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<!doctype html> <html lang="en" class="calculusandanalysis"> <head> <title>Elliptic Integral -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Elliptic Integral" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="An elliptic integral is an integral of the form int(A(x)+B(x)sqrt(S(x)))/(C(x)+D(x)sqrt(S(x)))dx, (1) or int(A(x)dx)/(B(x)sqrt(S(x))), (2) where A(x), B(x), C(x), and D(x) are polynomials in x, and S(x) is a polynomial of degree 3 or 4. Stated more simply, an elliptic integral is an integral of the form intR(w,x)dx, (3) where R(w,x) is a rational function of x and w, w^2 is a function of x that is cubic or quartic in x, R(w,x) contains at least one odd power of w, and w^2 has no..." /> <meta name="description" content="An elliptic integral is an integral of the form int(A(x)+B(x)sqrt(S(x)))/(C(x)+D(x)sqrt(S(x)))dx, (1) or int(A(x)dx)/(B(x)sqrt(S(x))), (2) where A(x), B(x), C(x), and D(x) are polynomials in x, and S(x) is a polynomial of degree 3 or 4. Stated more simply, an elliptic integral is an integral of the form intR(w,x)dx, (3) where R(w,x) is a rational function of x and w, w^2 is a function of x that is cubic or quartic in x, R(w,x) contains at least one odd power of w, and w^2 has no..." /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Calculus and Analysis:Special Functions:Elliptic Integrals" /> <meta name="DC.Subject" scheme="MSC_2000" content="33C" /> <meta name="DC.Subject" scheme="MSC_2000" content="33E" /> <meta name="DC.Rights" content="Copyright 1999-2025 Wolfram Research, Inc. See https://mathworld.wolfram.com/about/terms.html for a full terms of use statement." /> <meta name="DC.Format" scheme="IMT" content="text/html" /> <meta name="DC.Identifier" scheme="URI" content="https://mathworld.wolfram.com/EllipticIntegral.html" /> <meta name="DC.Language" scheme="RFC3066" content="en" /> <meta name="DC.Publisher" content="Wolfram Research, Inc." /> <meta name="DC.Relation.IsPartOf" scheme="URI" content="https://mathworld.wolfram.com/" /> <meta name="DC.Type" scheme="DCMIType" content="Text" /> <meta property="og:image" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_EllipticIntegral.png"> <meta property="og:url" content="https://mathworld.wolfram.com/EllipticIntegral.html"> <meta property="og:type" content="website"> <meta property="og:title" content="Elliptic Integral -- from Wolfram MathWorld"> <meta property="og:description" content="An elliptic integral is an integral of the form int(A(x)+B(x)sqrt(S(x)))/(C(x)+D(x)sqrt(S(x)))dx, (1) or int(A(x)dx)/(B(x)sqrt(S(x))), (2) where A(x), B(x), C(x), and D(x) are polynomials in x, and S(x) is a polynomial of degree 3 or 4. Stated more simply, an elliptic integral is an integral of the form intR(w,x)dx, (3) where R(w,x) is a rational function of x and w, w^2 is a function of x that is cubic or quartic in x, R(w,x) contains at least one odd power of w, and w^2 has no..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Elliptic Integral -- from Wolfram MathWorld"> <meta name="twitter:description" content="An elliptic integral is an integral of the form int(A(x)+B(x)sqrt(S(x)))/(C(x)+D(x)sqrt(S(x)))dx, (1) or int(A(x)dx)/(B(x)sqrt(S(x))), (2) where A(x), B(x), C(x), and D(x) are polynomials in x, and S(x) is a polynomial of degree 3 or 4. Stated more simply, an elliptic integral is an integral of the form intR(w,x)dx, (3) where R(w,x) is a rational function of x and w, w^2 is a function of x that is cubic or quartic in x, R(w,x) contains at least one odd power of w, and w^2 has no..."> <meta name="twitter:image:src" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_EllipticIntegral.png"> <link rel="canonical" href="https://mathworld.wolfram.com/EllipticIntegral.html" /> <meta http-equiv="x-ua-compatible" content="ie=edge"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta charset="utf-8"> <script async src="/common/javascript/analytics.js"></script> <script async src="//www.wolframcdn.com/consent/cookie-consent.js"></script> <script async src="/common/javascript/wal/latest/walLoad.js"></script> <link rel="stylesheet" href="/css/styles.css"> <link rel="preload" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css" as="style" 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href="/topics/SpecialFunctions.html">Special Functions</a> </li> <li> <a href="/topics/EllipticIntegrals.html">Elliptic Integrals</a> </li> </ul></nav>   <h1>Elliptic Integral</h1>  <hr class="margin-t-1-8 margin-b-3-4">   <div class="entry-content"> <p> An elliptic integral is an <a href="/Integral.html">integral</a> <a href="/OftheForm.html">of the form</a> </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation1.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="183" height="52" alt=" int(A(x)+B(x)sqrt(S(x)))/(C(x)+D(x)sqrt(S(x)))dx, " /></td><td align="right" width="3"> <div id="eqn1" class="eqnum"> (1) </div> </td></tr> </table> </div> <p> or </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation2.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="109" height="47" alt=" int(A(x)dx)/(B(x)sqrt(S(x))), " /></td><td align="right" width="3"> <div id="eqn2" class="eqnum"> (2) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/EllipticIntegral/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt="A(x)" />, <img src="/images/equations/EllipticIntegral/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="34" height="21" alt="B(x)" />, <img src="/images/equations/EllipticIntegral/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="36" height="21" alt="C(x)" />, and <img src="/images/equations/EllipticIntegral/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="D(x)" /> are <a href="/Polynomial.html">polynomials</a> in <img src="/images/equations/EllipticIntegral/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="x" />, and <img src="/images/equations/EllipticIntegral/Inline6.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="33" height="21" alt="S(x)" /> is a <a href="/Polynomial.html">polynomial</a> of degree 3 or 4. Stated more simply, an elliptic integral is an integral <a href="/OftheForm.html">of the form</a> </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation3.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="95" height="38" alt=" intR(w,x)dx, " /></td><td align="right" width="3"> <div id="eqn3" class="eqnum"> (3) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/EllipticIntegral/Inline7.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="56" height="21" alt="R(w,x)" /> is a <a href="/RationalFunction.html">rational function</a> of <img src="/images/equations/EllipticIntegral/Inline8.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="x" /> and <img src="/images/equations/EllipticIntegral/Inline9.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="w" />, <img src="/images/equations/EllipticIntegral/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="w^2" /> is a function of <img src="/images/equations/EllipticIntegral/Inline11.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="x" /> that is <a href="/CubicEquation.html">cubic</a> or <a href="/QuarticEquation.html">quartic</a> in <img src="/images/equations/EllipticIntegral/Inline12.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="x" />, <img src="/images/equations/EllipticIntegral/Inline13.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="56" height="21" alt="R(w,x)" /> contains at least one <a href="/OddNumber.html">odd</a> <a href="/Power.html">power</a> of <img src="/images/equations/EllipticIntegral/Inline14.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="w" />, and <img src="/images/equations/EllipticIntegral/Inline15.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="w^2" /> has no repeated factors (Abramowitz and Stegun 1972, p. 589). </p> <p> Elliptic integrals can be viewed as generalizations of the inverse <a href="/Trigonometry.html">trigonometric functions</a> and provide solutions to a wider class of problems. For instance, while the <a href="/ArcLength.html">arc length</a> of a <a href="/Circle.html">circle</a> is given as a simple function of the parameter, computing the <a href="/ArcLength.html">arc length</a> of an <a href="/Ellipse.html">ellipse</a> requires an elliptic integral. Similarly, the position of a pendulum is given by a <a href="/Trigonometry.html">trigonometric function</a> as a function of time for small angle oscillations, but the full solution for arbitrarily large displacements requires the use of elliptic integrals. Many other problems in electromagnetism and gravitation are solved by elliptic integrals. </p> <p> A very useful class of functions known as <a href="/EllipticFunction.html">elliptic functions</a> is obtained by inverting elliptic integrals to obtain generalizations of the trigonometric functions. <a href="/EllipticFunction.html">Elliptic functions</a> (among which the <a href="/JacobiEllipticFunctions.html">Jacobi elliptic functions</a> and <a href="/WeierstrassEllipticFunction.html">Weierstrass elliptic function</a> are the two most common forms) provide a powerful tool for analyzing many deep problems in <a href="/NumberTheory.html">number theory</a>, as well as other areas of mathematics. </p> <p> All elliptic integrals can be written in terms of three "standard" types. To see this, write </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline16.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="52" height="20" alt="R(w,x)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline17.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline18.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="59" height="42" alt="(P(w,x))/(Q(w,x))" /></td><td align="right" width="10"> <div id="eqn4" class="eqnum"> (4) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline19.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline20.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline21.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="144" height="43" alt="(wP(w,x)Q(-w,x))/(wQ(w,x)Q(-w,x))." /></td><td align="right" width="10"> <div id="eqn5" class="eqnum"> (5) </div> </td></tr> </table> </div> <p> But since <img src="/images/equations/EllipticIntegral/Inline22.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="75" height="21" alt="w^2=f(x)" />, </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline23.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="120" height="20" alt="Q(w,x)Q(-w,x)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline24.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline25.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="60" height="21" alt="Q_1(w,x)" /></td><td align="right" width="10"> <div id="eqn6" class="eqnum"> (6) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline26.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline27.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline28.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="75" height="21" alt="Q_1(-w,x)," /></td><td align="right" width="10"> <div id="eqn7" class="eqnum"> (7) </div> </td></tr> </table> </div> <p> then </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline29.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="133" height="20" alt="wP(w,x)Q(-w,x)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline30.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline31.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="361" height="20" alt="A+Bx+Cw+Dx^2+Ewx+Fw^2+Gw^2x+Hw^3x" /></td><td align="right" width="10"> <div id="eqn8" class="eqnum"> (8) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline32.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline33.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline34.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="395" height="23" alt="(A+Bx+Dx^2+Fw^2+Gw^2x)+w(c+Ex+Hw^2x+...)" /></td><td align="right" width="10"> <div id="eqn9" class="eqnum"> (9) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline35.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline36.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline37.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="112" height="20" alt="P_1(x)+wP_2(x)," /></td><td align="right" width="10"> <div id="eqn10" class="eqnum"> (10) </div> </td></tr> </table> </div> <p> so </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline38.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="52" height="20" alt="R(w,x)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline39.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline40.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="114" height="43" alt="(P_1(x)+wP_2(x))/(wQ_1(w))" /></td><td align="right" width="10"> <div id="eqn11" class="eqnum"> (11) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline41.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline42.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline43.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="104" height="39" alt="(R_1(x))/w+R_2(x)." /></td><td align="right" width="10"> <div id="eqn12" class="eqnum"> (12) </div> </td></tr> </table> </div> <p> But any function <img src="/images/equations/EllipticIntegral/Inline44.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="78" height="25" alt="intR_2(x)dx" /> can be evaluated in terms of <a href="/ElementaryFunction.html">elementary functions</a>, so the only portion that need be considered is </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation4.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="88" height="40" alt=" int(R_1(x))/wdx. " /></td><td align="right" width="3"> <div id="eqn13" class="eqnum"> (13) </div> </td></tr> </table> </div> <p> Now, any quartic can be expressed as <img src="/images/equations/EllipticIntegral/Inline45.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="36" height="21" alt="S_1S_2" /> where </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline46.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="20" alt="S_1" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline47.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline48.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="120" height="20" alt="a_1x^2+2b_1x+c_1" /></td><td align="right" width="10"> <div id="eqn14" class="eqnum"> (14) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline49.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="20" alt="S_2" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline50.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline51.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="124" height="20" alt="a_2x^2+2b_2x+c_2." /></td><td align="right" width="10"> <div id="eqn15" class="eqnum"> (15) </div> </td></tr> </table> </div> <p> The <a href="/Coefficient.html">coefficients</a> here are real, since pairs of <a href="/ComplexNumber.html">complex</a> <a href="/Root.html">roots</a> are <a href="/ComplexConjugate.html">complex conjugates</a> </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline52.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="177" height="20" alt="[x-(R+Ii)][x-(R-Ii)]" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline53.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline54.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="248" height="23" alt="x^2+x(-R+Ii-R-Ii)+(R^2-I^2i)" /></td><td align="right" width="10"> <div id="eqn16" class="eqnum"> (16) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline55.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline56.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline57.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="144" height="23" alt="x^2-2Rx+(R^2+I^2)." /></td><td align="right" width="10"> <div id="eqn17" class="eqnum"> (17) </div> </td></tr> </table> </div> <p> If all four <a href="/Root.html">roots</a> are real, they must be arranged so as not to interleave (Whittaker and Watson 1990, p. 514). Now define a quantity <img src="/images/equations/EllipticIntegral/Inline58.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="lambda" /> such that <img src="/images/equations/EllipticIntegral/Inline59.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="64" height="21" alt="S_1-lambdaS_2" /> </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation5.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="290" height="20" alt=" (a_1-lambdaa_2)x^2-(2b_1-2b_2lambda)x+(c_1-lambdac_2) " /></td><td align="right" width="3"> <div id="eqn18" class="eqnum"> (18) </div> </td></tr> </table> </div> <p> is a <a href="/SquareNumber.html">square number</a> and </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation6.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="272" height="24" alt=" 2sqrt((a_1-lambdaa_2)(c_1-lambdac_2))=2(b_1-b_2lambda) " /></td><td align="right" width="3"> <div id="eqn19" class="eqnum"> (19) </div> </td></tr> </table> </div> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation7.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="265" height="20" alt=" (a_1-lambdaa_2)(c_1-lambdac_2)-(b_1-lambdab_2)^2=0. " /></td><td align="right" width="3"> <div id="eqn20" class="eqnum"> (20) </div> </td></tr> </table> </div> <p> Call the <a href="/Root.html">roots</a> of this equation <img src="/images/equations/EllipticIntegral/Inline60.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="lambda_1" /> and <img src="/images/equations/EllipticIntegral/Inline61.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="lambda_2" />, then </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline62.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="64" height="20" alt="S_1-lambda_2S_2" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline63.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline64.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="230" height="33" alt="[sqrt((a_1-lambda_2a_2)x^2)+sqrt(c_1-lambda_2c_2)]^2" /></td><td align="right" width="10"> <div id="eqn21" class="eqnum"> (21) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline65.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline66.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline67.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="209" height="52" alt="(a_1-lambda_2a_2)(x+sqrt((c_1-lambda_2c_2)/(a_1-lambda_2a_2)))" /></td><td align="right" width="10"> <div id="eqn22" class="eqnum"> (22) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline68.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline69.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline70.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="130" height="21" alt="(a_1-lambda_2a_2)(x-beta)^2" /></td><td align="right" width="10"> <div id="eqn23" class="eqnum"> (23) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline71.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="64" height="20" alt="S_1-lambda_1S_2" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline72.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline73.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="230" height="33" alt="[sqrt((a_1-lambda_1a_2)x^2)+sqrt(c_1-lambda_1c_2)]^2" /></td><td align="right" width="10"> <div id="eqn24" class="eqnum"> (24) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline74.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline75.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline76.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="209" height="52" alt="(a_1-lambda_1a_2)(x+sqrt((c_1-lambda_1c_2)/(a_1-lambda_1a_2)))" /></td><td align="right" width="10"> <div id="eqn25" class="eqnum"> (25) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline77.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline78.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline79.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="133" height="20" alt="(a_1-lambda_1a_2)(x-alpha)^2." /></td><td align="right" width="10"> <div id="eqn26" class="eqnum"> (26) </div> </td></tr> </table> </div> <p> Taking (<a href="#eqn25">25</a>)<img src="/images/equations/EllipticIntegral/Inline80.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="-" />(<a href="#eqn26">26</a>) and <img src="/images/equations/EllipticIntegral/Inline81.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="97" height="21" alt="lambda_2(1)-lambda_1(2)" /> gives </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline82.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="74" height="20" alt="S_2(lambda_2-lambda_1)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline83.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline84.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="275" height="21" alt="(a_1-lambda_1a_2)(x-alpha)^2-(a_1-lambda_2a_2)(x-beta)^2" /></td><td align="right" width="10"> <div id="eqn27" class="eqnum"> (27) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline85.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="74" height="20" alt="S_1(lambda_2-lambda_1)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline86.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline87.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="315" height="21" alt="lambda_2(a_1-lambda_1a_2)(x-alpha)^2-lambda_1(a_1-lambda_2a_2)(x-beta)^2." /></td><td align="right" width="10"> <div id="eqn28" class="eqnum"> (28) </div> </td></tr> </table> </div> <p> Solving gives </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline88.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="20" alt="S_1" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline89.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline90.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="267" height="41" alt="(a_1-lambda_1a_2)/(lambda_2-lambda_1)(x-alpha)^2-(a_1-lambda_2a_2)/(lambda_2-lambda_1)(x-beta)^2" /></td><td align="right" width="10"> <div id="eqn29" class="eqnum"> (29) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline91.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline92.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline93.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="159" height="21" alt="A_1(x-alpha)^2+B_1(x-beta)^2" /></td><td align="right" width="10"> <div id="eqn30" class="eqnum"> (30) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline94.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="20" alt="S_2" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline95.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline96.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="323" height="41" alt="(lambda_2(a_1-lambda_1a_2))/(lambda_2-lambda_1)(x-alpha)^2-(lambda_1(a_1-lambda_2a_2))/(lambda_2-lambda_1)(x-beta)^2" /></td><td align="right" width="10"> <div id="eqn31" class="eqnum"> (31) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline97.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline98.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline99.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="163" height="21" alt="A_2(x-alpha)^2+B_2(x-beta)^2," /></td><td align="right" width="10"> <div id="eqn32" class="eqnum"> (32) </div> </td></tr> </table> </div> <p> so we have </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation8.svg" data-src-small="/images/equations/EllipticIntegral/NumberedEquation8_400.svg" data-src-default="/images/equations/EllipticIntegral/NumberedEquation8.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="438" data-big="438 23" data-small="270 50" border="0" alt=" w^2=S_1S_2=[A_1(x-alpha)^2+B_1(x-beta)^2][A^2(x-alpha)^2+B^2(x-beta)^2]. " /></td><td align="right" width="3"> <div id="eqn33" class="eqnum"> (33) </div> </td></tr> </table> </div> <p> Now let </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline100.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="5" height="20" alt="t" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline101.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline102.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="40" height="38" alt="(x-alpha)/(x-beta)" /></td><td align="right" width="10"> <div id="eqn34" class="eqnum"> (34) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline103.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="17" height="20" alt="dt" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline104.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline105.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="211" height="23" alt="[(x-beta)^(-1)-(x-alpha)(x-beta)^(-2)]dx" /></td><td align="right" width="10"> <div id="eqn35" class="eqnum"> (35) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline106.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline107.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline108.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="132" height="44" alt="((x-beta)-(x-alpha))/((x-beta)^2)dx" /></td><td align="right" width="10"> <div id="eqn36" class="eqnum"> (36) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline109.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline110.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline111.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="84" height="44" alt="(alpha-beta)/((x-beta)^2)dx," /></td><td align="right" width="10"> <div id="eqn37" class="eqnum"> (37) </div> </td></tr> </table> </div> <p> so </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline112.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="18" height="20" alt="w^2" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline113.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline114.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="302" height="43" alt="(x-beta)^4[A_1((x-alpha)/(x-beta))^2+B_1][A_2((x-alpha)/(x-beta))+B_2]" /></td><td align="right" width="10"> <div id="eqn38" class="eqnum"> (38) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline115.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline116.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline117.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="215" height="23" alt="(x-beta)^4(A_1t^2+B_1)(A_2t^2+B_2)," /></td><td align="right" width="10"> <div id="eqn39" class="eqnum"> (39) </div> </td></tr> </table> </div> <p> and </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline118.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="11" height="20" alt="w" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline119.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline120.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="230" height="30" alt="(x-beta)^2sqrt((A_1t^2+B_1)(A_2t^2+B_2))" /></td><td align="right" width="10"> <div id="eqn40" class="eqnum"> (40) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline121.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="26" height="39" alt="(dx)/w" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline122.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline123.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="327" height="56" alt="[((x-beta)^2)/(alpha-beta)dt]1/((x-beta)^2sqrt((A_1t^2+B_1)(A_2t^2+B_2)))" /></td><td align="right" width="10"> <div id="eqn41" class="eqnum"> (41) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline124.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline125.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline126.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="233" height="53" alt="(dt)/((alpha-beta)sqrt((A_1t^2+B_1)(A_2t^2+B_2)))." /></td><td align="right" width="10"> <div id="eqn42" class="eqnum"> (42) </div> </td></tr> </table> </div> <p> Now let </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation9.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="103" height="42" alt=" R_3(t)=(R_1(x))/(alpha-beta), " /></td><td align="right" width="3"> <div id="eqn43" class="eqnum"> (43) </div> </td></tr> </table> </div> <p> so </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation10.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="322" height="60" alt=" int(R_1(x)dx)/w=int(R_3(t)dt)/(sqrt((A_1t^2+B_1)(A_2t^2+B_2))). " /></td><td align="right" width="3"> <div id="eqn44" class="eqnum"> (44) </div> </td></tr> </table> </div> <p> Rewriting the <a href="/EvenNumber.html">even</a> and <a href="/OddNumber.html">odd</a> parts </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline127.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="99" height="20" alt="R_3(t)+R_3(-t)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline128.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline129.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="55" height="23" alt="2R_4(t^2)" /></td><td align="right" width="10"> <div id="eqn45" class="eqnum"> (45) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline130.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="99" height="20" alt="R_3(t)-R_3(-t)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline131.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline132.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="67" height="23" alt="2tR_5(t^2)," /></td><td align="right" width="10"> <div id="eqn46" class="eqnum"> (46) </div> </td></tr> </table> </div> <p> gives </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline133.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="36" height="20" alt="R_3(t)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline134.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline135.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="106" height="26" alt="1/2(R_(even)-R_(odd))" /></td><td align="right" width="10"> <div id="eqn47" class="eqnum"> (47) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline136.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline137.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline138.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="114" height="23" alt="R_4(t^2)+tR_5(t^2)," /></td><td align="right" width="10"> <div id="eqn48" class="eqnum"> (48) </div> </td></tr> </table> </div> <p> so we have </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation11.svg" data-src-small="/images/equations/EllipticIntegral/NumberedEquation11_400.svg" data-src-default="/images/equations/EllipticIntegral/NumberedEquation11.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="543" data-big="543 60" data-small="331 132" border="0" alt=" int(R_1(x)dx)/w=int(R_4(t^2)dt)/(sqrt((A_1t^2+B_1)(A_2t^2+B_2)))+int(R_5(t^2)tdt)/(sqrt((A_1t^2+B_1)(A_2t^2+B_2))). " /></td><td align="right" width="3"> <div id="eqn49" class="eqnum"> (49) </div> </td></tr> </table> </div> <p> Letting </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline139.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="9" height="20" alt="u" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline140.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline141.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="t^2" /></td><td align="right" width="10"> <div id="eqn50" class="eqnum"> (50) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline142.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="21" height="20" alt="du" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline143.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline144.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="37" height="20" alt="2tdt" /></td><td align="right" width="10"> <div id="eqn51" class="eqnum"> (51) </div> </td></tr> </table> </div> <p> reduces the second integral to </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation12.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="213" height="47" alt=" 1/2int(R_5(u)du)/(sqrt((A_1u+B_1)(A_2u+B_2))), " /></td><td align="right" width="3"> <div id="eqn52" class="eqnum"> (52) </div> </td></tr> </table> </div> <p> which can be evaluated using <a href="/ElementaryFunction.html">elementary functions</a>. The first integral can then be reduced by <a href="/IntegrationbyParts.html">integration by parts</a> to one of the three Legendre elliptic integrals (also called Legendre-Jacobi elliptic integrals), known as incomplete <a href="/EllipticIntegraloftheFirstKind.html">elliptic integrals of the first</a>, <a href="/EllipticIntegraloftheSecondKind.html">second</a>, and <a href="/EllipticIntegraloftheThirdKind.html">third kinds</a>, denoted <img src="/images/equations/EllipticIntegral/Inline145.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="F(phi,k)" />, <img src="/images/equations/EllipticIntegral/Inline146.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="E(phi,k)" />, and <img src="/images/equations/EllipticIntegral/Inline147.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="72" height="21" alt="Pi(n;phi,k)" />, respectively (von Kármán and Biot 1940, Whittaker and Watson 1990, p. 515). If <img src="/images/equations/EllipticIntegral/Inline148.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="59" height="21" alt="phi=pi/2" />, then the integrals are called complete elliptic integrals and are denoted <img src="/images/equations/EllipticIntegral/Inline149.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="K(k)" />, <img src="/images/equations/EllipticIntegral/Inline150.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt="E(k)" />, <img src="/images/equations/EllipticIntegral/Inline151.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="53" height="21" alt="Pi(n;k)" />. </p> <p> Incomplete elliptic integrals are denoted using a <a href="/EllipticModulus.html">elliptic modulus</a> <img src="/images/equations/EllipticIntegral/Inline152.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" />, <a href="/Parameter.html">parameter</a> <img src="/images/equations/EllipticIntegral/Inline153.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="48" height="21" alt="m=k^2" />, or <a href="/ModularAngle.html">modular angle</a> <img src="/images/equations/EllipticIntegral/Inline154.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="78" height="21" alt="alpha=sin^(-1)k" />. An elliptic integral is written <img src="/images/equations/EllipticIntegral/Inline155.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="56" height="24" alt="I(phi|m)" /> when the <a href="/Parameter.html">parameter</a> is used, <img src="/images/equations/EllipticIntegral/Inline156.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="50" height="21" alt="I(phi,k)" /> when the <a href="/EllipticModulus.html">elliptic modulus</a> is used, and <img src="/images/equations/EllipticIntegral/Inline157.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="46" height="21" alt="I(phi\alpha)" /> when the <a href="/ModularAngle.html">modular angle</a> is used. Complete elliptic integrals are defined when <img src="/images/equations/EllipticIntegral/Inline158.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="59" height="21" alt="phi=pi/2" /> and can be expressed using the expansion </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation13.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="311" height="49" alt=" (1-k^2sin^2theta)^(-1/2)=sum_(n=0)^infty((2n-1)!!)/((2n)!!)k^(2n)sin^(2n)theta. " /></td><td align="right" width="3"> <div id="eqn53" class="eqnum"> (53) </div> </td></tr> </table> </div> <p> An elliptic integral in standard form </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation14.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="83" height="48" alt=" int_a^x(dx)/(sqrt(f(x))), " /></td><td align="right" width="3"> <div id="eqn54" class="eqnum"> (54) </div> </td></tr> </table> </div> <p> where </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation15.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="266" height="20" alt=" f(x)=a_4x^4+a_3x^3+a_2x^2+a_1x+a_0, " /></td><td align="right" width="3"> <div id="eqn55" class="eqnum"> (55) </div> </td></tr> </table> </div> <p> can be computed analytically (Whittaker and Watson 1990, p. 453) in terms of the <a href="/WeierstrassEllipticFunction.html">Weierstrass elliptic function</a> with invariants </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline159.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="g_2" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline160.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline161.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="142" height="22" alt="a_0a_4-4a_1a_3+3a_2^2" /></td><td align="right" width="10"> <div id="eqn56" class="eqnum"> (56) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline162.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="g_3" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline163.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline164.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="242" height="22" alt="a_0a_2a_4-2a_1a_2a_3-a_4a_1^2-a_3^2a_0." /></td><td align="right" width="10"> <div id="eqn57" class="eqnum"> (57) </div> </td></tr> </table> </div> <p> If <img src="/images/equations/EllipticIntegral/Inline165.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="44" height="22" alt="a=x_0" /> is a root of <img src="/images/equations/EllipticIntegral/Inline166.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="65" height="21" alt="f(x)=0" />, then the solution is </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation16.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="315" height="28" alt=" x=x_0+1/4f^'(x_0)[P(z;g_2,g_3)-1/(24)f^('')(x_0)]^(-1). " /></td><td align="right" width="3"> <div id="eqn58" class="eqnum"> (58) </div> </td></tr> </table> </div> <p> For an arbitrary lower bound, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation17.svg" data-src-small="/images/equations/EllipticIntegral/NumberedEquation17_400.svg" data-src-default="/images/equations/EllipticIntegral/NumberedEquation17.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="462" data-big="462 61" data-small="305 97" border="0" alt=" x=a+(sqrt(f(a))P^'(z)+1/2f^'(a)[P(z)-1/(24)f^('')(a)]+1/(24)f(a)f^(''')(a))/(2[P(z)-1/(24)f^('')(a)]^2-1/(48)f(a)f^((iv))(a)), " /></td><td align="right" width="3"> <div id="eqn59" class="eqnum"> (59) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/EllipticIntegral/Inline167.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="138" height="22" alt="P(z)=P(z;g_2,g_3)" /> is a <a href="/WeierstrassEllipticFunction.html">Weierstrass elliptic function</a> (Whittaker and Watson 1990, p. 454). </p> <p> A generalized elliptic integral can be defined by the function </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline168.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="51" height="20" alt="T(a,b)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline169.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline170.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="205" height="49" alt="2/piint_0^(pi/2)(dtheta)/(sqrt(a^2cos^2theta+b^2sin^2theta))" /></td><td align="right" width="10"> <div id="eqn60" class="eqnum"> (60) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline171.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline172.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline173.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="199" height="49" alt="2/piint_0^(pi/2)(dtheta)/(costhetasqrt(a^2+b^2tan^2theta))" /></td><td align="right" width="10"> <div id="eqn61" class="eqnum"> (61) </div> </td></tr> </table> </div> <p> (Borwein and Borwein 1987). Now let </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline174.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="5" height="20" alt="t" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline175.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline176.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="43" height="20" alt="btantheta" /></td><td align="right" width="10"> <div id="eqn62" class="eqnum"> (62) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline177.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="17" height="20" alt="dt" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline178.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline179.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="77" height="20" alt="bsec^2thetadtheta." /></td><td align="right" width="10"> <div id="eqn63" class="eqnum"> (63) </div> </td></tr> </table> </div> <p> But </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation18.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="139" height="24" alt=" sectheta=sqrt(1+tan^2theta), " /></td><td align="right" width="3"> <div id="eqn64" class="eqnum"> (64) </div> </td></tr> </table> </div> <p> so </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline180.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="17" height="20" alt="dt" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline181.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline182.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="96" height="39" alt="b/(costheta)secthetadtheta" /></td><td align="right" width="10"> <div id="eqn65" class="eqnum"> (65) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline183.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline184.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline185.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="146" height="39" alt="b/(costheta)sqrt(1+tan^2theta)dtheta" /></td><td align="right" width="10"> <div id="eqn66" class="eqnum"> (66) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline186.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline187.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline188.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="144" height="47" alt="b/(costheta)sqrt(1+(t/b)^2)dtheta" /></td><td align="right" width="10"> <div id="eqn67" class="eqnum"> (67) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline189.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline190.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline191.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="109" height="39" alt="(dtheta)/(costheta)sqrt(b^2+t^2)," /></td><td align="right" width="10"> <div id="eqn68" class="eqnum"> (68) </div> </td></tr> </table> </div> <p> and </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation19.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="132" height="48" alt=" (dtheta)/(costheta)=(dt)/(sqrt(b^2+t^2)), " /></td><td align="right" width="3"> <div id="eqn69" class="eqnum"> (69) </div> </td></tr> </table> </div> <p> and the equation becomes </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline192.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="51" height="20" alt="T(a,b)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline193.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline194.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="179" height="53" alt="2/piint_0^infty(dt)/(sqrt((a^2+t^2)(b^2+t^2)))" /></td><td align="right" width="10"> <div id="eqn70" class="eqnum"> (70) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline195.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline196.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline197.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="183" height="53" alt="1/piint_(-infty)^infty(dt)/(sqrt((a^2+t^2)(b^2+t^2)))." /></td><td align="right" width="10"> <div id="eqn71" class="eqnum"> (71) </div> </td></tr> </table> </div> <p> Now we make the further substitution <img src="/images/equations/EllipticIntegral/Inline198.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="116" height="27" alt="u=1/2(t-ab/t)" />. The differential becomes </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation20.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="162" height="26" alt=" du=1/2(1+ab/t^2)dt, " /></td><td align="right" width="3"> <div id="eqn72" class="eqnum"> (72) </div> </td></tr> </table> </div> <p> but <img src="/images/equations/EllipticIntegral/Inline199.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="105" height="21" alt="2u=t-ab/t" />, so </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation21.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="131" height="23" alt=" 2u/t=1-ab/t^2 " /></td><td align="right" width="3"> <div id="eqn73" class="eqnum"> (73) </div> </td></tr> </table> </div> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation22.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="131" height="23" alt=" ab/t^2=1-2u/t " /></td><td align="right" width="3"> <div id="eqn74" class="eqnum"> (74) </div> </td></tr> </table> </div> <p> and </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation23.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="253" height="23" alt=" 1+ab/t^2=2-2u/t=2(1-u/t). " /></td><td align="right" width="3"> <div id="eqn75" class="eqnum"> (75) </div> </td></tr> </table> </div> <p> However, the left side is always positive, so </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation24.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="247" height="23" alt=" 1+ab/t^2=2-2u/t=2|1-u/t| " /></td><td align="right" width="3"> <div id="eqn76" class="eqnum"> (76) </div> </td></tr> </table> </div> <p> and the differential is </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation25.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="93" height="48" alt=" dt=(du)/(|1-u/t|). " /></td><td align="right" width="3"> <div id="eqn77" class="eqnum"> (77) </div> </td></tr> </table> </div> <p> We need to take some care with the limits of integration. Write (◇) as </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation26.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="269" height="44" alt=" int_(-infty)^inftyf(t)dt=int_(-infty)^(0^-)f(t)dt+int_(0^+)^inftyf(t)dt. " /></td><td align="right" width="3"> <div id="eqn78" class="eqnum"> (78) </div> </td></tr> </table> </div> <p> Now change the limits to those appropriate for the <img src="/images/equations/EllipticIntegral/Inline200.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="u" /> integration </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation27.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="296" height="39" alt=" int_(-infty)^inftyg(u)du+int_(-infty)^inftyg(u)du=2int_(-infty)^inftyg(u)du, " /></td><td align="right" width="3"> <div id="eqn79" class="eqnum"> (79) </div> </td></tr> </table> </div> <p> so we have picked up a factor of 2 which must be included. Using this fact and plugging (◇) in (◇) therefore gives </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation28.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="344" height="53" alt=" T(a,b)=2/piint_(-infty)^infty(du)/(|1-u/t|sqrt(a^2b^2+(a^2+b^2)t^2+t^4)). " /></td><td align="right" width="3"> <div id="eqn80" class="eqnum"> (80) </div> </td></tr> </table> </div> <p> Now note that </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline201.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="16" height="20" alt="u^2" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline202.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline203.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="133" height="45" alt="(t^4-2abt^2+a^2b^2)/(4t^2)" /></td><td align="right" width="10"> <div id="eqn81" class="eqnum"> (81) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline204.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="43" height="20" alt="4u^2t^2" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline205.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline206.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="127" height="20" alt="t^4-2abt^2+a^2b^2" /></td><td align="right" width="10"> <div id="eqn82" class="eqnum"> (82) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline207.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="63" height="20" alt="a^2b^2+t^4" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline208.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline209.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="111" height="20" alt="4u^2t^2+2abt^2." /></td><td align="right" width="10"> <div id="eqn83" class="eqnum"> (83) </div> </td></tr> </table> </div> <p> Plug (◇) into (◇) to obtain </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline210.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="51" height="20" alt="T(a,b)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline211.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline212.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="312" height="53" alt="2/piint_(-infty)^infty(du)/(|1-u/t|sqrt(4u^2t^2+2abt^2+(a^2+b^2)t^2))" /></td><td align="right" width="10"> <div id="eqn84" class="eqnum"> (84) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline213.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline214.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline215.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="208" height="50" alt="2/piint_(-infty)^infty(du)/(|t-u|sqrt(4u^2+(a+b)^2))." /></td><td align="right" width="10"> <div id="eqn85" class="eqnum"> (85) </div> </td></tr> </table> </div> <p> But </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline216.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="102" height="20" alt="2ut=t^2-ab " /></td><td align="right" width="10"> <div id="eqn86" class="eqnum"> (86) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline217.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="127" height="20" alt="t^2-2ut-ab=0 " /></td><td align="right" width="10"> <div id="eqn87" class="eqnum"> (87) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline218.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="187" height="31" alt="t=1/2(2u+/-sqrt(4u^2+4ab)) " /></td><td align="right" width="10"> <div id="eqn88" class="eqnum"> (88) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline219.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="127" height="24" alt="=u+/-sqrt(u^2+ab), " /></td><td align="right" width="10"> <div id="eqn89" class="eqnum"> (89) </div> </td></tr> </table> </div> <p> so </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation29.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="139" height="24" alt=" t-u=+/-sqrt(u^2+ab), " /></td><td align="right" width="3"> <div id="eqn90" class="eqnum"> (90) </div> </td></tr> </table> </div> <p> and (◇) becomes </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline220.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="51" height="20" alt="T(a,b)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline221.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline222.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="239" height="53" alt="2/piint_(-infty)^infty(du)/(sqrt([4u^2+(a+b)^2](u^2+ab)))" /></td><td align="right" width="10"> <div id="eqn91" class="eqnum"> (91) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline223.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline224.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline225.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="225" height="60" alt="1/piint_(-infty)^infty(du)/(sqrt([u^2+((a+b)/2)^2](u^2+ab)))." /></td><td align="right" width="10"> <div id="eqn92" class="eqnum"> (92) </div> </td></tr> </table> </div> <p> We have therefore demonstrated that </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation30.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="208" height="27" alt=" T(a,b)=T(1/2(a+b),sqrt(ab)). " /></td><td align="right" width="3"> <div id="eqn93" class="eqnum"> (93) </div> </td></tr> </table> </div> <p> We can thus iterate </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline226.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="26" height="20" alt="a_(i+1)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline227.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline228.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="67" height="26" alt="1/2(a_i+b_i)" /></td><td align="right" width="10"> <div id="eqn94" class="eqnum"> (94) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline229.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="26" height="20" alt="b_(i+1)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline230.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline231.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="52" height="24" alt="sqrt(a_ib_i)," /></td><td align="right" width="10"> <div id="eqn95" class="eqnum"> (95) </div> </td></tr> </table> </div> <p> as many times as we wish, without changing the value of the integral. But this iteration is the same as and therefore converges to the <a href="/Arithmetic-GeometricMean.html">arithmetic-geometric mean</a>, so the iteration terminates at <img src="/images/equations/EllipticIntegral/Inline232.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="143" height="22" alt="a_i=b_i=M(a_0,b_0)" />, and we have </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline233.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="65" height="20" alt="T(a_0,b_0)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline234.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline235.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="173" height="20" alt="T(M(a_0,b_0),M(a_0,b_0))" /></td><td align="right" width="10"> <div id="eqn96" class="eqnum"> (96) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline236.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline237.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline238.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="155" height="44" alt="1/piint_(-infty)^infty(dt)/(M^2(a_0,b_0)+t^2)" /></td><td align="right" width="10"> <div id="eqn97" class="eqnum"> (97) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline239.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline240.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline241.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="245" height="42" alt="1/(piM(a_0,b_0))[tan^(-1)(t/(M(a_0,b_0)))]_(-infty)^infty" /></td><td align="right" width="10"> <div id="eqn98" class="eqnum"> (98) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline242.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline243.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline244.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="171" height="42" alt="1/(piM(a_0,b_0))[pi/2-(-pi/2)]" /></td><td align="right" width="10"> <div id="eqn99" class="eqnum"> (99) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline245.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline246.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline247.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="80" height="42" alt="1/(M(a_0,b_0))." /></td><td align="right" width="10"> <div id="eqn100" class="eqnum"> (100) </div> </td></tr> </table> </div> <p> Complete elliptic integrals arise in finding the arc length of an <a href="/Ellipse.html">ellipse</a> and the period of a pendulum. They also arise in a natural way from the theory of theta functions. Complete elliptic integrals can be computed using a procedure involving the <a href="/Arithmetic-GeometricMean.html">arithmetic-geometric mean</a>. Note that </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline248.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="51" height="20" alt="T(a,b)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline249.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline250.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="205" height="49" alt="2/piint_0^(pi/2)(dtheta)/(sqrt(a^2cos^2theta+b^2sin^2theta))" /></td><td align="right" width="10"> <div id="eqn101" class="eqnum"> (101) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline251.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline252.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline253.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="212" height="59" alt="2/piint_0^(pi/2)(dtheta)/(asqrt(cos^2theta+(b/a)^2sin^2theta))" /></td><td align="right" width="10"> <div id="eqn102" class="eqnum"> (102) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline254.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline255.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline256.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="208" height="62" alt="2/(api)int_0^(pi/2)(dtheta)/(sqrt(1-(1-(b^2)/(a^2))sin^2theta))." /></td><td align="right" width="10"> <div id="eqn103" class="eqnum"> (103) </div> </td></tr> </table> </div> <p> So we have </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline257.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="51" height="20" alt="T(a,b)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline258.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline259.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="126" height="54" alt="2/(api)K(sqrt(1-(b^2)/(a^2)))" /></td><td align="right" width="10"> <div id="eqn104" class="eqnum"> (104) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline260.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline261.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline262.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="66" height="42" alt="1/(M(a,b))," /></td><td align="right" width="10"> <div id="eqn105" class="eqnum"> (105) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/EllipticIntegral/Inline263.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="K(k)" /> is the <a href="/CompleteEllipticIntegraloftheFirstKind.html">complete elliptic integral of the first kind</a>. We are free to let <img src="/images/equations/EllipticIntegral/Inline264.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="72" height="22" alt="a=a_0=1" /> and <img src="/images/equations/EllipticIntegral/Inline265.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="77" height="22" alt="b=b_0=k^'" />, so </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation31.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="273" height="42" alt=" 2/piK(sqrt(1-k^('2)))=2/piK(k)=1/(M(1,k^')), " /></td><td align="right" width="3"> <div id="eqn106" class="eqnum"> (106) </div> </td></tr> </table> </div> <p> since <img src="/images/equations/EllipticIntegral/Inline266.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="99" height="25" alt="k=sqrt(1-k^('2))" />, so </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation32.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="137" height="37" alt=" K(k)=pi/(2M(1,k^')). " /></td><td align="right" width="3"> <div id="eqn107" class="eqnum"> (107) </div> </td></tr> </table> </div> <p> But the <a href="/Arithmetic-GeometricMean.html">arithmetic-geometric mean</a> is defined by </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline267.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="13" height="20" alt="a_i" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline268.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline269.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="93" height="26" alt="1/2(a_(i-1)+b_(i-1))" /></td><td align="right" width="10"> <div id="eqn108" class="eqnum"> (108) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline270.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="13" height="20" alt="b_i" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline271.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline272.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="74" height="24" alt="sqrt(a_(i-1)b_(i-1))" /></td><td align="right" width="10"> <div id="eqn109" class="eqnum"> (109) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline273.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="c_i" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline274.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline275.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="150" height="65" alt="{1/2(a_(i-1)-b_(i-1)) i>0; sqrt(a_0^2-b_0^2) i=0," /></td><td align="right" width="10"> <div id="eqn110" class="eqnum"> (110) </div> </td></tr> </table> </div> <p> where </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation33.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="290" height="46" alt=" c_(n-1)=1/2a_n-b_n=(c_n^2)/(4a_(n+1))<=(c_n^2)/(4M(a_0,b_0)), " /></td><td align="right" width="3"> <div id="eqn111" class="eqnum"> (111) </div> </td></tr> </table> </div> <p> so we have </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation34.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="97" height="37" alt=" K(k)=pi/(2a_N), " /></td><td align="right" width="3"> <div id="eqn112" class="eqnum"> (112) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/EllipticIntegral/Inline276.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="22" alt="a_N" /> is the value to which <img src="/images/equations/EllipticIntegral/Inline277.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="a_n" /> converges. Similarly, taking instead <img src="/images/equations/EllipticIntegral/Inline278.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="46" height="23" alt="a_0^'=1" /> and <img src="/images/equations/EllipticIntegral/Inline279.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="46" height="23" alt="b_0^'=k" /> gives </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation35.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="102" height="39" alt=" K^'(k)=pi/(2a_N^'). " /></td><td align="right" width="3"> <div id="eqn113" class="eqnum"> (113) </div> </td></tr> </table> </div> <p> Borwein and Borwein (1987) also show that defining </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline280.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="54" height="20" alt="U(a,b)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline281.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline282.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="222" height="42" alt="pi/2int_0^(pi/2)sqrt(a^2cos^2theta+b^2sin^2theta)dtheta" /></td><td align="right" width="10"> <div id="eqn114" class="eqnum"> (114) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline283.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline284.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline285.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="58" height="40" alt="aE^'(b/a)" /></td><td align="right" width="10"> <div id="eqn115" class="eqnum"> (115) </div> </td></tr> </table> </div> <p> leads to </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation36.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="318" height="20" alt=" 2U(a_(n+1),b_(n+1))-U(a_n,b_n)=a_nb_nT(a_n,b_n), " /></td><td align="right" width="3"> <div id="eqn116" class="eqnum"> (116) </div> </td></tr> </table> </div> <p> so </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation37.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="324" height="42" alt=" (K(k)-E(k))/(K(k))=1/2(c_0^2+2c_1^2+2^2c_2^2+...+2^nc_n^2) " /></td><td align="right" width="3"> <div id="eqn117" class="eqnum"> (117) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/EllipticIntegral/Inline286.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="44" height="22" alt="a_0=1" /> and <img src="/images/equations/EllipticIntegral/Inline287.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="49" height="22" alt="b_0=k^'" />, and </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation38.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="366" height="42" alt=" (K^'(k)-E^'(k))/(K^'(k))=1/2(c_0^'^2+2c_1^'^2+2^2c_2^'^2+...+2^nc_n^'^2). " /></td><td align="right" width="3"> <div id="eqn118" class="eqnum"> (118) </div> </td></tr> </table> </div> <p> The elliptic integrals satisfy a large number of identities. The complementary functions and moduli are defined by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation39.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="206" height="31" alt=" K^'(k)=K(sqrt(1-k^2))=K(k^'). " /></td><td align="right" width="3"> <div id="eqn119" class="eqnum"> (119) </div> </td></tr> </table> </div> <p> Use the identity of generalized elliptic integrals </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation40.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="204" height="27" alt=" T(a,b)=T(1/2(a+b),sqrt(ab)) " /></td><td align="right" width="3"> <div id="eqn120" class="eqnum"> (120) </div> </td></tr> </table> </div> <p> to write </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline288.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="115" height="54" alt="1/aK(sqrt(1-(b^2)/(a^2)))" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline289.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline290.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="175" height="52" alt="2/(a+b)K(sqrt(1-(4ab)/((a+b)^2)))" /></td><td align="right" width="10"> <div id="eqn121" class="eqnum"> (121) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline291.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline292.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline293.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="196" height="60" alt="2/(a+b)K(sqrt((a^2+b^2-2ab)/((a+b)^2)))" /></td><td align="right" width="10"> <div id="eqn122" class="eqnum"> (122) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline294.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline295.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline296.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="111" height="40" alt="2/(a+b)K((a-b)/(a+b))" /></td><td align="right" width="10"> <div id="eqn123" class="eqnum"> (123) </div> </td></tr> </table> </div> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation41.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="239" height="58" alt=" K(sqrt(1-(b^2)/(a^2)))=2/(1+b/a)K((1-b/a)/(1+b/a)). " /></td><td align="right" width="3"> <div id="eqn124" class="eqnum"> (124) </div> </td></tr> </table> </div> <p> Define </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation42.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="50" height="39" alt=" k^'=b/a, " /></td><td align="right" width="3"> <div id="eqn125" class="eqnum"> (125) </div> </td></tr> </table> </div> <p> and use </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation43.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="95" height="24" alt=" k=sqrt(1-k^('2)), " /></td><td align="right" width="3"> <div id="eqn126" class="eqnum"> (126) </div> </td></tr> </table> </div> <p> so </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation44.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="178" height="40" alt=" K(k)=2/(1+k^')K((1-k^')/(1+k^')). " /></td><td align="right" width="3"> <div id="eqn127" class="eqnum"> (127) </div> </td></tr> </table> </div> <p> Now letting <img src="/images/equations/EllipticIntegral/Inline297.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="139" height="21" alt="l=(1-k^')/(1+k^')" /> gives </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation45.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="243" height="20" alt=" l(1+k^')=1-k^'=>k^'(l+1)=1-l " /></td><td align="right" width="3"> <div id="eqn128" class="eqnum"> (128) </div> </td></tr> </table> </div> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation46.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="70" height="39" alt=" k^'=(1-l)/(1+l) " /></td><td align="right" width="3"> <div id="eqn129" class="eqnum"> (129) </div> </td></tr> </table> </div> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline298.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="8" height="20" alt="k" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline299.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline300.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="65" height="24" alt="sqrt(1-k^('2))" /></td><td align="right" width="10"> <div id="eqn130" class="eqnum"> (130) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline301.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline302.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline303.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="102" height="50" alt="sqrt(1-((1-l)/(1+l))^2)" /></td><td align="right" width="10"> <div id="eqn131" class="eqnum"> (131) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline304.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline305.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline306.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="138" height="58" alt="sqrt(((1+l)^2-(1-l)^2)/((1+l)^2))" /></td><td align="right" width="10"> <div id="eqn132" class="eqnum"> (132) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline307.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline308.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline309.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="45" height="44" alt="(2sqrt(l))/(1+l)," /></td><td align="right" width="10"> <div id="eqn133" class="eqnum"> (133) </div> </td></tr> </table> </div> <p> and </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline310.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="63" height="26" alt="1/2(1+k^')" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline311.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline312.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="91" height="40" alt="1/2(1+(1-l)/(1+l))" /></td><td align="right" width="10"> <div id="eqn134" class="eqnum"> (134) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline313.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline314.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline315.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="132" height="40" alt="1/2[((1+l)+(1-l))/(1+l)]" /></td><td align="right" width="10"> <div id="eqn135" class="eqnum"> (135) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline316.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline317.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline318.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="40" height="39" alt="1/(1+l)." /></td><td align="right" width="10"> <div id="eqn136" class="eqnum"> (136) </div> </td></tr> </table> </div> <p> Writing <img src="/images/equations/EllipticIntegral/Inline319.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" /> instead of <img src="/images/equations/EllipticIntegral/Inline320.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="5" height="21" alt="l" />, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation47.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="173" height="50" alt=" K(k)=1/(k+1)K((2sqrt(k))/(1+k)). " /></td><td align="right" width="3"> <div id="eqn137" class="eqnum"> (137) </div> </td></tr> </table> </div> <p> Similarly, from Borwein and Borwein (1987), </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation48.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="245" height="50" alt=" E(k)=(1+k)/2E((2sqrt(k))/(1+k))+(k^('2))/2K(k) " /></td><td align="right" width="3"> <div id="eqn138" class="eqnum"> (138) </div> </td></tr> </table> </div> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation49.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="244" height="40" alt=" E(k)=(1+k^')E((1-k^')/(1+k^'))-k^'K(k). " /></td><td align="right" width="3"> <div id="eqn139" class="eqnum"> (139) </div> </td></tr> </table> </div> <p> Expressions in terms of the complementary function can be derived from interchanging the moduli and their complements in (◇), (◇), (◇), and (◇). </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline321.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="39" height="20" alt="K^'(k)" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline322.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline323.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="39" height="20" alt="K(k^')" /></td><td align="right" width="10"> <div id="eqn140" class="eqnum"> (140) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline324.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline325.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline326.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="109" height="40" alt="2/(1+k)K((1-k)/(1+k))" /></td><td align="right" width="10"> <div id="eqn141" class="eqnum"> (141) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline327.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline328.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline329.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="180" height="52" alt="2/(1+k)K^'(sqrt(1-((1-k)/(1+k))^2))" /></td><td align="right" width="10"> <div id="eqn142" class="eqnum"> (142) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline330.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline331.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline332.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="119" height="50" alt="2/(1+k)K^'((2sqrt(k))/(1+k))" /></td><td align="right" width="10"> <div id="eqn143" class="eqnum"> (143) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline333.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline334.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline335.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="124" height="50" alt="1/(1+k^')K((2sqrt(k^'))/(1+k^'))" /></td><td align="right" width="10"> <div id="eqn144" class="eqnum"> (144) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EllipticIntegral/Inline336.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EllipticIntegral/Inline337.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EllipticIntegral/Inline338.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="128" height="40" alt="1/(1+k^')K^'((1-k^')/(1+k^'))," /></td><td align="right" width="10"> <div id="eqn145" class="eqnum"> (145) </div> </td></tr> </table> </div> <p> and </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation50.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="245" height="50" alt=" E^'(k)=(1+k)E^'((2sqrt(k))/(1+k))-kK^'(k) " /></td><td align="right" width="3"> <div id="eqn146" class="eqnum"> (146) </div> </td></tr> </table> </div> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation51.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="275" height="43" alt=" E^'(k)=((1+k^')/2)E^'((1-k^')/(1+k^'))+(k^2)/2K^'(k). " /></td><td align="right" width="3"> <div id="eqn147" class="eqnum"> (147) </div> </td></tr> </table> </div> <p> Taking the ratios </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation52.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="253" height="66" alt=" (K^'(k))/(K(k))=2(K^'((2sqrt(k))/(1+k)))/(K((2sqrt(k))/(1+k)))=1/2(K^'((1-k^')/(1+k^')))/(K((1-k^')/(1+k^'))) " /></td><td align="right" width="3"> <div id="eqn148" class="eqnum"> (148) </div> </td></tr> </table> </div> <p> gives the <a href="/ModularEquation.html">modular equation</a> of degree 2. It is also true that </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EllipticIntegral/NumberedEquation53.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="277" height="61" alt=" K(x)=4/((1+sqrt(x^'))^2)K([(1-RadicalBox[{1, -, {x, ^, 4}}, 4])/(1+RadicalBox[{1, -, {x, ^, 4}}, 4])]^2). " /></td><td align="right" width="3"> <div id="eqn149" class="eqnum"> (149) </div> </td></tr> </table> </div> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/AbelianIntegral.html">Abelian Integral</a>, <a href="/CarlsonEllipticIntegrals.html">Carlson Elliptic Integrals</a>, <a href="/CompleteEllipticIntegraloftheFirstKind.html">Complete Elliptic Integral of the First Kind</a>, <a href="/CompleteEllipticIntegraloftheSecondKind.html">Complete Elliptic Integral of the Second Kind</a>, <a href="/CompleteEllipticIntegraloftheThirdKind.html">Complete Elliptic Integral of the Third Kind</a>, <a href="/DeltaAmplitude.html">Delta Amplitude</a>, <a href="/EllipticArgument.html">Elliptic Argument</a>, <a href="/EllipticCharacteristic.html">Elliptic Characteristic</a>, <a href="/EllipticFunction.html">Elliptic Function</a>, <a href="/EllipticIntegraloftheFirstKind.html">Elliptic Integral of the First Kind</a>, <a href="/EllipticIntegraloftheSecondKind.html">Elliptic Integral of the Second Kind</a>, <a href="/EllipticIntegraloftheThirdKind.html">Elliptic Integral of the Third Kind</a>, <a href="/EllipticIntegralSingularValue.html">Elliptic Integral Singular Value</a>, <a href="/EllipticModulus.html">Elliptic Modulus</a>, <a href="/HeumanLambdaFunction.html">Heuman Lambda Function</a>, <a href="/JacobiAmplitude.html">Jacobi Amplitude</a>, <a href="/JacobiEllipticFunctions.html">Jacobi Elliptic Functions</a>, <a href="/JacobiZetaFunction.html">Jacobi Zeta Function</a>, <a href="/ModularAngle.html">Modular Angle</a>, <a href="/Nome.html">Nome</a>, <a href="/Parameter.html">Parameter</a>, <a href="/WeierstrassEllipticFunction.html">Weierstrass Elliptic Function</a>       <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=elliptic+integral"> elliptic integral </a> </li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=area+between+the+curves+y%3D1-x%5E2+and+y%3Dx">area between the curves y=1-x^2 and y=x</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=colorize+image+of+Poe">colorize image of Poe</a></li> </ul> </div> </div>   <h2>References</h2><cite>Abramowitz, M. and Stegun, I. A. (Eds.). "Elliptic Integrals." Ch. 17 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486612724/ref=nosim/ericstreasuretro">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.</a></i> New York: Dover, pp. 587-607, 1972.</cite><cite>Arfken, G. "Elliptic Integrals." §5.8 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0120598760/ref=nosim/ericstreasuretro">Mathematical Methods for Physicists, 3rd ed.</a></i> Orlando, FL: Academic Press, pp. 321-327, 1985.</cite><cite>Borwein, J. M. and Borwein, P. B. <i><a href="http://www.amazon.com/exec/obidos/ASIN/047131515X/ref=nosim/ericstreasuretro">Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.</a></i> New York: Wiley, 1987.</cite><cite>Hancock, H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/B00085HTBU/ref=nosim/ericstreasuretro">Elliptic Integrals.</a></i> New York: Wiley, 1917.</cite><cite>Kármán, T. von and Biot, M. A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/B0006AOTLK/ref=nosim/ericstreasuretro">Mathematical Methods in Engineering: An Introduction to the Mathematical Treatment of Engineering Problems.</a></i> New York: McGraw-Hill, p. 121, 1940.</cite><cite>King, L. V. <i>The Direct Numerical Calculation of Elliptic Functions and Integrals.</i> London: Cambridge University Press, 1924.</cite><cite>Prasolov, V. and Solovyev, Y. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0821805878/ref=nosim/ericstreasuretro">Elliptic Functions and Elliptic Integrals.</a></i> Providence, RI: Amer. Math. Soc., 1997.</cite><cite>Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Elliptic Integrals and Jacobi Elliptic Functions." §6.11 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/052143064X/ref=nosim/ericstreasuretro">Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.</a></i> Cambridge, England: Cambridge University Press, pp. 254-263, 1992.</cite><cite>Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. <i><a href="http://www.amazon.com/exec/obidos/ASIN/2881247369/ref=nosim/ericstreasuretro">Integrals and Series, Vol. 1: Elementary Functions.</a></i> New York: Gordon & Breach, 1986.</cite><cite>Timofeev, A. F. <i>Integration of Functions.</i> Moscow and Leningrad: GTTI, 1948.</cite><cite>Weisstein, E. W. "Books about Elliptic Integrals." <a href="http://www.ericweisstein.com/encyclopedias/books/EllipticIntegrals.html">http://www.ericweisstein.com/encyclopedias/books/EllipticIntegrals.html</a>.</cite><cite>Whittaker, E. T. and Watson, G. N. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521091896/ref=nosim/ericstreasuretro">A Course in Modern Analysis, 4th ed.</a></i> Cambridge, England: Cambridge University Press, 1990.</cite><cite>Woods, F. S. "Elliptic Integrals." Ch. 16 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/B00085L67S/ref=nosim/ericstreasuretro">Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathematics.</a></i> Boston, MA: Ginn, pp. 365-386, 1926.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/elliptic_integral/o7/4a/5k/" title="Elliptic Integral" target="_blank">Elliptic Integral</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Elliptic Integral." 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