18(ii)</a> </li> <li id="Airyfunctions.tables.zeros" class="ltx_indexentry"> zeros <a href=".././9.18#iv" title="§9.18(iv) Zeros ‣ §9.18 Tables ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.18(iv)</a>, <a href=".././9.9#v" title="§9.9(v) Tables ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.9(v)</a>—<a href=".././9.9#v" title="§9.9(v) Tables ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.9(v)</a> </li> </ul> </li> <li id="Airyfunctions.Wronskians" class="ltx_indexentry"> Wronskians <a href=".././9.2#iv" title="§9.2(iv) Wronskians ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.2(iv)</a> </li> <li id="Airyfunctions.zeros" class="ltx_indexentry"> zeros <ul class="ltx_indexlist"> <li id="Airyfunctions.zeros.asymptoticexpansions" class="ltx_indexentry"> asymptotic expansions <a href=".././9.9#iv" title="§9.9(iv) Asymptotic Expansions ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.9(iv)</a> </li> <li id="Airyfunctions.zeros.computation" class="ltx_indexentry"> computation <a href=".././3.3#Px8" title="Example ‣ §3.3(v) Inverse Interpolation ‣ §3.3 Interpolation ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.3(v)</a>, <a href=".././9.17#v" title="§9.17(v) Zeros ‣ §9.17 Methods of Computation ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.17(v)</a> </li> <li id="Airyfunctions.zeros.differentiation" class="ltx_indexentry"> differentiation <a href=".././9.9#iii" title="§9.9(iii) Derivatives With Respect to 𝑘 ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.9(iii)</a> </li> <li id="Airyfunctions.zeros.relationtomodulusandphase" class="ltx_indexentry"> relation to modulus and phase <a href=".././9.9#ii" title="§9.9(ii) Relation to Modulus and Phase ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.9(ii)</a> </li> <li id="Airyfunctions.zeros.tables" class="ltx_indexentry"> tables <a href=".././9.18#iv" title="§9.18(iv) Zeros ‣ §9.18 Tables ‣ Computation ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.18(iv)</a>, <a href=".././9.9#v" title="§9.9(v) Tables ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.9(v)</a>—<a href=".././9.9#v" title="§9.9(v) Tables ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.9(v)</a> </li> </ul> </li> </ul> </li> <li id="Airytransform" class="ltx_indexentry"> Airy transform <a href=".././9.10#ix" title="§9.10(ix) Compendia ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref">§9.10(ix)</a> </li> <li id="Airysequation" class="ltx_indexentry"> Airy’s equation, see <a href="#Airyfunctions.differentialequation" title="Index" class="ltx_ref">Airy functions, differential equation</a>. </li> <li id="AitkensDelta2process" class="ltx_indexentry"> Aitken’s <math class="ltx_Math" altimg="m1.png" altimg-height="21px" altimg-valign="-2px" altimg-width="31px" alttext="\Delta^{2}" display="inline"><msup><mi mathvariant="normal">Δ</mi><mn>2</mn></msup></math>-process <ul class="ltx_indexlist"> <li id="AitkensDelta2process.forsequences" class="ltx_indexentry"> for sequences <a href=".././3.9#iii" title="§3.9(iii) Aitken’s Δ²-Process ‣ §3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.9(iii)</a> </li> <li id="AitkensDelta2process.iterated" class="ltx_indexentry"> iterated <a href=".././3.9#iii.p2" title="§3.9(iii) Aitken’s Δ²-Process ‣ §3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.9(iii)</a> </li> </ul> </li> <li id="AlSalamChiharapolynomials" class="ltx_indexentry"> Al-Salam–Chihara polynomials <a href=".././18.28#iv" title="§18.28(iv) 𝑞⁻¹-Al-Salam–Chihara Polynomials ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(iv)</a> </li> <li id="algebraiccurves" class="ltx_indexentry"> algebraic curves <ul class="ltx_indexlist"> <li id="algebraiccurves.Riemannsurface" class="ltx_indexentry"> Riemann surface <a href=".././21.10#I1.i2.p1" title="In §21.10(ii) Riemann Theta Functions Associated with a Riemann Surface ‣ §21.10 Methods of Computation ‣ Computation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref">2nd item</a>, <a href=".././21.7#i" title="§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref">§21.7(i)</a>, <a href=".././21.7#iii" title="§21.7(iii) Frobenius’ Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref">§21.7(iii)</a> </li> </ul> </li> <li id="algebraicequations" class="ltx_indexentry"> algebraic equations <ul class="ltx_indexlist"> <li id="algebraicequations.parametrizationviaJacobianellipticfunctions" class="ltx_indexentry"> parametrization via Jacobian elliptic functions <a href=".././22.18#i" title="§22.18(i) Lengths and Parametrization of Plane Curves ‣ §22.18 Mathematical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.18(i)</a> <ul class="ltx_indexlist"> <li id="algebraicequations.parametrizationviaJacobianellipticfunctions.sphericaltrigonometry" class="ltx_indexentry"> spherical trigonometry <a href=".././22.18#iii.p1" title="§22.18(iii) Uniformization and Other Parametrizations ‣ §22.18 Mathematical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.18(iii)</a> </li> <li id="algebraicequations.parametrizationviaJacobianellipticfunctions.uniformization" class="ltx_indexentry"> uniformization <a href=".././22.18#iii" title="§22.18(iii) Uniformization and Other Parametrizations ‣ §22.18 Mathematical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.18(iii)</a> </li> </ul> </li> </ul> </li> <li id="algebraicLamefunctions" class="ltx_indexentry"> algebraic Lamé functions <a href=".././29.17#ii" title="§29.17(ii) Algebraic Lamé Functions ‣ §29.17 Other Solutions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref">§29.17(ii)</a> </li> <li id="almostMathiewequation" class="ltx_indexentry"> almost Mathiew equation <ul class="ltx_indexlist"> <li id="almostMathiewequation.singularcontinuousspectra" class="ltx_indexentry"> singular continuous spectra <a href=".././1.18#vii.p1" title="§1.18(vii) Continuous Spectra: More General Cases ‣ §1.18 Linear Second Order Differential Operators and Eigenfunction Expansions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.18(vii)</a> </li> </ul> </li> <li id="alternant" class="ltx_indexentry"> alternant <ul class="ltx_indexlist"> <li id="alternant.determinant" class="ltx_indexentry"> determinant <a href=".././1.3#ii" title="§1.3(ii) Special Determinants ‣ §1.3 Determinants, Linear Operators, and Spectral Expansions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.3(ii)</a> </li> </ul> </li> <li id="amplitudeamfunction" class="ltx_indexentry"> amplitude (<math class="ltx_Math" altimg="m6.png" altimg-height="13px" altimg-valign="-2px" altimg-width="32px" alttext="\operatorname{am}" display="inline"><mi href=".././22.16#E1" title="Jacobi’s amplitude function">am</mi></math>) function <a href=".././22.16#i" title="§22.16(i) Jacobi’s Amplitude (am) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.16(i)</a> <ul class="ltx_indexlist"> <li id="amplitudeamfunction.applications" class="ltx_indexentry"> applications <a href=".././22.19#i" title="§22.19(i) Classical Dynamics: The Pendulum ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.19(i)</a> </li> <li id="amplitudeamfunction.approximations" class="ltx_indexentry"> approximations <ul class="ltx_indexlist"> <li id="amplitudeamfunction.approximations.smallkk" class="ltx_indexentry"> small <math class="ltx_Math" altimg="m7.png" altimg-height="24px" altimg-valign="-6px" altimg-width="42px" alttext="k,k^{\prime}" display="inline"><mrow><mi href=".././22.1#t1.r3" title="modulus">k</mi><mo>,</mo><msup><mi href=".././22.1#t1.r4" title="complementary modulus">k</mi><mo href=".././22.1#t1.r4" title="complementary modulus">′</mo></msup></mrow></math> <a href=".././22.16#Px6" title="Approximations for Small 𝑘, 𝑘' ‣ §22.16(i) Jacobi’s Amplitude (am) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.16(i)</a> </li> <li id="amplitudeamfunction.approximations.smallx" class="ltx_indexentry"> small <math class="ltx_Math" altimg="m9.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href=".././22.1#t1.r1" title="real">x</mi></math> <a href=".././22.16#Px5" title="Approximation for Small 𝑥 ‣ §22.16(i) Jacobi’s Amplitude (am) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.16(i)</a> </li> </ul> </li> <li id="amplitudeamfunction.computation" class="ltx_indexentry"> computation <a href=".././22.20#vi" title="§22.20(vi) Related Functions ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.20(vi)</a> </li> <li id="amplitudeamfunction.definition" class="ltx_indexentry"> definition <a href=".././22.16#Px1" title="Definition ‣ §22.16(i) Jacobi’s Amplitude (am) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.16(i)</a> </li> <li id="amplitudeamfunction.Fourierseries" class="ltx_indexentry"> Fourier series <a href=".././22.16#Px7" title="Fourier Series ‣ §22.16(i) Jacobi’s Amplitude (am) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.16(i)</a> </li> <li id="amplitudeamfunction.integralrepresentation" class="ltx_indexentry"> integral representation <a href=".././22.16#Px3" title="Integral Representation ‣ §22.16(i) Jacobi’s Amplitude (am) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.16(i)</a> </li> <li id="amplitudeamfunction.quasiperiodicity" class="ltx_indexentry"> quasi-periodicity <a href=".././22.16#Px2" title="Quasi-Periodicity ‣ §22.16(i) Jacobi’s Amplitude (am) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.16(i)</a> </li> <li id="amplitudeamfunction.relationtoellipticintegrals" class="ltx_indexentry"> relation to elliptic integrals <a href=".././22.16#Px8" title="Relation to Elliptic Integrals ‣ §22.16(i) Jacobi’s Amplitude (am) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.16(i)</a> </li> <li id="amplitudeamfunction.relationtoGudermannianfunction" class="ltx_indexentry"> relation to Gudermannian function <a href=".././22.16#Px6.p1" title="Approximations for Small 𝑘, 𝑘' ‣ §22.16(i) Jacobi’s Amplitude (am) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.16(i)</a> </li> <li id="amplitudeamfunction.specialvalues" class="ltx_indexentry"> special values <a href=".././22.16#Px4" title="Special Values ‣ §22.16(i) Jacobi’s Amplitude (am) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.16(i)</a> </li> <li id="amplitudeamfunction.tables" class="ltx_indexentry"> tables <a href=".././22.21" title="§22.21 Tables ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.21</a> </li> </ul> </li> <li id="analyticcontinuation" class="ltx_indexentry"> analytic continuation <a href=".././1.10#ii" title="§1.10(ii) Analytic Continuation ‣ §1.10 Functions of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.10(ii)</a> <ul class="ltx_indexlist"> <li id="analyticcontinuation.byreflection" class="ltx_indexentry"> by reflection <a href=".././1.10#Px3" title="Schwarz Reflection Principle ‣ §1.10(ii) Analytic Continuation ‣ §1.10 Functions of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.10(ii)</a> </li> </ul> </li> <li id="analyticcontinuationofmatrixelementsoftheresolventontohigherRiemannsheets" class="ltx_indexentry"> analytic continuation of matrix elements of the resolvent onto higher Riemann sheets <ul class="ltx_indexlist"> <li id="analyticcontinuationofmatrixelementsoftheresolventontohigherRiemannsheets.dilatationtransformations" class="ltx_indexentry"> dilatation transformations <a href=".././1.18#viii.p2" title="§1.18(viii) Mixed Spectra and Eigenfunction Expansions ‣ §1.18 Linear Second Order Differential Operators and Eigenfunction Expansions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.18(viii)</a> </li> </ul> </li> <li id="analyticcontinuationontohigherRiemannsheets" class="ltx_indexentry"> analytic continuation onto higher Riemann sheets <ul class="ltx_indexlist"> <li id="analyticcontinuationontohigherRiemannsheets.matrixelementsoftheresolvent" class="ltx_indexentry"> matrix elements of the resolvent <a href=".././1.18#viii.p2" title="§1.18(viii) Mixed Spectra and Eigenfunction Expansions ‣ §1.18 Linear Second Order Differential Operators and Eigenfunction Expansions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.18(viii)</a> </li> </ul> </li> <li id="analyticfunction" class="ltx_indexentry"> analytic function <a href=".././1.9#Px13" title="Analyticity ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.9(ii)</a> <ul class="ltx_indexlist"> <li id="analyticfunction.atinfinity" class="ltx_indexentry"> at infinity <a href=".././1.9#iv.p1" title="§1.9(iv) Conformal Mapping ‣ §1.9 Calculus of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.9(iv)</a> </li> <li id="analyticfunction.inadomain" class="ltx_indexentry"> in a domain <a href=".././1.9#Px13.p2" title="Analyticity ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.9(ii)</a> </li> <li id="analyticfunction.singularities" class="ltx_indexentry"> singularities <a href=".././1.10#iii.p2" title="§1.10(iii) Laurent Series ‣ §1.10 Functions of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.10(iii)</a> </li> <li id="analyticfunction.zeros" class="ltx_indexentry"> zeros <a href=".././1.10#Px2.p1" title="Zeros ‣ §1.10(i) Taylor’s Theorem for Complex Variables ‣ §1.10 Functions of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.10(i)</a> </li> </ul> </li> <li id="Andersonlocalization" class="ltx_indexentry"> Anderson localization <ul class="ltx_indexlist"> <li id="Andersonlocalization.randompotentials" class="ltx_indexentry"> random potentials <a href=".././1.18#vii.p1" title="§1.18(vii) Continuous Spectra: More General Cases ‣ §1.18 Linear Second Order Differential Operators and Eigenfunction Expansions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.18(vii)</a> </li> </ul> </li> <li id="Angerfunction" class="ltx_indexentry"> Anger function, see <a href="#AngerWeberfunctions" title="Index" class="ltx_ref">Anger–Weber functions</a>. </li> <li id="AngerWeberfunctions" class="ltx_indexentry"> Anger–Weber functions <a href=".././11.10" title="§11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10</a> <ul class="ltx_indexlist"> <li id="AngerWeberfunctions.analyticproperties" class="ltx_indexentry"> analytic properties <a href=".././11.10#i.p1" title="§11.10(i) Definitions ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(i)</a> </li> <li id="AngerWeberfunctions.asymptoticexpansions" class="ltx_indexentry"> asymptotic expansions <ul class="ltx_indexlist"> <li id="AngerWeberfunctions.asymptoticexpansions.largeargument" class="ltx_indexentry"> large argument <a href=".././11.11#i" title="§11.11(i) Large |𝑧|, Fixed 𝜈 ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.11(i)</a>—<a href=".././11.11#i.p1" title="§11.11(i) Large |𝑧|, Fixed 𝜈 ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.11(i)</a> </li> <li id="AngerWeberfunctions.asymptoticexpansions.largeorder" class="ltx_indexentry"> large order <a href=".././11.11#ii" title="§11.11(ii) Large |𝜈|, Fixed 𝑧 ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.11(ii)</a> </li> </ul> </li> <li id="AngerWeberfunctions.computation" class="ltx_indexentry"> computation <a href=".././11.13#i" title="§11.13(i) Introduction ‣ §11.13 Methods of Computation ‣ Computation ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.13(i)</a>, <a href=".././3.6#Px2" title="Example 2. Weber Function ‣ §3.6(vi) Examples ‣ §3.6 Linear Difference Equations ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.6(vi)</a> </li> <li id="AngerWeberfunctions.definitions" class="ltx_indexentry"> definitions <a href=".././11.10#i" title="§11.10(i) Definitions ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(i)</a> </li> <li id="AngerWeberfunctions.derivatives" class="ltx_indexentry"> derivatives <a href=".././11.10#ix" title="§11.10(ix) Recurrence Relations and Derivatives ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(ix)</a> </li> <li id="AngerWeberfunctions.differentialequation" class="ltx_indexentry"> differential equation <a href=".././11.10#ii" title="§11.10(ii) Differential Equations ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(ii)</a> </li> <li id="AngerWeberfunctions.graphics" class="ltx_indexentry"> graphics <a href=".././11.10#F1" title="In §11.10(iv) Graphics ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">Figure 11.10.1</a>, <a href=".././11.10.F1.mag" title="In §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">Figure 11.10.1</a>, <a href=".././11.10.F1.mag#F1.thumb" title="In Figure 11.10.1 ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">Figure 11.10.1</a>, <a href=".././11.10#F2" title="In §11.10(iv) Graphics ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">Figure 11.10.2</a>, <a href=".././11.10.F2.mag" title="In §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">Figure 11.10.2</a>, <a href=".././11.10.F2.mag#F2.thumb" title="In Figure 11.10.2 ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">Figure 11.10.2</a>, <a href=".././11.10#F3" title="In §11.10(iv) Graphics ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">Figure 11.10.3</a>, <a href=".././11.10.F3.mag" title="In §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">Figure 11.10.3</a>, <a href=".././11.10.F3.mag#F3.thumb" title="In Figure 11.10.3 ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">Figure 11.10.3</a>, <a href=".././11.10#F4" title="In §11.10(iv) Graphics ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">Figure 11.10.4</a>, <a href=".././11.10.F4.mag" title="In §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">Figure 11.10.4</a>, <a href=".././11.10.F4.mag#F4.thumb" title="In Figure 11.10.4 ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">Figure 11.10.4</a> </li> <li id="AngerWeberfunctions.incomplete" class="ltx_indexentry"> incomplete <a href=".././11.14#v" title="§11.14(v) Incomplete Functions ‣ §11.14 Tables ‣ Computation ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.14(v)</a> </li> <li id="AngerWeberfunctions.integralrepresentations" class="ltx_indexentry"> integral representations <a href=".././11.10#i" title="§11.10(i) Definitions ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(i)</a> </li> <li id="AngerWeberfunctions.integrals" class="ltx_indexentry"> integrals <a href=".././11.10#x" title="§11.10(x) Integrals and Sums ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(x)</a> </li> <li id="AngerWeberfunctions.interrelations" class="ltx_indexentry"> interrelations <a href=".././11.10#v" title="§11.10(v) Interrelations ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(v)</a> </li> <li id="AngerWeberfunctions.Maclaurinseries" class="ltx_indexentry"> Maclaurin series <a href=".././11.10#iii" title="§11.10(iii) Maclaurin Series ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(iii)</a> </li> <li id="AngerWeberfunctions.notation" class="ltx_indexentry"> notation <a href=".././11.1" title="§11.1 Special Notation ‣ Notation ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.1</a> </li> <li id="AngerWeberfunctions.order" class="ltx_indexentry"> order <a href=".././11.1" title="§11.1 Special Notation ‣ Notation ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.1</a> </li> <li id="AngerWeberfunctions.recurrencerelations" class="ltx_indexentry"> recurrence relations <a href=".././11.10#ix" title="§11.10(ix) Recurrence Relations and Derivatives ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(ix)</a> </li> <li id="AngerWeberfunctions.relationstootherfunctions" class="ltx_indexentry"> relations to other functions <ul class="ltx_indexlist"> <li id="AngerWeberfunctions.relationstootherfunctions.Fresnelintegrals" class="ltx_indexentry"> Fresnel integrals <a href=".././11.10#vi.p1" title="§11.10(vi) Relations to Other Functions ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(vi)</a> </li> <li id="AngerWeberfunctions.relationstootherfunctions.Lommelfunctions" class="ltx_indexentry"> Lommel functions <a href=".././11.10#vi" title="§11.10(vi) Relations to Other Functions ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(vi)</a>—<a href=".././11.10#vi.p1" title="§11.10(vi) Relations to Other Functions ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(vi)</a> </li> <li id="AngerWeberfunctions.relationstootherfunctions.Struvefunctions" class="ltx_indexentry"> Struve functions <a href=".././11.10#vi.p2" title="§11.10(vi) Relations to Other Functions ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(vi)</a> </li> </ul> </li> <li id="AngerWeberfunctions.seriesexpansions" class="ltx_indexentry"> series expansions <ul class="ltx_indexlist"> <li id="AngerWeberfunctions.seriesexpansions.powerseries" class="ltx_indexentry"> power series <a href=".././11.10#iii" title="§11.10(iii) Maclaurin Series ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(iii)</a> </li> <li id="AngerWeberfunctions.seriesexpansions.productsofBesselfunctions" class="ltx_indexentry"> products of Bessel functions <a href=".././11.10#viii" title="§11.10(viii) Expansions in Series of Products of Bessel Functions ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(viii)</a> </li> </ul> </li> <li id="AngerWeberfunctions.specialvalues" class="ltx_indexentry"> special values <a href=".././11.10#vii" title="§11.10(vii) Special Values ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(vii)</a> </li> <li id="AngerWeberfunctions.sums" class="ltx_indexentry"> sums <a href=".././11.10#x" title="§11.10(x) Integrals and Sums ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.10(x)</a> </li> <li id="AngerWeberfunctions.tables" class="ltx_indexentry"> tables <a href=".././11.14#iv" title="§11.14(iv) Anger–Weber Functions ‣ §11.14 Tables ‣ Computation ‣ Chapter 11 Struve and Related Functions" class="ltx_ref">§11.14(iv)</a> </li> </ul> </li> <li id="anglebetweenarcs" class="ltx_indexentry"> angle between arcs <a href=".././1.9#Px22" title="Conformal Transformation ‣ §1.9(iv) Conformal Mapping ‣ §1.9 Calculus of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.9(iv)</a> </li> <li id="angularmomenta" class="ltx_indexentry"> angular momenta <a href=".././34.2#p1" title="§34.2 Definition: 3⁢𝑗 Symbol ‣ Properties ‣ Chapter 34 3⁢𝑗,6⁢𝑗,9⁢𝑗 Symbols" class="ltx_ref">§34.2</a> </li> <li id="angularmomentum" class="ltx_indexentry"> angular momentum <ul class="ltx_indexlist"> <li id="angularmomentum.generalizedhypergeometricfunctions" class="ltx_indexentry"> generalized hypergeometric functions <a href=".././16.24#iii" title="§16.24(iii) 3⁢𝑗, 6⁢𝑗, and 9⁢𝑗 Symbols ‣ §16.24 Physical Applications ‣ Applications ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.24(iii)</a> </li> </ul> </li> <li id="angularmomentumcouplingcoefficients" class="ltx_indexentry"> angular momentum coupling coefficients, see <a href=".././idx/T#threejsymbols" title="Index T ‣ Index" class="ltx_ref"><math class="ltx_Math" altimg="m3.png" altimg-height="21px" altimg-valign="-6px" altimg-width="24px" alttext="\mathit{3j}" display="inline"><mrow><mn class="ltx_mathvariant_italic" href=".././34.2#E4" mathvariant="italic" title="3⁢𝑗 symbol">3</mn><mo href=".././34.2#E4" title="3⁢𝑗 symbol">⁢</mo><mi href=".././34.2#E4" title="3⁢𝑗 symbol">j</mi></mrow></math> symbols</a>, <a href=".././idx/S#sixjsymbols" title="Index S ‣ Index" class="ltx_ref"><math class="ltx_Math" altimg="m4.png" altimg-height="21px" altimg-valign="-6px" altimg-width="24px" alttext="\mathit{6j}" display="inline"><mrow><mn class="ltx_mathvariant_italic" href=".././34.4#E1" mathvariant="italic" title="6⁢𝑗 symbol">6</mn><mo href=".././34.4#E1" title="6⁢𝑗 symbol">⁢</mo><mi href=".././34.4#E1" title="6⁢𝑗 symbol">j</mi></mrow></math> symbols</a>, and <a href=".././idx/N#ninejsymbols" title="Index N ‣ Index" class="ltx_ref"><math class="ltx_Math" altimg="m5.png" altimg-height="21px" altimg-valign="-6px" altimg-width="24px" alttext="\mathit{9j}" display="inline"><mrow><mn class="ltx_mathvariant_italic" href=".././34.6#E1" mathvariant="italic" title="9⁢𝑗 symbol">9</mn><mo href=".././34.6#E1" title="9⁢𝑗 symbol">⁢</mo><mi href=".././34.6#E1" title="9⁢𝑗 symbol">j</mi></mrow></math> symbols</a>. </li> <li id="angularmomentumoperator" class="ltx_indexentry"> angular momentum operator <ul class="ltx_indexlist"> <li id="angularmomentumoperator.sphericalcoordinates" class="ltx_indexentry"> spherical coordinates <a href=".././14.30#iv.p2" title="§14.30(iv) Applications ‣ §14.30 Spherical and Spheroidal Harmonics ‣ Applications ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.30(iv)</a> </li> </ul> </li> <li id="annulus" class="ltx_indexentry"> annulus <a href=".././1.10#iii" title="§1.10(iii) Laurent Series ‣ §1.10 Functions of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.10(iii)</a> </li> <li id="antennaresearch" class="ltx_indexentry"> antenna research <ul class="ltx_indexlist"> <li id="antennaresearch.Lamefunctions" class="ltx_indexentry"> Lamé functions <a href=".././29.19#i" title="§29.19(i) Lamé Functions ‣ §29.19 Physical Applications ‣ Applications ‣ Chapter 29 Lamé Functions" class="ltx_ref">§29.19(i)</a> </li> </ul> </li> <li id="Appellfunctions" class="ltx_indexentry"> Appell functions <a href=".././16.13" title="§16.13 Appell Functions ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.13</a> <ul class="ltx_indexlist"> <li id="Appellfunctions.analyticcontinuation" class="ltx_indexentry"> analytic continuation <a href=".././16.15#p1" title="§16.15 Integral Representations and Integrals ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.15</a> </li> <li id="Appellfunctions.applications" class="ltx_indexentry"> applications <ul class="ltx_indexlist"> <li id="Appellfunctions.applications.physical" class="ltx_indexentry"> physical <a href=".././16.24" title="§16.24 Physical Applications ‣ Applications ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.24</a> </li> </ul> </li> <li id="Appellfunctions.computation" class="ltx_indexentry"> computation <a href=".././16#PT6" title="Computation ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">Ch.16</a> </li> <li id="Appellfunctions.definition" class="ltx_indexentry"> definition <a href=".././16.13" title="§16.13 Appell Functions ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.13</a>—<a href=".././16.13#p1" title="§16.13 Appell Functions ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.13</a> </li> <li id="Appellfunctions.integralrepresentations" class="ltx_indexentry"> integral representations <a href=".././16.15" title="§16.15 Integral Representations and Integrals ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.15</a> </li> <li id="Appellfunctions.integrals" class="ltx_indexentry"> integrals <a href=".././16.15" title="§16.15 Integral Representations and Integrals ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.15</a> <ul class="ltx_indexlist"> <li id="Appellfunctions.integrals.inverseLaplacetransform" class="ltx_indexentry"> inverse Laplace transform <a href=".././16.15#p1" title="§16.15 Integral Representations and Integrals ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.15</a> </li> </ul> </li> <li id="Appellfunctions.notation" class="ltx_indexentry"> notation <a href=".././16.13" title="§16.13 Appell Functions ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.13</a> </li> <li id="Appellfunctions.partialdifferentialequations" class="ltx_indexentry"> partial differential equations <a href=".././16.14#i" title="§16.14(i) Appell Functions ‣ §16.14 Partial Differential Equations ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.14(i)</a> </li> <li id="Appellfunctions.relationtoLegendresellipticintegrals" class="ltx_indexentry"> relation to Legendre’s elliptic integrals <a href=".././19.5#p1" title="§19.5 Maclaurin and Related Expansions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref">§19.5</a> </li> <li id="Appellfunctions.relationtosymmetricellipticintegrals" class="ltx_indexentry"> relation to symmetric elliptic integrals <a href=".././19.25#vii.p1" title="§19.25(vii) Hypergeometric Function ‣ §19.25 Relations to Other Functions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref">§19.25(vii)</a> </li> <li id="Appellfunctions.relationstohypergeometricfunctions" class="ltx_indexentry"> relations to hypergeometric functions <a href=".././16.16#i" title="§16.16(i) Reduction Formulas ‣ §16.16 Transformations of Variables ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.16(i)</a> </li> <li id="Appellfunctions.transformationsofvariables" class="ltx_indexentry"> transformations of variables <a href=".././16.16" title="§16.16 Transformations of Variables ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.16</a>—<a href=".././16.16#ii.p2" title="§16.16(ii) Other Transformations ‣ §16.16 Transformations of Variables ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.16(ii)</a> <ul class="ltx_indexlist"> <li id="Appellfunctions.transformationsofvariables.quadratic" class="ltx_indexentry"> quadratic <a href=".././16.16#ii" title="§16.16(ii) Other Transformations ‣ §16.16 Transformations of Variables ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.16(ii)</a> </li> <li id="Appellfunctions.transformationsofvariables.reductionformulas" class="ltx_indexentry"> reduction formulas <a href=".././16.16#i" title="§16.16(i) Reduction Formulas ‣ §16.16 Transformations of Variables ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function" class="ltx_ref">§16.16(i)</a> </li> </ul> </li> </ul> </li> <li id="approximationtechniques" class="ltx_indexentry"> approximation techniques <ul class="ltx_indexlist"> <li id="approximationtechniques.Chebyshevseriesexpansions" class="ltx_indexentry"> Chebyshev-series expansions <a href=".././3.11#Px1" title="Chebyshev Expansions ‣ §3.11(ii) Chebyshev-Series Expansions ‣ §3.11 Approximation Techniques ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.11(ii)</a> </li> <li id="approximationtechniques.leastsquares" class="ltx_indexentry"> least squares <a href=".././3.11#Px8" title="The Fast Fourier Transform ‣ §3.11(v) Least Squares Approximations ‣ §3.11 Approximation Techniques ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.11(v)</a>, <a href=".././3.11#v" title="§3.11(v) Least Squares Approximations ‣ §3.11 Approximation Techniques ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.11(v)</a>—<a href=".././3.11#v" title="§3.11(v) Least Squares Approximations ‣ §3.11 Approximation Techniques ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.11(v)</a> </li> <li id="approximationtechniques.minimaxpolynomials" class="ltx_indexentry"> minimax polynomials <a href=".././3.11#i" title="§3.11(i) Minimax Polynomial Approximations ‣ §3.11 Approximation Techniques ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.11(i)</a> </li> <li id="approximationtechniques.minimaxrationalfunctions" class="ltx_indexentry"> minimax rational functions <a href=".././3.11#iii" title="§3.11(iii) Minimax Rational Approximations ‣ §3.11 Approximation Techniques ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.11(iii)</a> </li> <li id="approximationtechniques.Pade" class="ltx_indexentry"> Padé <a href=".././3.11#iv" title="§3.11(iv) Padé Approximations ‣ §3.11 Approximation Techniques ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.11(iv)</a>—<a href=".././3.11#Px6" title="Laplace Transform Inversion ‣ §3.11(iv) Padé Approximations ‣ §3.11 Approximation Techniques ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.11(iv)</a> </li> <li id="approximationtechniques.splines" class="ltx_indexentry"> splines <a href=".././3.11#vi" title="§3.11(vi) Splines ‣ §3.11 Approximation Techniques ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.11(vi)</a>—<a href=".././3.11#vi" title="§3.11(vi) Splines ‣ §3.11 Approximation Techniques ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.11(vi)</a> </li> </ul> </li> <li id="arclength" class="ltx_indexentry"> arc length <ul class="ltx_indexlist"> <li id="arclength.Jacobianellipticfunctions" class="ltx_indexentry"> Jacobian elliptic functions <a href=".././22.18#i" title="§22.18(i) Lengths and Parametrization of Plane Curves ‣ §22.18 Mathematical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.18(i)</a> </li> </ul> </li> <li id="arcs" class="ltx_indexentry"> arc(s) <a href=".././1.9#iii" title="§1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.9(iii)</a> <ul class="ltx_indexlist"> <li id="arcs.anglebetween" class="ltx_indexentry"> angle between <a href=".././1.9#Px22" title="Conformal Transformation ‣ §1.9(iv) Conformal Mapping ‣ §1.9 Calculus of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.9(iv)</a> </li> </ul> </li> <li id="areaoftriangle" class="ltx_indexentry"> area of triangle <a href=".././10.22#Px10.p2" title="Triple Products ‣ §10.22(iv) Integrals over the Interval (0,∞) ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref">§10.22(iv)</a> </li> <li id="argumentprinciple" class="ltx_indexentry"> argument principle, see <a href=".././idx/P#phaseprinciple" title="Index P ‣ Index" class="ltx_ref">phase principle</a>. </li> <li id="arithmeticFouriertransform" class="ltx_indexentry"> arithmetic Fourier transform <a href=".././27.17" title="§27.17 Other Applications ‣ Applications ‣ Chapter 27 Functions of Number Theory" class="ltx_ref">§27.17</a> </li> <li id="arithmeticmean" class="ltx_indexentry"> arithmetic mean <a href=".././1.2#iv" title="§1.2(iv) Means ‣ §1.2 Elementary Algebra ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.2(iv)</a>, <a href=".././1.7#iii" title="§1.7(iii) Means ‣ §1.7 Inequalities ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.7(iii)</a> </li> <li id="arithmeticprogression" class="ltx_indexentry"> arithmetic progression <a href=".././1.2#Px2" title="Arithmetic Progression ‣ §1.2(ii) Finite Series ‣ §1.2 Elementary Algebra ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref">§1.2(ii)</a> </li> <li id="arithmeticgeometricmean" class="ltx_indexentry"> arithmetic-geometric mean <a href=".././19.8#i" title="§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM) ‣ §19.8 Quadratic Transformations ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref">§19.8(i)</a> <ul class="ltx_indexlist"> <li id="arithmeticgeometricmean.hypergeometricfunction" class="ltx_indexentry"> hypergeometric function <a href=".././15.17#iv.p2" title="§15.17(iv) Combinatorics ‣ §15.17 Mathematical Applications ‣ Applications ‣ Chapter 15 Hypergeometric Function" class="ltx_ref">§15.17(iv)</a> </li> <li id="arithmeticgeometricmean.integralrepresentations" class="ltx_indexentry"> integral representations <a href=".././19.8#i" title="§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM) ‣ §19.8 Quadratic Transformations ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref">§19.8(i)</a> </li> <li id="arithmeticgeometricmean.Jacobianellipticfunctions" class="ltx_indexentry"> Jacobian elliptic functions <a href=".././22.20#ii" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref">§22.20(ii)</a> </li> <li id="arithmeticgeometricmean.Legendresellipticintegrals" class="ltx_indexentry"> Legendre’s elliptic integrals <a href=".././19.8#i" title="§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM) ‣ §19.8 Quadratic Transformations ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref">§19.8(i)</a>—<a href=".././19.8#i.p3" title="§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM) ‣ §19.8 Quadratic Transformations ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref">§19.8(i)</a> </li> <li id="arithmeticgeometricmean.symmetricellipticintegrals" class="ltx_indexentry"> symmetric elliptic integrals <a href=".././19.22#ii" title="§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM) ‣ §19.22 Quadratic Transformations ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref">§19.22(ii)</a> </li> </ul> </li> </ul> </div> <div class="ltx_page_column2"> <ul class="ltx_indexlist"> <li id="arithmetics" class="ltx_indexentry"> arithmetics <ul class="ltx_indexlist"> <li id="arithmetics.complex" class="ltx_indexentry"> complex <a href=".././3.1#v.p3" title="§3.1(v) Error Measures ‣ §3.1 Arithmetics and Error Measures ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.1(v)</a> </li> <li id="arithmetics.exactrational" class="ltx_indexentry"> exact rational <a href=".././3.1#iii" title="§3.1(iii) Rational Arithmetics ‣ §3.1 Arithmetics and Error Measures ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.1(iii)</a> </li> <li id="arithmetics.floatingpoint" class="ltx_indexentry"> floating-point <a href=".././3.1#i" title="§3.1(i) Floating-Point Arithmetic ‣ §3.1 Arithmetics and Error Measures ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.1(i)</a> </li> <li id="arithmetics.interval" class="ltx_indexentry"> interval <a href=".././3.1#ii" title="§3.1(ii) Interval Arithmetic ‣ §3.1 Arithmetics and Error Measures ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.1(ii)</a> </li> <li id="arithmetics.levelindex" class="ltx_indexentry"> level-index <a href=".././3.1#iv" title="§3.1(iv) Level-Index Arithmetic ‣ §3.1 Arithmetics and Error Measures ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref">§3.1(iv)</a> </li> </ul> </li> <li id="Askeypolynomials" class="ltx_indexentry"> Askey polynomials <a href=".././18.33#Px3" title="Askey ‣ §18.33(iv) Special Cases ‣ §18.33 Polynomials Orthogonal on the Unit Circle ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.33(iv)</a> </li> <li id="Askeyschemefororthogonalpolynomials" class="ltx_indexentry"> Askey scheme for orthogonal polynomials <a href=".././18.21#F1" title="In Meixner–Pollaczek → Hermite ‣ §18.21(ii) Limit Relations and Special Cases ‣ §18.21 Hahn Class: Interrelations ‣ Askey Scheme ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">Figure 18.21.1</a>, <a href=".././18.21.F1.mag" title="In §18.21 Hahn Class: Interrelations ‣ Askey Scheme ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">Figure 18.21.1</a>, <a href=".././18.21.F1.mag#F1.thumb" title="In Figure 18.21.1 ‣ §18.21 Hahn Class: Interrelations ‣ Askey Scheme ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">Figure 18.21.1</a> </li> <li id="AskeyGasperinequality" class="ltx_indexentry"> Askey–Gasper inequality <a href=".././18.14#Px13.p1" title="Jacobi ‣ §18.14(iv) Positive Sums ‣ §18.14 Inequalities ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.14(iv)</a>, <a href=".././18.38#Px6.p1" title="Complex Function Theory ‣ §18.38(ii) Classical OP’s: Mathematical Developments and Applications ‣ §18.38 Mathematical Applications ‣ Applications ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.38(ii)</a> </li> <li id="AskeyWilsonclassorthogonalpolynomials" class="ltx_indexentry"> Askey–Wilson class orthogonal polynomials <a href=".././18.28" title="§18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28</a>—<a href=".././18.28#xi" title="§18.28(xi) Limits for 𝑞↓-1 ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(xi)</a> <ul class="ltx_indexlist"> <li id="AskeyWilsonclassorthogonalpolynomials.aseigenfunctionsofaqdifferenceoperator" class="ltx_indexentry"> as eigenfunctions of a <math class="ltx_Math" altimg="m8.png" altimg-height="17px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href=".././18.1#i.t1.r3" title="real variable">q</mi></math>-difference operator <a href=".././18.28#i.p1" title="§18.28(i) Introduction ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(i)</a> </li> <li id="AskeyWilsonclassorthogonalpolynomials.asymptoticapproximations" class="ltx_indexentry"> asymptotic approximations <a href=".././18.29" title="§18.29 Asymptotic Approximations for 𝑞-Hahn and Askey–Wilson Classes ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.29</a> </li> <li id="AskeyWilsonclassorthogonalpolynomials.interrelationswithotherorthogonalpolynomials" class="ltx_indexentry"> interrelations with other orthogonal polynomials <a href=".././18.21#F1" title="In Meixner–Pollaczek → Hermite ‣ §18.21(ii) Limit Relations and Special Cases ‣ §18.21 Hahn Class: Interrelations ‣ Askey Scheme ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">Figure 18.21.1</a>, <a href=".././18.21.F1.mag" title="In §18.21 Hahn Class: Interrelations ‣ Askey Scheme ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">Figure 18.21.1</a>, <a href=".././18.21.F1.mag#F1.thumb" title="In Figure 18.21.1 ‣ §18.21 Hahn Class: Interrelations ‣ Askey Scheme ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">Figure 18.21.1</a> </li> <li id="AskeyWilsonclassorthogonalpolynomials.limitrelations" class="ltx_indexentry"> limit relations <a href=".././18.28#x" title="§18.28(x) Limit Relations ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(x)</a> </li> <li id="AskeyWilsonclassorthogonalpolynomials.orthogonalityproperties" class="ltx_indexentry"> orthogonality properties <a href=".././18.28#i.p1" title="§18.28(i) Introduction ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(i)</a> </li> <li id="AskeyWilsonclassorthogonalpolynomials.representationasqhypergeometricfunctions" class="ltx_indexentry"> representation as <math class="ltx_Math" altimg="m8.png" altimg-height="17px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href=".././18.1#i.t1.r3" title="real variable">q</mi></math>-hypergeometric functions <a href=".././18.28#i.p1" title="§18.28(i) Introduction ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(i)</a>—<a href=".././18.28#xi" title="§18.28(xi) Limits for 𝑞↓-1 ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(xi)</a> </li> </ul> </li> <li id="AskeyWilsonpolynomials" class="ltx_indexentry"> Askey–Wilson polynomials <a href=".././18.28#ii" title="§18.28(ii) Askey–Wilson Polynomials ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(ii)</a>, see also Askey–Wilson class orthogonal polynomials. <ul class="ltx_indexlist"> <li id="AskeyWilsonpolynomials.andZhedanovalgebra" class="ltx_indexentry"> and Zhedanov algebra <a href=".././18.38#Px14.p2" title="Zhedanov Algebra ‣ §18.38(iii) Other OP’s ‣ §18.38 Mathematical Applications ‣ Applications ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.38(iii)</a> </li> <li id="AskeyWilsonpolynomials.asymptoticapproximations" class="ltx_indexentry"> asymptotic approximations <a href=".././18.29" title="§18.29 Asymptotic Approximations for 𝑞-Hahn and Askey–Wilson Classes ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.29</a> </li> <li id="AskeyWilsonpolynomials.duality" class="ltx_indexentry"> duality <a href=".././18.28#Px4" title="Duality ‣ §18.28(ii) Askey–Wilson Polynomials ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(ii)</a> </li> <li id="AskeyWilsonpolynomials.nonsymmetric" class="ltx_indexentry"> nonsymmetric <a href=".././18.38#Px15.p3" title="Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials ‣ §18.38(iii) Other OP’s ‣ §18.38 Mathematical Applications ‣ Applications ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.38(iii)</a> </li> <li id="AskeyWilsonpolynomials.qdifferenceequation" class="ltx_indexentry"> <math class="ltx_Math" altimg="m8.png" altimg-height="17px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href=".././18.1#i.t1.r3" title="real variable">q</mi></math>-difference equation <a href=".././18.28#Px2" title="𝑞-Difference Equation ‣ §18.28(ii) Askey–Wilson Polynomials ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(ii)</a> </li> <li id="AskeyWilsonpolynomials.recurrencerelation" class="ltx_indexentry"> recurrence relation <a href=".././18.28#Px3" title="Recurrence Relation ‣ §18.28(ii) Askey–Wilson Polynomials ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(ii)</a> </li> <li id="AskeyWilsonpolynomials.relationtoqhypergeometricfunctions" class="ltx_indexentry"> relation to <math class="ltx_Math" altimg="m8.png" altimg-height="17px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href=".././18.1#i.t1.r3" title="real variable">q</mi></math>-hypergeometric functions <a href=".././18.28#ii" title="§18.28(ii) Askey–Wilson Polynomials ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(ii)</a>—<a href=".././18.28#xi" title="§18.28(xi) Limits for 𝑞↓-1 ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.28(xi)</a> </li> </ul> </li> <li id="associatedAngerWeberfunction" class="ltx_indexentry"> associated Anger–Weber function, see <a href="#AngerWeberfunctions" title="Index" class="ltx_ref">Anger–Weber functions</a>. </li> <li id="associatedHermitepolynomials" class="ltx_indexentry"> associated Hermite polynomials <a href=".././18.30#iv" title="§18.30(iv) Associated Hermite Polynomials ‣ §18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30(iv)</a> </li> <li id="associatedLaguerrefunctions" class="ltx_indexentry"> associated Laguerre functions <a href=".././33.22#v.p3" title="§33.22(v) Asymptotic Solutions ‣ §33.22 Particle Scattering and Atomic and Molecular Spectra ‣ Physical Applications ‣ Chapter 33 Coulomb Functions" class="ltx_ref">§33.22(v)</a> </li> <li id="associatedLaguerrepolynomials" class="ltx_indexentry"> associated Laguerre polynomials <a href=".././18.30#iii" title="§18.30(iii) Associated Laguerre Polynomials ‣ §18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30(iii)</a> </li> <li id="associatedLegendreequation" class="ltx_indexentry"> associated Legendre equation <a href=".././14.2#ii" title="§14.2(ii) Associated Legendre Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.2(ii)</a>, <a href=".././14.21#i" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(i)</a> <ul class="ltx_indexlist"> <li id="associatedLegendreequation.exponentpairs" class="ltx_indexentry"> exponent pairs <a href=".././14.2#iii" title="§14.2(iii) Numerically Satisfactory Solutions ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.2(iii)</a> </li> <li id="associatedLegendreequation.numericallysatisfactorysolutions" class="ltx_indexentry"> numerically satisfactory solutions <a href=".././14.2#iii" title="§14.2(iii) Numerically Satisfactory Solutions ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.2(iii)</a>, <a href=".././14.21#ii" title="§14.21(ii) Numerically Satisfactory Solutions ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(ii)</a> </li> <li id="associatedLegendreequation.singularities" class="ltx_indexentry"> singularities <a href=".././14.2#iii" title="§14.2(iii) Numerically Satisfactory Solutions ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.2(iii)</a> </li> <li id="associatedLegendreequation.standardsolutions" class="ltx_indexentry"> standard solutions <a href=".././14.2#ii" title="§14.2(ii) Associated Legendre Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.2(ii)</a>, <a href=".././14.21#i.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(i)</a>, <a href=".././14.3#Px4" title="Associated Legendre Function of the Second Kind ‣ §14.3(ii) Interval 1<𝑥<∞ ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.3(ii)</a> </li> </ul> </li> <li id="associatedLegendrefunctions" class="ltx_indexentry"> associated Legendre functions <a href=".././14.1#p3" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.1</a>, see also <a href=".././idx/F#Ferrersfunctions" title="Index F ‣ Index" class="ltx_ref">Ferrers functions</a>. <ul class="ltx_indexlist"> <li id="associatedLegendrefunctions.additiontheorems" class="ltx_indexentry"> addition theorems <a href=".././14.18#ii" title="§14.18(ii) Addition Theorems ‣ §14.18 Sums ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.18(ii)</a>, <a href=".././14.28#i" title="§14.28(i) Addition Theorem ‣ §14.28 Sums ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.28(i)</a> </li> <li id="associatedLegendrefunctions.analyticcontinuation" class="ltx_indexentry"> analytic continuation <a href=".././14.24" title="§14.24 Analytic Continuation ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.24</a> </li> <li id="associatedLegendrefunctions.analyticproperties" class="ltx_indexentry"> analytic properties <a href=".././14.21#i.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(i)</a> </li> <li id="associatedLegendrefunctions.applications" class="ltx_indexentry"> applications <a href=".././14#PT4" title="Applications ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">Ch.14</a>—<a href=".././14.31#iii.p2" title="§14.31(iii) Miscellaneous ‣ §14.31 Other Applications ‣ Applications ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.31(iii)</a> </li> <li id="associatedLegendrefunctions.asymptoticapproximations" class="ltx_indexentry"> asymptotic approximations, see <a href="#associatedLegendrefunctions.uniformasymptoticapproximations" title="Index" class="ltx_ref">uniform asymptotic approximations</a>. </li> <li id="associatedLegendrefunctions.behavioratsingularities" class="ltx_indexentry"> behavior at singularities <a href=".././14.21#iii" title="§14.21(iii) Properties ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(iii)</a>, <a href=".././14.8#i.p2" title="§14.8(i) 𝑥→limit-from1- or 𝑥→limit-from{-1}+ ‣ §14.8 Behavior at Singularities ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.8(i)</a> </li> <li id="associatedLegendrefunctions.computation" class="ltx_indexentry"> computation <a href=".././14.32" title="§14.32 Methods of Computation ‣ Computation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.32</a> </li> <li id="associatedLegendrefunctions.connectionformulas" class="ltx_indexentry"> connection formulas <a href=".././14.21#iii" title="§14.21(iii) Properties ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(iii)</a>, <a href=".././14.9#iii" title="§14.9(iii) Connections Between 𝑃^{±𝜇}_𝜈(𝑥), 𝑃^{±𝜇}_{-𝜈-1}(𝑥), 𝑸^{±𝜇}_𝜈(𝑥), 𝑸^𝜇_{-𝜈-1}(𝑥) ‣ §14.9 Connection Formulas ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.9(iii)</a> </li> <li id="associatedLegendrefunctions.continuedfractions" class="ltx_indexentry"> continued fractions <a href=".././14.14" title="§14.14 Continued Fractions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.14</a> </li> <li id="associatedLegendrefunctions.crossproducts" class="ltx_indexentry"> cross-products <a href=".././14.2#iv" title="§14.2(iv) Wronskians and Cross-Products ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.2(iv)</a> </li> <li id="associatedLegendrefunctions.definitions" class="ltx_indexentry"> definitions <a href=".././14.21#i.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(i)</a>, <a href=".././14.3#ii" title="§14.3(ii) Interval 1<𝑥<∞ ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.3(ii)</a>—<a href=".././14.3#iii.p2" title="§14.3(iii) Alternative Hypergeometric Representations ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.3(iii)</a> </li> <li id="associatedLegendrefunctions.degree" class="ltx_indexentry"> degree <a href=".././14.1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.1</a> </li> <li id="associatedLegendrefunctions.derivatives" class="ltx_indexentry"> derivatives <a href=".././14.10" title="§14.10 Recurrence Relations and Derivatives ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.10</a> <ul class="ltx_indexlist"> <li id="associatedLegendrefunctions.derivatives.withrespecttodegreeororder" class="ltx_indexentry"> with respect to degree or order <a href=".././14.11" title="§14.11 Derivatives with Respect to Degree or Order ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.11</a> </li> </ul> </li> <li id="associatedLegendrefunctions.differentialequation" class="ltx_indexentry"> differential equation, see <a href="#associatedLegendreequation" title="Index" class="ltx_ref">associated Legendre equation</a>. </li> <li id="associatedLegendrefunctions.expansionsinseriesof" class="ltx_indexentry"> expansions in series of <a href=".././14.18#i.p1" title="§14.18(i) Expansion Theorem ‣ §14.18 Sums ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.18(i)</a> </li> <li id="associatedLegendrefunctions.generalized" class="ltx_indexentry"> generalized <a href=".././14.29#p1" title="§14.29 Generalizations ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.29</a> </li> <li id="associatedLegendrefunctions.generatingfunctions" class="ltx_indexentry"> generating functions <a href=".././14.21#iii" title="§14.21(iii) Properties ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(iii)</a>, <a href=".././14.7#iv" title="§14.7(iv) Generating Functions ‣ §14.7 Integer Degree and Order ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.7(iv)</a> </li> <li id="associatedLegendrefunctions.graphics" class="ltx_indexentry"> graphics <a href=".././14.22" title="§14.22 Graphics ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.22</a>—<a href=".././14.4#iii" title="§14.4(iii) Associated Legendre Functions: 2D Graphs ‣ §14.4 Graphics ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.4(iii)</a>, <a href=".././14.4#iv" title="§14.4(iv) Associated Legendre Functions: 3D Surfaces ‣ §14.4 Graphics ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.4(iv)</a> </li> <li id="associatedLegendrefunctions.Heinesformula" class="ltx_indexentry"> Heine’s formula <a href=".././14.28#ii" title="§14.28(ii) Heine’s Formula ‣ §14.28 Sums ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.28(ii)</a> </li> <li id="associatedLegendrefunctions.hypergeometricrepresentations" class="ltx_indexentry"> hypergeometric representations <a href=".././14.21#iii" title="§14.21(iii) Properties ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(iii)</a>, <a href=".././14.3" title="§14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.3</a>—<a href=".././14.3#iii.p2" title="§14.3(iii) Alternative Hypergeometric Representations ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.3(iii)</a> </li> <li id="associatedLegendrefunctions.integerdegreeandorder" class="ltx_indexentry"> integer degree and order <a href=".././14.21#iii" title="§14.21(iii) Properties ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(iii)</a>, <a href=".././14.7" title="§14.7 Integer Degree and Order ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.7</a>—<a href=".././14.7#iv.p5" title="§14.7(iv) Generating Functions ‣ §14.7 Integer Degree and Order ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.7(iv)</a> </li> <li id="associatedLegendrefunctions.integerorder" class="ltx_indexentry"> integer order <a href=".././14.21#i.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(i)</a>, <a href=".././14.6" title="§14.6 Integer Order ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.6</a>—<a href=".././14.6#ii.p2" title="§14.6(ii) Negative Integer Orders ‣ §14.6 Integer Order ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.6(ii)</a> </li> <li id="associatedLegendrefunctions.integralrepresentations" class="ltx_indexentry"> integral representations <a href=".././14.12#ii" title="§14.12(ii) 1<𝑥<∞ ‣ §14.12 Integral Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.12(ii)</a>, <a href=".././14.25" title="§14.25 Integral Representations ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.25</a> </li> <li id="associatedLegendrefunctions.integrals" class="ltx_indexentry"> integrals <ul class="ltx_indexlist"> <li id="associatedLegendrefunctions.integrals.definite" class="ltx_indexentry"> definite <a href=".././14.17#ii" title="§14.17(ii) Barnes’ Integral ‣ §14.17 Integrals ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.17(ii)</a>, <a href=".././14.17#iii" title="§14.17(iii) Orthogonality Properties ‣ §14.17 Integrals ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.17(iii)</a>, <a href=".././14.17#iv" title="§14.17(iv) Definite Integrals of Products ‣ §14.17 Integrals ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.17(iv)</a> </li> <li id="associatedLegendrefunctions.integrals.Laplacetransforms" class="ltx_indexentry"> Laplace transforms <a href=".././14.17#v" title="§14.17(v) Laplace Transforms ‣ §14.17 Integrals ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.17(v)</a> </li> <li id="associatedLegendrefunctions.integrals.Mellintransforms" class="ltx_indexentry"> Mellin transforms <a href=".././14.17#vi" title="§14.17(vi) Mellin Transforms ‣ §14.17 Integrals ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.17(vi)</a> </li> <li id="associatedLegendrefunctions.integrals.products" class="ltx_indexentry"> products <a href=".././14.17#iv" title="§14.17(iv) Definite Integrals of Products ‣ §14.17 Integrals ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.17(iv)</a> </li> </ul> </li> <li id="associatedLegendrefunctions.notation" class="ltx_indexentry"> notation <a href=".././14.1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.1</a> </li> <li id="associatedLegendrefunctions.ofthefirstkind" class="ltx_indexentry"> of the first kind <a href=".././14.3#Px3" title="Associated Legendre Function of the First Kind ‣ §14.3(ii) Interval 1<𝑥<∞ ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.3(ii)</a> </li> <li id="associatedLegendrefunctions.ofthesecondkind" class="ltx_indexentry"> of the second kind <a href=".././14.3#Px4" title="Associated Legendre Function of the Second Kind ‣ §14.3(ii) Interval 1<𝑥<∞ ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.3(ii)</a> </li> <li id="associatedLegendrefunctions.Olvers" class="ltx_indexentry"> Olver’s <a href=".././14.21#i.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(i)</a>, <a href=".././14.3#Px4.p3" title="Associated Legendre Function of the Second Kind ‣ §14.3(ii) Interval 1<𝑥<∞ ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.3(ii)</a> </li> <li id="associatedLegendrefunctions.order" class="ltx_indexentry"> order <a href=".././14.1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.1</a> </li> <li id="associatedLegendrefunctions.orthogonality" class="ltx_indexentry"> orthogonality <a href=".././14.17#iii" title="§14.17(iii) Orthogonality Properties ‣ §14.17 Integrals ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.17(iii)</a> </li> <li id="associatedLegendrefunctions.principalvaluesorbranches" class="ltx_indexentry"> principal values (or branches) <a href=".././14.21#i.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(i)</a> </li> <li id="associatedLegendrefunctions.recurrencerelations" class="ltx_indexentry"> recurrence relations <a href=".././14.10" title="§14.10 Recurrence Relations and Derivatives ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.10</a>, <a href=".././14.21#iii" title="§14.21(iii) Properties ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(iii)</a> </li> <li id="associatedLegendrefunctions.relationstootherfunctions" class="ltx_indexentry"> relations to other functions <ul class="ltx_indexlist"> <li id="associatedLegendrefunctions.relationstootherfunctions.ellipticintegrals" class="ltx_indexentry"> elliptic integrals <a href=".././14.5#v.p1" title="§14.5(v) 𝜇=0, 𝜈=±{1/2} ‣ §14.5 Special Values ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.5(v)</a> </li> <li id="associatedLegendrefunctions.relationstootherfunctions.Gegenbauerfunction" class="ltx_indexentry"> Gegenbauer function <a href=".././14.3#iv" title="§14.3(iv) Relations to Other Functions ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.3(iv)</a> </li> <li id="associatedLegendrefunctions.relationstootherfunctions.hypergeometricfunction" class="ltx_indexentry"> hypergeometric function <a href=".././14.3#ii" title="§14.3(ii) Interval 1<𝑥<∞ ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.3(ii)</a>, <a href=".././14.3#iii.p1" title="§14.3(iii) Alternative Hypergeometric Representations ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.3(iii)</a>, <a href=".././15.9#iv" title="§15.9(iv) Associated Legendre Functions; Ferrers Functions ‣ §15.9 Relations to Other Functions ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref">§15.9(iv)</a> </li> <li id="associatedLegendrefunctions.relationstootherfunctions.Jacobifunction" class="ltx_indexentry"> Jacobi function <a href=".././14.3#iv" title="§14.3(iv) Relations to Other Functions ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.3(iv)</a> </li> <li id="associatedLegendrefunctions.relationstootherfunctions.Legendrepolynomials" class="ltx_indexentry"> Legendre polynomials <a href=".././14.7#i" title="§14.7(i) 𝜇=0 ‣ §14.7 Integer Degree and Order ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.7(i)</a> </li> </ul> </li> <li id="associatedLegendrefunctions.Rodriguestypeformulas" class="ltx_indexentry"> Rodrigues-type formulas <a href=".././14.7#ii" title="§14.7(ii) Rodrigues-Type Formulas ‣ §14.7 Integer Degree and Order ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.7(ii)</a> </li> <li id="associatedLegendrefunctions.specialvalues" class="ltx_indexentry"> special values <a href=".././14.5#iii.p1" title="§14.5(iii) 𝜇=±{1/2} ‣ §14.5 Special Values ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.5(iii)</a>, <a href=".././14.5#v.p1" title="§14.5(v) 𝜇=0, 𝜈=±{1/2} ‣ §14.5 Special Values ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.5(v)</a> </li> <li id="associatedLegendrefunctions.sums" class="ltx_indexentry"> sums <a href=".././14.18" title="§14.18 Sums ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.18</a>—<a href=".././14.18#iv.p1" title="§14.18(iv) Compendia ‣ §14.18 Sums ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.18(iv)</a>, <a href=".././14.28" title="§14.28 Sums ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.28</a> </li> <li id="associatedLegendrefunctions.tables" class="ltx_indexentry"> tables <a href=".././14.33" title="§14.33 Tables ‣ Computation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.33</a> </li> <li id="associatedLegendrefunctions.uniformasymptoticapproximations" class="ltx_indexentry"> uniform asymptotic approximations <ul class="ltx_indexlist"> <li id="associatedLegendrefunctions.uniformasymptoticapproximations.largedegree" class="ltx_indexentry"> large degree <a href=".././14.15#iii" title="§14.15(iii) Large 𝜈, Fixed 𝜇 ‣ §14.15 Uniform Asymptotic Approximations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.15(iii)</a>—<a href=".././14.15#v.p6" title="§14.15(v) Large 𝜈, (𝜈+{1/2})⁢𝛿≤𝜇≤(𝜈+{1/2})/𝛿 ‣ §14.15 Uniform Asymptotic Approximations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.15(v)</a>, <a href=".././14.26" title="§14.26 Uniform Asymptotic Expansions ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.26</a> </li> <li id="associatedLegendrefunctions.uniformasymptoticapproximations.largeorder" class="ltx_indexentry"> large order <a href=".././14.15#i" title="§14.15(i) Large 𝜇, Fixed 𝜈 ‣ §14.15 Uniform Asymptotic Approximations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.15(i)</a>—<a href=".././14.15#ii" title="§14.15(ii) Large 𝜇, 0≤𝜈+{1/2}≤(1-𝛿)⁢𝜇 ‣ §14.15 Uniform Asymptotic Approximations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.15(ii)</a>, <a href=".././14.26" title="§14.26 Uniform Asymptotic Expansions ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.26</a> </li> </ul> </li> <li id="associatedLegendrefunctions.valuesonthecut" class="ltx_indexentry"> values on the cut <a href=".././14.23" title="§14.23 Values on the Cut ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.23</a> </li> <li id="associatedLegendrefunctions.Whipplesformula" class="ltx_indexentry"> Whipple’s formula <a href=".././14.9#iv" title="§14.9(iv) Whipple’s Formula ‣ §14.9 Connection Formulas ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.9(iv)</a> </li> <li id="associatedLegendrefunctions.Wronskians" class="ltx_indexentry"> Wronskians <a href=".././14.2#iv.p1" title="§14.2(iv) Wronskians and Cross-Products ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.2(iv)</a>—<a href=".././14.2#iv.p2" title="§14.2(iv) Wronskians and Cross-Products ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.2(iv)</a>, <a href=".././14.21#iii" title="§14.21(iii) Properties ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.21(iii)</a> </li> <li id="associatedLegendrefunctions.zeros" class="ltx_indexentry"> zeros <a href=".././14.16#iii" title="§14.16(iii) Interval 1<𝑥<∞ ‣ §14.16 Zeros ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.16(iii)</a>, <a href=".././14.27" title="§14.27 Zeros ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref">§14.27</a> </li> </ul> </li> <li id="associatedorthogonalpolynomials" class="ltx_indexentry"> associated orthogonal polynomials <a href=".././18.2#x" title="§18.2(x) Orthogonal Polynomials and Continued Fractions ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.2(x)</a>, <a href=".././18.30" title="§18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30</a>—<a href=".././18.30" title="§18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30</a> <ul class="ltx_indexlist"> <li id="associatedorthogonalpolynomials.andcorecursiveOPs" class="ltx_indexentry"> and corecursive OP’s <a href=".././18.2#x.p2" title="§18.2(x) Orthogonal Polynomials and Continued Fractions ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.2(x)</a>, <a href=".././18.30#vi" title="§18.30(vi) Corecursive Orthogonal Polynomials ‣ §18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30(vi)</a> </li> <li id="associatedorthogonalpolynomials.Hermite" class="ltx_indexentry"> Hermite <a href=".././18.30#iv" title="§18.30(iv) Associated Hermite Polynomials ‣ §18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30(iv)</a> </li> <li id="associatedorthogonalpolynomials.Jacobi" class="ltx_indexentry"> Jacobi <a href=".././18.30#i" title="§18.30(i) Associated Jacobi Polynomials ‣ §18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30(i)</a> </li> <li id="associatedorthogonalpolynomials.Laguerre" class="ltx_indexentry"> Laguerre <a href=".././18.30#iii" title="§18.30(iii) Associated Laguerre Polynomials ‣ §18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30(iii)</a> </li> <li id="associatedorthogonalpolynomials.Legendre" class="ltx_indexentry"> Legendre <a href=".././18.30#ii" title="§18.30(ii) Associated Legendre Polynomials ‣ §18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30(ii)</a> </li> <li id="associatedorthogonalpolynomials.MeixnerPollaczek" class="ltx_indexentry"> Meixner–Pollaczek <a href=".././18.30#v" title="§18.30(v) Associated Meixner–Pollaczek Polynomials ‣ §18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30(v)</a> </li> <li id="associatedorthogonalpolynomials.monic" class="ltx_indexentry"> monic <a href=".././18.2#x" title="§18.2(x) Orthogonal Polynomials and Continued Fractions ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.2(x)</a>, <a href=".././18.30#vii" title="§18.30(vii) Corecursive and Associated Monic Orthogonal Polynomials ‣ §18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30(vii)</a> </li> <li id="associatedorthogonalpolynomials.type2Pollaczek" class="ltx_indexentry"> type 2 Pollaczek <a href=".././18.30#viii.p1" title="§18.30(viii) Other Associated Polynomials ‣ §18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30(viii)</a> </li> <li id="associatedorthogonalpolynomials.ultraspherical" class="ltx_indexentry"> ultraspherical <a href=".././18.30#viii.p1" title="§18.30(viii) Other Associated Polynomials ‣ §18.30 Associated OP’s ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref">§18.30(viii)</a> </li> </ul> </li> <li id="astrophysics" class="ltx_indexentry"> astrophysics <ul class="ltx_indexlist"> <li id="astrophysics.errorfunctionsandVoigtfunctions" class="ltx_indexentry"> error functions and Voigt functions <a href=".././7.21#p3" title="§7.21 Physical Applications ‣ Applications ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref">§7.21</a> </li> <li id="astrophysics.HeunfunctionsandHeunsequation" class="ltx_indexentry"> Heun functions and Heun’s equation <a href=".././31.17#ii" title="§31.17(ii) Other Applications ‣ §31.17 Physical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref">§31.17(ii)</a> </li> </ul> </li> <li id="asymptoticandordersymbols" class="ltx_indexentry"> asymptotic and order symbols <a href=".././2.1#i" title="§2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(i)</a> <ul class="ltx_indexlist"> <li id="asymptoticandordersymbols.definition" class="ltx_indexentry"> definition <a href=".././2.1#i.p1" title="§2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(i)</a> </li> <li id="asymptoticandordersymbols.differentiation" class="ltx_indexentry"> differentiation <a href=".././2.1#ii" title="§2.1(ii) Integration and Differentiation ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(ii)</a> </li> <li id="asymptoticandordersymbols.integration" class="ltx_indexentry"> integration <a href=".././2.1#ii" title="§2.1(ii) Integration and Differentiation ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(ii)</a> </li> </ul> </li> <li id="asymptoticapproximationsandexpansions" class="ltx_indexentry"> asymptotic approximations and expansions, see also <a href="#asymptoticapproximationsofintegrals" title="Index" class="ltx_ref">asymptotic approximations of integrals</a>, <a href="#asymptoticapproximationsofsumsandsequences" title="Index" class="ltx_ref">asymptotic approximations of sums and sequences</a>, <a href="#asymptoticsolutionsofdifferenceequations" title="Index" class="ltx_ref">asymptotic solutions of difference equations</a>, <a href="#asymptoticsolutionsofdifferentialequations" title="Index" class="ltx_ref">asymptotic solutions of differential equations</a>, and <a href="#asymptoticsolutionsoftranscendentalequations" title="Index" class="ltx_ref">asymptotic solutions of transcendental equations</a>. <ul class="ltx_indexlist"> <li id="asymptoticapproximationsandexpansions.algebraicoperations" class="ltx_indexentry"> algebraic operations <a href=".././2.1#iii" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(iii)</a> </li> <li id="asymptoticapproximationsandexpansions.casesoffailure" class="ltx_indexentry"> cases of failure <a href=".././2.11#i" title="§2.11(i) Numerical Use of Asymptotic Expansions ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(i)</a>—<a href=".././2.11#i.p2" title="§2.11(i) Numerical Use of Asymptotic Expansions ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(i)</a>, <a href=".././2.6#i.p1" title="§2.6(i) Divergent Integrals ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.6(i)</a> </li> <li id="asymptoticapproximationsandexpansions.differentiation" class="ltx_indexentry"> differentiation <a href=".././2.1#iii" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(iii)</a> </li> <li id="asymptoticapproximationsandexpansions.doubleasymptoticproperties" class="ltx_indexentry"> double asymptotic properties <ul class="ltx_indexlist"> <li id="asymptoticapproximationsandexpansions.doubleasymptoticproperties.Besselfunctions" class="ltx_indexentry"> Bessel functions <a href=".././10.41#v" title="§10.41(v) Double Asymptotic Properties (Continued) ‣ §10.41 Asymptotic Expansions for Large Order ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref">§10.41(v)</a> </li> <li id="asymptoticapproximationsandexpansions.doubleasymptoticproperties.Hankelfunctions" class="ltx_indexentry"> Hankel functions <a href=".././10.41#v" title="§10.41(v) Double Asymptotic Properties (Continued) ‣ §10.41 Asymptotic Expansions for Large Order ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref">§10.41(v)</a> </li> <li id="asymptoticapproximationsandexpansions.doubleasymptoticproperties.Kelvinfunctions" class="ltx_indexentry"> Kelvin functions <a href=".././10.69" title="§10.69 Uniform Asymptotic Expansions for Large Order ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref">§10.69</a> </li> <li id="asymptoticapproximationsandexpansions.doubleasymptoticproperties.modifiedBesselfunctions" class="ltx_indexentry"> modified Bessel functions <a href=".././10.41#iv" title="§10.41(iv) Double Asymptotic Properties ‣ §10.41 Asymptotic Expansions for Large Order ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref">§10.41(iv)</a> </li> <li id="asymptoticapproximationsandexpansions.doubleasymptoticproperties.paraboliccylinderfunctions" class="ltx_indexentry"> parabolic cylinder functions <a href=".././12.10#vi" title="§12.10(vi) Modifications of Expansions in Elementary Functions ‣ §12.10 Uniform Asymptotic Expansions for Large Parameter ‣ Properties ‣ Chapter 12 Parabolic Cylinder Functions" class="ltx_ref">§12.10(vi)</a> </li> </ul> </li> <li id="asymptoticapproximationsandexpansions.exponentiallyimprovedexpansions" class="ltx_indexentry"> exponentially-improved expansions <a href=".././2.11#iii" title="§2.11(iii) Exponentially-Improved Expansions ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(iii)</a>—<a href=".././2.11#v" title="§2.11(v) Exponentially-Improved Expansions (continued) ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(v)</a> </li> <li id="asymptoticapproximationsandexpansions.generalized" class="ltx_indexentry"> generalized <a href=".././2.1#v" title="§2.1(v) Generalized Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(v)</a> </li> <li id="asymptoticapproximationsandexpansions.hyperasymptoticexpansions" class="ltx_indexentry"> hyperasymptotic expansions <a href=".././2.11#v" title="§2.11(v) Exponentially-Improved Expansions (continued) ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(v)</a> </li> <li id="asymptoticapproximationsandexpansions.improvedaccuracyvianumericaltransformations" class="ltx_indexentry"> improved accuracy via numerical transformations <a href=".././2.11#vi" title="§2.11(vi) Direct Numerical Transformations ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(vi)</a>—<a href=".././2.11#vi" title="§2.11(vi) Direct Numerical Transformations ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(vi)</a> </li> <li id="asymptoticapproximationsandexpansions.integration" class="ltx_indexentry"> integration <a href=".././2.1#iii" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(iii)</a> </li> <li id="asymptoticapproximationsandexpansions.logarithmsof" class="ltx_indexentry"> logarithms of <a href=".././2.1#iii" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(iii)</a> </li> <li id="asymptoticapproximationsandexpansions.null" class="ltx_indexentry"> null <a href=".././2.1#iii.p6" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(iii)</a> </li> <li id="asymptoticapproximationsandexpansions.numericaluseof" class="ltx_indexentry"> numerical use of <a href=".././2.11#i" title="§2.11(i) Numerical Use of Asymptotic Expansions ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(i)</a>—<a href=".././2.11#vi" title="§2.11(vi) Direct Numerical Transformations ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(vi)</a> </li> <li id="asymptoticapproximationsandexpansions.Poincaretype" class="ltx_indexentry"> Poincaré type <a href=".././2.1#iii.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(iii)</a> </li> <li id="asymptoticapproximationsandexpansions.powersof" class="ltx_indexentry"> powers of <a href=".././2.1#iii" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(iii)</a> </li> <li id="asymptoticapproximationsandexpansions.reexpansionofremainderterms" class="ltx_indexentry"> re-expansion of remainder terms <a href=".././2.11#iii" title="§2.11(iii) Exponentially-Improved Expansions ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(iii)</a>—<a href=".././2.11#vi" title="§2.11(vi) Direct Numerical Transformations ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(vi)</a> </li> <li id="asymptoticapproximationsandexpansions.reversionof" class="ltx_indexentry"> reversion of <a href=".././2.2#Px1" title="Example ‣ §2.2 Transcendental Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.2</a> </li> <li id="asymptoticapproximationsandexpansions.Stokesphenomenon" class="ltx_indexentry"> Stokes phenomenon <a href=".././2.11#iv" title="§2.11(iv) Stokes Phenomenon ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(iv)</a> </li> <li id="asymptoticapproximationsandexpansions.substitutionof" class="ltx_indexentry"> substitution of <a href=".././2.1#iii" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(iii)</a> </li> <li id="asymptoticapproximationsandexpansions.uniform" class="ltx_indexentry"> uniform <a href=".././2.1#iv" title="§2.1(iv) Uniform Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(iv)</a> </li> <li id="asymptoticapproximationsandexpansions.uniqueness" class="ltx_indexentry"> uniqueness <a href=".././2.1#iii" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(iii)</a> </li> <li id="asymptoticapproximationsandexpansions.viaconnectionformulas" class="ltx_indexentry"> via connection formulas <a href=".././2.11#ii" title="§2.11(ii) Connection Formulas ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(ii)</a> </li> </ul> </li> <li id="asymptoticapproximationsofintegrals" class="ltx_indexentry"> asymptotic approximations of integrals <a href=".././2.2#Px1" title="Example ‣ §2.2 Transcendental Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.2</a>—<a href=".././2.6#iv" title="§2.6(iv) Regularization ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.6(iv)</a> <ul class="ltx_indexlist"> <li id="asymptoticapproximationsofintegrals.Bleisteinsmethod" class="ltx_indexentry"> Bleistein’s method <a href=".././2.3#v" title="§2.3(v) Coalescing Peak and Endpoint: Bleistein’s Method ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.3(v)</a> </li> <li id="asymptoticapproximationsofintegrals.ChesterFriedmanUrsellmethod" class="ltx_indexentry"> Chester–Friedman–Ursell method <a href=".././2.4#v" title="§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.4(v)</a> </li> <li id="asymptoticapproximationsofintegrals.coalescingcriticalpoints" class="ltx_indexentry"> coalescing critical points <a href=".././2.4#v" title="§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.4(v)</a>, <a href=".././2.4#vi" title="§2.4(vi) Other Coalescing Critical Points ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.4(vi)</a> </li> <li id="asymptoticapproximationsofintegrals.coalescingpeakandendpoint" class="ltx_indexentry"> coalescing peak and endpoint <a href=".././2.3#v" title="§2.3(v) Coalescing Peak and Endpoint: Bleistein’s Method ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.3(v)</a> </li> <li id="asymptoticapproximationsofintegrals.coalescingsaddlepoints" class="ltx_indexentry"> coalescing saddle points <a href=".././2.4#v" title="§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.4(v)</a> </li> <li id="asymptoticapproximationsofintegrals.distributionalmethods" class="ltx_indexentry"> distributional methods <a href=".././2.6" title="§2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.6</a>—<a href=".././2.6#iv" title="§2.6(iv) Regularization ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.6(iv)</a> </li> <li id="asymptoticapproximationsofintegrals.Fourierintegrals" class="ltx_indexentry"> Fourier integrals <a href=".././2.3#i.p3" title="§2.3(i) Integration by Parts ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.3(i)</a> </li> <li id="asymptoticapproximationsofintegrals.Haarsmethod" class="ltx_indexentry"> Haar’s method <a href=".././2.4#ii" title="§2.4(ii) Inverse Laplace Transforms ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.4(ii)</a> </li> <li id="asymptoticapproximationsofintegrals.integrationbyparts" class="ltx_indexentry"> integration by parts <a href=".././2.3#i" title="§2.3(i) Integration by Parts ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.3(i)</a> </li> <li id="asymptoticapproximationsofintegrals.inverseLaplacetransforms" class="ltx_indexentry"> inverse Laplace transforms <a href=".././2.4#i" title="§2.4(i) Watson’s Lemma ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.4(i)</a>—<a href=".././2.4#ii" title="§2.4(ii) Inverse Laplace Transforms ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.4(ii)</a> </li> <li id="asymptoticapproximationsofintegrals.Laplacetransforms" class="ltx_indexentry"> Laplace transforms <a href=".././2.3#i.p1" title="§2.3(i) Integration by Parts ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.3(i)</a> </li> <li id="asymptoticapproximationsofintegrals.Laplacesmethod" class="ltx_indexentry"> Laplace’s method <a href=".././2.3#iii" title="§2.3(iii) Laplace’s Method ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.3(iii)</a>—<a href=".././2.4#iii" title="§2.4(iii) Laplace’s Method ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.4(iii)</a> </li> <li id="asymptoticapproximationsofintegrals.Mellintransform" class="ltx_indexentry"> Mellin transform <a href=".././2.3#vi" title="§2.3(vi) Asymptotics of Mellin Transforms ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.3(vi)</a> </li> <li id="asymptoticapproximationsofintegrals.Mellintransformmethods" class="ltx_indexentry"> Mellin transform methods <a href=".././2.5" title="§2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.5</a> <ul class="ltx_indexlist"> <li id="asymptoticapproximationsofintegrals.Mellintransformmethods.extensions" class="ltx_indexentry"> extensions <a href=".././2.5#ii" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.5(ii)</a>—<a href=".././2.5#ii" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.5(ii)</a> </li> </ul> </li> <li id="asymptoticapproximationsofintegrals.methodofstationaryphase" class="ltx_indexentry"> method of stationary phase <a href=".././2.3#iv" title="§2.3(iv) Method of Stationary Phase ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.3(iv)</a> <ul class="ltx_indexlist"> <li id="asymptoticapproximationsofintegrals.methodofstationaryphase.extensions" class="ltx_indexentry"> extensions <a href=".././2.3#iv.p5" title="§2.3(iv) Method of Stationary Phase ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.3(iv)</a> </li> </ul> </li> <li id="asymptoticapproximationsofintegrals.methodofsteepestdescents" class="ltx_indexentry"> method of steepest descents <a href=".././2.4#iv.p2" title="§2.4(iv) Saddle Points ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.4(iv)</a> </li> <li id="asymptoticapproximationsofintegrals.multidimensionalintegrals" class="ltx_indexentry"> multidimensional integrals <a href=".././2.5#ii" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.5(ii)</a> </li> <li id="asymptoticapproximationsofintegrals.Stieltjestransforms" class="ltx_indexentry"> Stieltjes transforms <a href=".././2.6#ii" title="§2.6(ii) Stieltjes Transform ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.6(ii)</a>—<a href=".././2.6#ii" title="§2.6(ii) Stieltjes Transform ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.6(ii)</a> <ul class="ltx_indexlist"> <li id="asymptoticapproximationsofintegrals.Stieltjestransforms.generalized" class="ltx_indexentry"> generalized <a href=".././2.6#ii.p7" title="§2.6(ii) Stieltjes Transform ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.6(ii)</a> </li> </ul> </li> <li id="asymptoticapproximationsofintegrals.Watsonslemma" class="ltx_indexentry"> Watson’s lemma <a href=".././2.3#ii" title="§2.3(ii) Watson’s Lemma ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.3(ii)</a>, <a href=".././2.4#i" title="§2.4(i) Watson’s Lemma ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.4(i)</a> <ul class="ltx_indexlist"> <li id="asymptoticapproximationsofintegrals.Watsonslemma.generalized" class="ltx_indexentry"> generalized <a href=".././2.3#ii" title="§2.3(ii) Watson’s Lemma ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.3(ii)</a> </li> </ul> </li> </ul> </li> <li id="asymptoticapproximationsofsumsandsequences" class="ltx_indexentry"> asymptotic approximations of sums and sequences <a href=".././2.10" title="§2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.10</a>—<a href=".././2.10#Px4" title="Example ‣ §2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.10(iv)</a> <ul class="ltx_indexlist"> <li id="asymptoticapproximationsofsumsandsequences.AbelPlanaformula" class="ltx_indexentry"> Abel–Plana formula <a href=".././2.10#i.p2" title="§2.10(i) Euler–Maclaurin Formula ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.10(i)</a>—<a href=".././2.10#i.p2" title="§2.10(i) Euler–Maclaurin Formula ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.10(i)</a> </li> <li id="asymptoticapproximationsofsumsandsequences.Darbouxsmethod" class="ltx_indexentry"> Darboux’s method <a href=".././2.10#iv" title="§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.10(iv)</a>—<a href=".././2.10#Px4" title="Example ‣ §2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.10(iv)</a> </li> <li id="asymptoticapproximationsofsumsandsequences.entirefunctions" class="ltx_indexentry"> entire functions <a href=".././2.10#iii" title="§2.10(iii) Asymptotic Expansions of Entire Functions ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.10(iii)</a> </li> <li id="asymptoticapproximationsofsumsandsequences.EulerMaclaurinformula" class="ltx_indexentry"> Euler–Maclaurin formula <a href=".././2.10#i" title="§2.10(i) Euler–Maclaurin Formula ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.10(i)</a>—<a href=".././2.10#Px1.p5" title="Example ‣ §2.10(i) Euler–Maclaurin Formula ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.10(i)</a> </li> <li id="asymptoticapproximationsofsumsandsequences.summationbyparts" class="ltx_indexentry"> summation by parts <a href=".././2.10#ii" title="§2.10(ii) Summation by Parts ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.10(ii)</a> </li> </ul> </li> <li id="asymptoticscaleorsequence" class="ltx_indexentry"> asymptotic scale or sequence <a href=".././2.1#v.p1" title="§2.1(v) Generalized Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.1(v)</a> </li> <li id="asymptoticsolutionsofdifferenceequations" class="ltx_indexentry"> asymptotic solutions of difference equations <a href=".././2.9" title="§2.9 Difference Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.9</a>—<a href=".././2.9#iii.p4" title="§2.9(iii) Other Approximations ‣ §2.9 Difference Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.9(iii)</a> <ul class="ltx_indexlist"> <li id="asymptoticsolutionsofdifferenceequations.characteristicequation" class="ltx_indexentry"> characteristic equation <a href=".././2.9#i.p2" title="§2.9(i) Distinct Characteristic Values ‣ §2.9 Difference Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.9(i)</a> </li> <li id="asymptoticsolutionsofdifferenceequations.coincidentcharacteristicvalues" class="ltx_indexentry"> coincident characteristic values <a href=".././2.9#ii" title="§2.9(ii) Coincident Characteristic Values ‣ §2.9 Difference Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.9(ii)</a> </li> <li id="asymptoticsolutionsofdifferenceequations.LiouvilleGreenorWKBJtypeapproximations" class="ltx_indexentry"> Liouville–Green (or WKBJ) type approximations <a href=".././2.9#iii" title="§2.9(iii) Other Approximations ‣ §2.9 Difference Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.9(iii)</a> </li> <li id="asymptoticsolutionsofdifferenceequations.transitionpoints" class="ltx_indexentry"> transition points <a href=".././2.9#iii.p2" title="§2.9(iii) Other Approximations ‣ §2.9 Difference Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.9(iii)</a> </li> <li id="asymptoticsolutionsofdifferenceequations.turningpoints" class="ltx_indexentry"> turning points <a href=".././2.9#iii.p2" title="§2.9(iii) Other Approximations ‣ §2.9 Difference Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.9(iii)</a> </li> <li id="asymptoticsolutionsofdifferenceequations.withaparameter" class="ltx_indexentry"> with a parameter <a href=".././2.9#ii" title="§2.9(ii) Coincident Characteristic Values ‣ §2.9 Difference Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.9(ii)</a>—<a href=".././2.9#iii.p4" title="§2.9(iii) Other Approximations ‣ §2.9 Difference Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.9(iii)</a> </li> </ul> </li> <li id="asymptoticsolutionsofdifferentialequations" class="ltx_indexentry"> asymptotic solutions of differential equations <a href=".././2.6#iv" title="§2.6(iv) Regularization ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.6(iv)</a>—<a href=".././2.8#vi.p8" title="§2.8(vi) Coalescing Transition Points ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(vi)</a> <ul class="ltx_indexlist"> <li id="asymptoticsolutionsofdifferentialequations.characteristicequation" class="ltx_indexentry"> characteristic equation <a href=".././2.7#ii.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.7(ii)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.coincidentcharacteristicvalues" class="ltx_indexentry"> coincident characteristic values <a href=".././2.7#ii" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.7(ii)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.errorcontrolfunction" class="ltx_indexentry"> error-control function <a href=".././2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.7(iii)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.Fabrystransformation" class="ltx_indexentry"> Fabry’s transformation <a href=".././2.7#ii" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.7(ii)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.irregularsingularitiesofrank1" class="ltx_indexentry"> irregular singularities of rank 1 <a href=".././2.7#ii" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.7(ii)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.LiouvilleGreenor" class="ltx_indexentry"> Liouville–Green (or WKBJ) approximations <a href=".././2.7#iii" title="§2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.7(iii)</a>—<a href=".././2.7#Px2" title="Example ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.7(iii)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.LiouvilleGreenapproximationtheorem" class="ltx_indexentry"> Liouville–Green approximation theorem <a href=".././2.7#iii" title="§2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.7(iii)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.numericallysatisfactorysolutions" class="ltx_indexentry"> numerically satisfactory solutions <a href=".././2.7#iv" title="§2.7(iv) Numerically Satisfactory Solutions ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.7(iv)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.resurgence" class="ltx_indexentry"> resurgence <a href=".././2.11#v" title="§2.11(v) Exponentially-Improved Expansions (continued) ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.11(v)</a>, <a href=".././2.7#ii.p5" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.7(ii)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.withaparameter" class="ltx_indexentry"> with a parameter <a href=".././2.8" title="§2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8</a>—<a href=".././2.8#vi.p8" title="§2.8(vi) Coalescing Transition Points ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(vi)</a> <ul class="ltx_indexlist"> <li id="asymptoticsolutionsofdifferentialequations.withaparameter.classificationofcases" class="ltx_indexentry"> classification of cases <a href=".././2.8#i" title="§2.8(i) Classification of Cases ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(i)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.withaparameter.coalescingtransitionpoints" class="ltx_indexentry"> coalescing transition points <a href=".././2.8#vi" title="§2.8(vi) Coalescing Transition Points ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(vi)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.withaparameter.connectionformulasacrosstransitionpoints" class="ltx_indexentry"> connection formulas across transition points <a href=".././2.8#v" title="§2.8(v) Multiple and Fractional Turning Points ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(v)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.withaparameter.intermsofAiryfunctions" class="ltx_indexentry"> in terms of Airy functions <a href=".././2.8#iii" title="§2.8(iii) Case II: Simple Turning Point ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(iii)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.withaparameter.intermsofBesselfunctionsoffixedorder" class="ltx_indexentry"> in terms of Bessel functions of fixed order <a href=".././2.8#iv" title="§2.8(iv) Case III: Simple Pole ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(iv)</a>—<a href=".././2.8#iv" title="§2.8(iv) Case III: Simple Pole ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(iv)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.withaparameter.intermsofBesselfunctionsofvariableorder" class="ltx_indexentry"> in terms of Bessel functions of variable order <a href=".././2.8#vi" title="§2.8(vi) Coalescing Transition Points ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(vi)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.withaparameter.intermsofelementaryfunctions" class="ltx_indexentry"> in terms of elementary functions <a href=".././2.8#ii.p1" title="§2.8(ii) Case I: No Transition Points ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(ii)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.withaparameter.Liouvilletransformation" class="ltx_indexentry"> Liouville transformation <a href=".././2.8#i" title="§2.8(i) Classification of Cases ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(i)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.withaparameter.transitionpoints" class="ltx_indexentry"> transition points <a href=".././2.8#i.p1" title="§2.8(i) Classification of Cases ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(i)</a> </li> <li id="asymptoticsolutionsofdifferentialequations.withaparameter.turningpoints" class="ltx_indexentry"> turning points <a href=".././2.8#i.p1" title="§2.8(i) Classification of Cases ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.8(i)</a> </li> </ul> </li> </ul> </li> <li id="asymptoticsolutionsoftranscendentalequations" class="ltx_indexentry"> asymptotic solutions of transcendental equations <a href=".././2.2" title="§2.2 Transcendental Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.2</a> <ul class="ltx_indexlist"> <li id="asymptoticsolutionsoftranscendentalequations.Lagrangesformula" class="ltx_indexentry"> Lagrange’s formula <a href=".././2.2#Px1.p2" title="Example ‣ §2.2 Transcendental Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref">§2.2</a> </li> </ul> </li> <li id="atomicphotoionization" class="ltx_indexentry"> atomic photo-ionization <ul class="ltx_indexlist"> <li id="atomicphotoionization.Coulombfunctions" class="ltx_indexentry"> Coulomb functions <a href=".././33.22#Px1" title="𝗄 Scaling ‣ §33.22(ii) Definitions of Variables ‣ §33.22 Particle Scattering and Atomic and Molecular Spectra ‣ Physical Applications ‣ Chapter 33 Coulomb Functions" class="ltx_ref">§33.22(ii)</a> </li> </ul> </li> <li id="atomicphysics" class="ltx_indexentry"> atomic physics <ul class="ltx_indexlist"> <li id="atomicphysics.Coulombfunctions" class="ltx_indexentry"> Coulomb functions <a href=".././33.22#Px3" title="i⁢𝗄 Scaling ‣ §33.22(ii) Definitions of Variables ‣ §33.22 Particle Scattering and Atomic and Molecular Spectra ‣ Physical Applications ‣ Chapter 33 Coulomb Functions" class="ltx_ref">§33.22(ii)</a> </li> <li id="atomicphysics.errorfunctions" class="ltx_indexentry"> error functions <a href=".././7.21#p3" title="§7.21 Physical Applications ‣ Applications ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref">§7.21</a> </li> </ul> </li> <li id="atomicspectra" class="ltx_indexentry"> atomic spectra <ul class="ltx_indexlist"> <li id="atomicspectra.Coulombfunctions" class="ltx_indexentry"> Coulomb functions <a href=".././33.22#Px1" title="𝗄 Scaling ‣ §33.22(ii) Definitions of Variables ‣ §33.22 Particle Scattering and Atomic and Molecular Spectra ‣ Physical Applications ‣ Chapter 33 Coulomb Functions" class="ltx_ref">§33.22(ii)</a> </li> </ul> </li> <li id="atomicspectroscopy" class="ltx_indexentry"> atomic spectroscopy <ul class="ltx_indexlist"> <li id="atomicspectroscopy.3j6j9jsymbols" class="ltx_indexentry"> <math class="ltx_Math" altimg="m2.png" altimg-height="21px" altimg-valign="-6px" altimg-width="83px" alttext="\mathit{3j},\mathit{6j},\mathit{9j}" display="inline"><mrow><mrow><mn class="ltx_mathvariant_italic" href=".././34.2#E4" mathvariant="italic" title="3⁢𝑗 symbol">3</mn><mo href=".././34.2#E4" title="3⁢𝑗 symbol">⁢</mo><mi href=".././34.2#E4" title="3⁢𝑗 symbol">j</mi></mrow><mo>,</mo><mrow><mn class="ltx_mathvariant_italic" href=".././34.4#E1" mathvariant="italic" title="6⁢𝑗 symbol">6</mn><mo href=".././34.4#E1" title="6⁢𝑗 symbol">⁢</mo><mi href=".././34.4#E1" title="6⁢𝑗 symbol">j</mi></mrow><mo>,</mo><mrow><mn class="ltx_mathvariant_italic" href=".././34.6#E1" mathvariant="italic" title="9⁢𝑗 symbol">9</mn><mo href=".././34.6#E1" title="9⁢𝑗 symbol">⁢</mo><mi href=".././34.6#E1" title="9⁢𝑗 symbol">j</mi></mrow></mrow></math> symbols <a href=".././34.12" title="§34.12 Physical Applications ‣ Applications ‣ Chapter 34 3⁢𝑗,6⁢𝑗,9⁢𝑗 Symbols" class="ltx_ref">§34.12</a> </li> </ul> </li> <li id="attractivepotentials" class="ltx_indexentry"> attractive potentials <ul class="ltx_indexlist"> <li id="attractivepotentials.Coulombfunctions" class="ltx_indexentry"> Coulomb functions <a href=".././33.22#Px1.tab1" title="𝗄 Scaling ‣ §33.22(ii) Definitions of Variables ‣ §33.22 Particle Scattering and Atomic and Molecular Spectra ‣ Physical Applications ‣ Chapter 33 Coulomb Functions" class="ltx_ref">§33.22(ii)</a>, <a href=".././33.22#Px2.tab1" title="𝑍 Scaling ‣ §33.22(ii) Definitions of Variables ‣ §33.22 Particle Scattering and Atomic and Molecular Spectra ‣ Physical Applications ‣ Chapter 33 Coulomb Functions" class="ltx_ref">§33.22(ii)</a>, <a href=".././33.22#Px3.tab1" title="i⁢𝗄 Scaling ‣ §33.22(ii) Definitions of Variables ‣ §33.22 Particle Scattering and Atomic and Molecular Spectra ‣ Physical Applications ‣ Chapter 33 Coulomb Functions" class="ltx_ref">§33.22(ii)</a> </li> </ul> </li> <li id="auxiliaryfunctionsforFresnelintegrals" class="ltx_indexentry"> auxiliary functions for Fresnel integrals <ul class="ltx_indexlist"> <li id="auxiliaryfunctionsforFresnelintegrals.approximations" class="ltx_indexentry"> approximations <a href=".././7.24#i" title="§7.24(i) Approximations in Terms of Elementary Functions ‣ §7.24 Approximations ‣ Computation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref">§7.24(i)</a> </li> <li id="auxiliaryfunctionsforFresnelintegrals.asymptoticexpansions" class="ltx_indexentry"> asymptotic expansions <a href=".././7.12#ii" title="§7.12(ii) Fresnel Integrals ‣ §7.12 Asymptotic Expansions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref">§7.12(ii)</a> </li> <li id="auxiliaryfunctionsforFresnelintegrals.computation" class="ltx_indexentry"> computation <a href=".././7.22#i" title="§7.22(i) Main Functions ‣ §7.22 Methods of Computation ‣ Computation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref">§7.22(i)</a> </li> <li id="auxiliaryfunctionsforFresnelintegrals.definitions" class="ltx_indexentry"> definitions <a href=".././7.2#iv" title="§7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref">§7.2(iv)</a> </li> <li id="auxiliaryfunctionsforFresnelintegrals.derivatives" class="ltx_indexentry"> derivatives <a href=".././7.10#p1" title="§7.10 Derivatives ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref">§7.10</a> </li> <li id="auxiliaryfunctionsforFresnelintegrals.integralrepresentations" class="ltx_indexentry"> integral representations <a href=".././7.7#ii" title="§7.7(ii) Auxiliary Functions ‣ §7.7 Integral Representations ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref">§7.7(ii)</a> <ul class="ltx_indexlist"> <li id="auxiliaryfunctionsforFresnelintegrals.integralrepresentations.MellinBarnesintegrals" class="ltx_indexentry"> Mellin–Barnes integrals <a href=".././7.7#Px1" title="Mellin–Barnes Integrals ‣ §7.7(ii) Auxiliary Functions ‣ §7.7 Integral Representations ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref">§7.7(ii)</a> </li> </ul> </li> <li id="auxiliaryfunctionsforFresnelintegrals.symmetry" class="ltx_indexentry"> symmetry <a href=".././7.4#p1" title="§7.4 Symmetry ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref">§7.4</a> </li> </ul> </li> <li id="auxiliaryfunctionsforsineandcosineintegrals" class="ltx_indexentry"> auxiliary functions for sine and cosine integrals <ul class="ltx_indexlist"> <li id="auxiliaryfunctionsforsineandcosineintegrals.analyticcontinuation" class="ltx_indexentry"> analytic continuation <a href=".././6.4#p2" title="§6.4 Analytic Continuation ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref">§6.4</a> </li> <li id="auxiliaryfunctionsforsineandcosineintegrals.approximations" class="ltx_indexentry"> approximations <a href=".././6.20#I1.i4.p1" title="In §6.20(i) Approximations in Terms of Elementary Functions ‣ §6.20 Approximations ‣ Computation ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref">4th item</a> </li> <li id="auxiliaryfunctionsforsineandcosineintegrals.asymptoticexpansions" class="ltx_indexentry"> asymptotic expansions <a href=".././6.12#ii.p1" title="§6.12(ii) Sine and Cosine Integrals ‣ §6.12 Asymptotic Expansions ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref">§6.12(ii)</a> <ul class="ltx_indexlist"> <li id="auxiliaryfunctionsforsineandcosineintegrals.asymptoticexpansions.exponentiallyimproved" class="ltx_indexentry"> exponentially-improved <a href=".././6.12#ii.p3" title="§6.12(ii) Sine and Cosine Integrals ‣ §6.12 Asymptotic Expansions ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref">§6.12(ii)</a> </li> </ul> </li> <li id="auxiliaryfunctionsforsineandcosineintegrals.Chebyshevseriesexpansions" class="ltx_indexentry"> Chebyshev-series expansions <a href=".././6.20#I2.i5.p1" title="In §6.20(ii) Expansions in Chebyshev Series ‣ §6.20 Approximations ‣ Computation ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref">5th item</a> </li> <li id="auxiliaryfunctionsforsineandcosineintegrals.computation" class="ltx_indexentry"> computation <a href=".././6.18#ii" title="§6.18(ii) Auxiliary Functions ‣ §6.18 Methods of Computation ‣ Computation ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref">§6.18(ii)</a> </li> <li id="auxiliaryfunctionsforsineandcosineintegrals.definition" class="ltx_indexentry"> definition <a href=".././6.2#iii" title="§6.2(iii) Auxiliary Functions ‣ §6.2 Definitions and Interrelations ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref">§6.2(iii)</a> </li> <li id="auxiliaryfunctionsforsineandcosineintegrals.integralrepresentations" class="ltx_indexentry"> integral representations <a href=".././6.7#iii" title="§6.7(iii) Auxiliary Functions ‣ §6.7 Integral Representations ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref">§6.7(iii)</a> </li> <li id="auxiliaryfunctionsforsineandcosineintegrals.principalvalues" class="ltx_indexentry"> principal values <a href=".././6.4#p3" title="§6.4 Analytic Continuation ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref">§6.4</a> </li> <li id="auxiliaryfunctionsforsineandcosineintegrals.relationtoconfluenthypergeometricfunctions" class="ltx_indexentry"> relation to confluent hypergeometric functions <a href=".././6.11#Px2.p1" title="Confluent Hypergeometric Function ‣ §6.11 Relations to Other Functions ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref">§6.11</a> </li> <li id="auxiliaryfunctionsforsineandcosineintegrals.tables" class="ltx_indexentry"> tables <a href=".././6.19#ii" title="§6.19(ii) Real Variables ‣ §6.19 Tables ‣ Computation ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref">§6.19(ii)</a> </li> </ul> </li> <li id="axiallysymmetricpotentialtheory" class="ltx_indexentry"> axially symmetric potential theory <a href=".././19.18#ii.p4" title="§19.18(ii) Differential Equations ‣ §19.18 Derivatives and Differential Equations ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref">§19.18(ii)</a> </li> </ul> </div> </div> </section> </div> <div class="ltx_page_footer"> <div class="ltx_siblings"> <a href=".././bib/Z" title="In Bibliography" class="ltx_ref" rel="prev">Bibliography Z</a><a href=".././idx/B" title="In Index" class="ltx_ref" rel="next">Index B</a> </div> <div class="ltx_footer_links ltx_centering"> <a href=".././about/notices">© 2010–2025 NIST</a> / <a href=".././about/notices#S2">Disclaimer</a> / <a href="mailto:DLMF-feedback@nist.gov">Feedback</a>; Version 1.2.4; Release date 2025-03-15.</div> <div class="ltx_nist_logo"><a href="http://www.nist.gov/"><img src=".././style/nistidc.png" width="500" alt="NIST"></a></div> <div class="ltx_nist_links ltx_align_right"> <a href="https://www.nist.gov/privacy-policy">Site Privacy</a> <a href="https://www.nist.gov/oism/accessibility">Accessibility</a> <a href="https://www.nist.gov/privacy">Privacy Program</a> <a href="https://www.nist.gov/oism/copyrights">Copyrights</a> <a href="https://www.commerce.gov/vulnerability-disclosure-policy">Vulnerability Disclosure</a> <a href="https://www.nist.gov/no-fear-act-policy">No Fear Act Policy</a> <a href="https://www.nist.gov/foia">FOIA</a> <a href="https://www.nist.gov/environmental-policy-statement">Environmental Policy</a> <a href="https://www.nist.gov/summary-report-scientific-integrity">Scientific Integrity</a> <a href="https://www.nist.gov/nist-information-quality-standards">Information Quality Standards</a> <a href="https://www.commerce.gov/">Commerce.gov</a> <a href="http://www.science.gov/">Science.gov</a> <a href="http://www.usa.gov/">USA.gov</a> </div> </div> </div> </body> </html>