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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.ellint.ellint_intro"></a><a class="link" href="ellint_intro.html" title="Elliptic Integral Overview">Elliptic Integral Overview</a>
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</h3></div></div></div>
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<p>
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The main reference for the elliptic integrals is:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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M. Abramowitz and I. A. Stegun (Eds.) (1964) Handbook of Mathematical Functions
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with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards
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Applied Mathematics Series, U.S. Government Printing Office, Washington,
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D.C.
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</p></blockquote></div>
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<p>
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and its recently revised version <a href="http://dlmf.nist.gov/" target="_top">NIST
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Digital Library of Mathematical Functions (DMLF)</a>, in particular
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<a href="https://dlmf.nist.gov/19" target="_top">Elliptic Integrals, B. C. Carlson</a>
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</p></blockquote></div>
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<p>
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Mathworld also contain a lot of useful background information:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<a href="http://mathworld.wolfram.com/EllipticIntegral.html" target="_top">Weisstein,
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Eric W. "Elliptic Integral." From MathWorld--A Wolfram Web Resource.</a>
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</p></blockquote></div>
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<p>
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As does <a href="http://en.wikipedia.org/wiki/Elliptic_integral" target="_top">Wikipedia
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Elliptic integral</a>.
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</p>
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<h5>
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<a name="math_toolkit.ellint.ellint_intro.h0"></a>
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<span class="phrase"><a name="math_toolkit.ellint.ellint_intro.notation"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.notation">Notation</a>
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</h5>
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<p>
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All variables are real numbers unless otherwise noted.
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</p>
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<h5>
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<a name="math_toolkit.ellint.ellint_intro.h1"></a>
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<span class="phrase"><a name="math_toolkit.ellint.ellint_intro.definition"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.definition">Definition</a>
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</h5>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint1.svg"></span>
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</p></blockquote></div>
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<p>
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is called elliptic integral if <span class="emphasis"><em>R(t, s)</em></span> is a rational
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function of <span class="emphasis"><em>t</em></span> and <span class="emphasis"><em>s</em></span>, and <span class="emphasis"><em>s<sup>2</sup></em></span>
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is a cubic or quartic polynomial in <span class="emphasis"><em>t</em></span>.
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</p>
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<p>
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Elliptic integrals generally cannot be expressed in terms of elementary functions.
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However, Legendre showed that all elliptic integrals can be reduced to the
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following three canonical forms:
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</p>
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<p>
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Elliptic Integral of the First Kind (Legendre form)
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint2.svg"></span>
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</p></blockquote></div>
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<p>
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Elliptic Integral of the Second Kind (Legendre form)
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint3.svg"></span>
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</p></blockquote></div>
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<p>
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Elliptic Integral of the Third Kind (Legendre form)
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint4.svg"></span>
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</p></blockquote></div>
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<p>
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where
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint5.svg"></span>
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</p></blockquote></div>
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<div class="note"><table border="0" summary="Note">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
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<th align="left">Note</th>
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</tr>
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<tr><td align="left" valign="top">
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<p>
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<span class="emphasis"><em>φ</em></span> is called the amplitude.
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</p>
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<p>
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<span class="emphasis"><em>k</em></span> is called the elliptic modulus or eccentricity.
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</p>
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<p>
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<span class="emphasis"><em>α</em></span> is called the modular angle.
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</p>
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<p>
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<span class="emphasis"><em>n</em></span> is called the characteristic.
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</p>
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</td></tr>
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</table></div>
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<div class="caution"><table border="0" summary="Caution">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
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<th align="left">Caution</th>
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</tr>
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<tr><td align="left" valign="top">
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<p>
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Perhaps more than any other special functions the elliptic integrals are
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expressed in a variety of different ways. In particular, the final parameter
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<span class="emphasis"><em>k</em></span> (the modulus) may be expressed using a modular angle
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α, or a parameter <span class="emphasis"><em>m</em></span>. These are related by:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">k = sin  α</span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">m = k<sup>2</sup> = sin<sup>2</sup>α</span>
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</p></blockquote></div>
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<p>
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So that the integral of the third kind (for example) may be expressed as
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either:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">Π(n, φ, k)</span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">Π(n, φ \ α)</span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">Π(n, φ | m)</span>
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</p></blockquote></div>
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<p>
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To further complicate matters, some texts refer to the <span class="emphasis"><em>complement
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of the parameter m</em></span>, or 1 - m, where:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">1 - m = 1 - k<sup>2</sup> = cos<sup>2</sup>α</span>
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</p></blockquote></div>
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<p>
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This implementation uses <span class="emphasis"><em>k</em></span> throughout: this matches
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the requirements of the <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf" target="_top">Technical
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Report on C++ Library Extensions</a>.<br>
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</p>
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<p>
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So you should be extra careful when using these functions!
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</p>
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</td></tr>
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</table></div>
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<div class="warning"><table border="0" summary="Warning">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../doc/src/images/warning.png"></td>
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<th align="left">Warning</th>
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</tr>
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<tr><td align="left" valign="top"><p>
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Boost.Math order of arguments differs from other implementations: <span class="emphasis"><em>k</em></span>
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is always the <span class="bold"><strong>first</strong></span> argument.
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</p></td></tr>
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</table></div>
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<p>
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A simple example comparing use of <a href="http://www.wolframalpha.com/" target="_top">Wolfram
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Alpha</a> with Boost.Math (including much higher precision using <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>)
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is <a href="../../../../example/jacobi_zeta_example.cpp" target="_top">jacobi_zeta_example.cpp</a>.
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</p>
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<p>
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When <span class="emphasis"><em>φ</em></span> = <span class="emphasis"><em>π</em></span> / 2, the elliptic integrals
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are called <span class="emphasis"><em>complete</em></span>.
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</p>
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<p>
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Complete Elliptic Integral of the First Kind (Legendre form)
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint6.svg"></span>
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</p></blockquote></div>
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<p>
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Complete Elliptic Integral of the Second Kind (Legendre form)
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint7.svg"></span>
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</p></blockquote></div>
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<p>
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Complete Elliptic Integral of the Third Kind (Legendre form)
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint8.svg"></span>
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</p></blockquote></div>
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<p>
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Legendre also defined a fourth integral /D(φ,k)/ which is a combination of
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the other three:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span>
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</p></blockquote></div>
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<p>
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Like the other Legendre integrals this comes in both complete and incomplete
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forms.
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</p>
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<h5>
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<a name="math_toolkit.ellint.ellint_intro.h2"></a>
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<span class="phrase"><a name="math_toolkit.ellint.ellint_intro.carlson_elliptic_integrals"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.carlson_elliptic_integrals">Carlson
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Elliptic Integrals</a>
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</h5>
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<p>
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Carlson [<a class="link" href="ellint_intro.html#ellint_ref_carlson77">Carlson77</a>] [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>]
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gives an alternative definition of elliptic integral's canonical forms:
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</p>
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<p>
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Carlson's Elliptic Integral of the First Kind
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint9.svg"></span>
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</p></blockquote></div>
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<p>
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where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, <span class="emphasis"><em>z</em></span>
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are nonnegative and at most one of them may be zero.
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</p>
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<p>
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Carlson's Elliptic Integral of the Second Kind
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint10.svg"></span>
|
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</p></blockquote></div>
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<p>
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where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span> are nonnegative, at
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most one of them may be zero, and <span class="emphasis"><em>z</em></span> must be positive.
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</p>
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<p>
|
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Carlson's Elliptic Integral of the Third Kind
|
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint11.svg"></span>
|
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</p></blockquote></div>
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<p>
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where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, <span class="emphasis"><em>z</em></span>
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are nonnegative, at most one of them may be zero, and <span class="emphasis"><em>p</em></span>
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must be nonzero.
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</p>
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<p>
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Carlson's Degenerate Elliptic Integral
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint12.svg"></span>
|
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</p></blockquote></div>
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<p>
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where <span class="emphasis"><em>x</em></span> is nonnegative and <span class="emphasis"><em>y</em></span> is
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nonzero.
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</p>
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<div class="note"><table border="0" summary="Note">
|
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
|
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<th align="left">Note</th>
|
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</tr>
|
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<tr><td align="left" valign="top">
|
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<p>
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<span class="emphasis"><em>R<sub>C</sub>(x, y) = R<sub>F</sub>(x, y, y)</em></span>
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</p>
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<p>
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<span class="emphasis"><em>R<sub>D</sub>(x, y, z) = R<sub>J</sub>(x, y, z, z)</em></span>
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</p>
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</td></tr>
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</table></div>
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<p>
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Carlson's Symmetric Integral
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/ellint27.svg"></span>
|
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</p></blockquote></div>
|
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<h5>
|
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<a name="math_toolkit.ellint.ellint_intro.h3"></a>
|
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<span class="phrase"><a name="math_toolkit.ellint.ellint_intro.duplication_theorem"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.duplication_theorem">Duplication
|
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Theorem</a>
|
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</h5>
|
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<p>
|
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Carlson proved in [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>]
|
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that
|
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</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
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<span class="inlinemediaobject"><img src="../../../equations/ellint13.svg"></span>
|
|
|
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</p></blockquote></div>
|
|
<h5>
|
|
<a name="math_toolkit.ellint.ellint_intro.h4"></a>
|
|
<span class="phrase"><a name="math_toolkit.ellint.ellint_intro.carlson_s_formulas"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.carlson_s_formulas">Carlson's
|
|
Formulas</a>
|
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</h5>
|
|
<p>
|
|
The Legendre form and Carlson form of elliptic integrals are related by equations:
|
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</p>
|
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<div class="blockquote"><blockquote class="blockquote"><p>
|
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<span class="inlinemediaobject"><img src="../../../equations/ellint14.svg"></span>
|
|
|
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</p></blockquote></div>
|
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<p>
|
|
In particular,
|
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</p>
|
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<div class="blockquote"><blockquote class="blockquote"><p>
|
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<span class="inlinemediaobject"><img src="../../../equations/ellint15.svg"></span>
|
|
|
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</p></blockquote></div>
|
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<h5>
|
|
<a name="math_toolkit.ellint.ellint_intro.h5"></a>
|
|
<span class="phrase"><a name="math_toolkit.ellint.ellint_intro.miscellaneous_elliptic_integrals"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.miscellaneous_elliptic_integrals">Miscellaneous
|
|
Elliptic Integrals</a>
|
|
</h5>
|
|
<p>
|
|
There are two functions related to the elliptic integrals which otherwise
|
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defy categorisation, these are the Jacobi Zeta function:
|
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</p>
|
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<div class="blockquote"><blockquote class="blockquote"><p>
|
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<span class="inlinemediaobject"><img src="../../../equations/jacobi_zeta.svg"></span>
|
|
|
|
</p></blockquote></div>
|
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<p>
|
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and the Heuman Lambda function:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
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<span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>
|
|
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</p></blockquote></div>
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<p>
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Both of these functions are easily implemented in terms of Carlson's integrals,
|
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and are provided in this library as <a class="link" href="jacobi_zeta.html" title="Jacobi Zeta Function">jacobi_zeta</a>
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and <a class="link" href="heuman_lambda.html" title="Heuman Lambda Function">heuman_lambda</a>.
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</p>
|
|
<h5>
|
|
<a name="math_toolkit.ellint.ellint_intro.h6"></a>
|
|
<span class="phrase"><a name="math_toolkit.ellint.ellint_intro.numerical_algorithms"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.numerical_algorithms">Numerical
|
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Algorithms</a>
|
|
</h5>
|
|
<p>
|
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The conventional methods for computing elliptic integrals are Gauss and Landen
|
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transformations, which converge quadratically and work well for elliptic
|
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integrals of the first and second kinds. Unfortunately they suffer from loss
|
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of significant digits for the third kind.
|
|
</p>
|
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<p>
|
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Carlson's algorithm [<a class="link" href="ellint_intro.html#ellint_ref_carlson79">Carlson79</a>]
|
|
[<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>], by contrast, provides
|
|
a unified method for all three kinds of elliptic integrals with satisfactory
|
|
precisions.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.ellint.ellint_intro.h7"></a>
|
|
<span class="phrase"><a name="math_toolkit.ellint.ellint_intro.references"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.references">References</a>
|
|
</h5>
|
|
<p>
|
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Special mention goes to:
|
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</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
A. M. Legendre, <span class="emphasis"><em>Traité des Fonctions Elliptiques et des Integrales
|
|
Euleriennes</em></span>, Vol. 1. Paris (1825).
|
|
</p></blockquote></div>
|
|
<p>
|
|
However the main references are:
|
|
</p>
|
|
<div class="orderedlist"><ol class="orderedlist" type="1">
|
|
<li class="listitem">
|
|
<a name="ellint_ref_AS"></a>M. Abramowitz and I. A. Stegun (Eds.) (1964)
|
|
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical
|
|
Tables, National Bureau of Standards Applied Mathematics Series, U.S.
|
|
Government Printing Office, Washington, D.C.
|
|
</li>
|
|
<li class="listitem">
|
|
<a href="https://dlmf.nist.gov/19" target="_top">NIST Digital Library of Mathematical
|
|
Functions, Elliptic Integrals, B. C. Carlson</a>
|
|
</li>
|
|
<li class="listitem">
|
|
<a name="ellint_ref_carlson79"></a>B.C. Carlson, <span class="emphasis"><em>Computing
|
|
elliptic integrals by duplication</em></span>, Numerische Mathematik,
|
|
vol 33, 1 (1979).
|
|
</li>
|
|
<li class="listitem">
|
|
<a name="ellint_ref_carlson77"></a>B.C. Carlson, <span class="emphasis"><em>Elliptic Integrals
|
|
of the First Kind</em></span>, SIAM Journal on Mathematical Analysis,
|
|
vol 8, 231 (1977).
|
|
</li>
|
|
<li class="listitem">
|
|
<a name="ellint_ref_carlson78"></a>B.C. Carlson, <span class="emphasis"><em>Short Proofs
|
|
of Three Theorems on Elliptic Integrals</em></span>, SIAM Journal on Mathematical
|
|
Analysis, vol 9, 524 (1978).
|
|
</li>
|
|
<li class="listitem">
|
|
<a name="ellint_ref_carlson81"></a>B.C. Carlson and E.M. Notis, <span class="emphasis"><em>ALGORITHM
|
|
577: Algorithms for Incomplete Elliptic Integrals</em></span>, ACM Transactions
|
|
on Mathematmal Software, vol 7, 398 (1981).
|
|
</li>
|
|
<li class="listitem">
|
|
B. C. Carlson, <span class="emphasis"><em>On computing elliptic integrals and functions</em></span>.
|
|
J. Math. and Phys., 44 (1965), pp. 36-51.
|
|
</li>
|
|
<li class="listitem">
|
|
B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals of the second
|
|
kind</em></span>. Math. Comp., 49 (1987), pp. 595-606. (Supplement, ibid.,
|
|
pp. S13-S17.)
|
|
</li>
|
|
<li class="listitem">
|
|
B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals of the third kind</em></span>.
|
|
Math. Comp., 51 (1988), pp. 267-280. (Supplement, ibid., pp. S1-S5.)
|
|
</li>
|
|
<li class="listitem">
|
|
B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: cubic cases</em></span>.
|
|
Math. Comp., 53 (1989), pp. 327-333.
|
|
</li>
|
|
<li class="listitem">
|
|
B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: one quadratic
|
|
factor</em></span>. Math. Comp., 56 (1991), pp. 267-280.
|
|
</li>
|
|
<li class="listitem">
|
|
B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: two quadratic
|
|
factors</em></span>. Math. Comp., 59 (1992), pp. 165-180.
|
|
</li>
|
|
<li class="listitem">
|
|
B. C. Carlson, <span class="emphasis"><em><a href="http://arxiv.org/abs/math.CA/9409227" target="_top">Numerical
|
|
computation of real or complex elliptic integrals</a></em></span>.
|
|
Numerical Algorithms, Volume 10, Number 1 / March, 1995, p13-26.
|
|
</li>
|
|
<li class="listitem">
|
|
B. C. Carlson and John L. Gustafson, <span class="emphasis"><em><a href="http://arxiv.org/abs/math.CA/9310223" target="_top">Asymptotic
|
|
Approximations for Symmetric Elliptic Integrals</a></em></span>, SIAM
|
|
Journal on Mathematical Analysis, Volume 25, Issue 2 (March 1994), 288-303.
|
|
</li>
|
|
</ol></div>
|
|
<p>
|
|
The following references, while not directly relevent to our implementation,
|
|
may also be of interest:
|
|
</p>
|
|
<div class="orderedlist"><ol class="orderedlist" type="1">
|
|
<li class="listitem">
|
|
R. Burlisch, <span class="emphasis"><em>Numerical Compuation of Elliptic Integrals and
|
|
Elliptic Functions.</em></span> Numerical Mathematik 7, 78-90.
|
|
</li>
|
|
<li class="listitem">
|
|
R. Burlisch, <span class="emphasis"><em>An extension of the Bartky Transformation to Incomplete
|
|
Elliptic Integrals of the Third Kind</em></span>. Numerical Mathematik
|
|
13, 266-284.
|
|
</li>
|
|
<li class="listitem">
|
|
R. Burlisch, <span class="emphasis"><em>Numerical Compuation of Elliptic Integrals and
|
|
Elliptic Functions. III</em></span>. Numerical Mathematik 13, 305-315.
|
|
</li>
|
|
<li class="listitem">
|
|
T. Fukushima and H. Ishizaki, <span class="emphasis"><em><a href="http://adsabs.harvard.edu/abs/1994CeMDA..59..237F" target="_top">Numerical
|
|
Computation of Incomplete Elliptic Integrals of a General Form.</a></em></span>
|
|
Celestial Mechanics and Dynamical Astronomy, Volume 59, Number 3 / July,
|
|
1994, 237-251.
|
|
</li>
|
|
</ol></div>
|
|
</div>
|
|
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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