<title>Elliptic Integral Overview</title>
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D.C.
</p></blockquote></div>
<p>
+ and its recently revised version <a href="http://dlmf.nist.gov/" target="_top">NIST
+ Digital Library of Mathematical Functions (DMLF)</a>, in particular
+ </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <a href="https://dlmf.nist.gov/19" target="_top">Elliptic Integrals, B. C. Carlson</a>
+ </p></blockquote></div>
+<p>
Mathworld also contain a lot of useful background information:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<a name="math_toolkit.ellint.ellint_intro.h1"></a>
<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>
</h5>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint1.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint1.svg"></span>
+
+ </p></blockquote></div>
<p>
is called elliptic integral if <span class="emphasis"><em>R(t, s)</em></span> is a rational
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>
is a cubic or quartic polynomial in <span class="emphasis"><em>t</em></span>.
</p>
<p>
- Elliptic integrals generally can not be expressed in terms of elementary
- functions. However, Legendre showed that all elliptic integrals can be reduced
- to the following three canonical forms:
+ Elliptic integrals generally cannot be expressed in terms of elementary functions.
+ However, Legendre showed that all elliptic integrals can be reduced to the
+ following three canonical forms:
</p>
<p>
Elliptic Integral of the First Kind (Legendre form)
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint2.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint2.svg"></span>
+
+ </p></blockquote></div>
<p>
Elliptic Integral of the Second Kind (Legendre form)
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint3.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint3.svg"></span>
+
+ </p></blockquote></div>
<p>
Elliptic Integral of the Third Kind (Legendre form)
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint4.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint4.svg"></span>
+
+ </p></blockquote></div>
<p>
where
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint5.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint5.svg"></span>
+
+ </p></blockquote></div>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
<span class="emphasis"><em>φ</em></span> is called the amplitude.
</p>
<p>
- <span class="emphasis"><em>k</em></span> is called the modulus.
+ <span class="emphasis"><em>k</em></span> is called the elliptic modulus or eccentricity.
</p>
<p>
<span class="emphasis"><em>α</em></span> is called the modular angle.
<span class="emphasis"><em>k</em></span> (the modulus) may be expressed using a modular angle
α, or a parameter <span class="emphasis"><em>m</em></span>. These are related by:
</p>
-<p>
- k = sinα
- </p>
-<p>
- m = k<sup>2</sup> = sin<sup>2</sup>α
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">k = sin  α</span>
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">m = k<sup>2</sup> = sin<sup>2</sup>α</span>
+ </p></blockquote></div>
<p>
So that the integral of the third kind (for example) may be expressed as
either:
</p>
-<p>
- Π(n, φ, k)
- </p>
-<p>
- Π(n, φ \ α)
- </p>
-<p>
- Π(n, φ| m)
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">Π(n, φ, k)</span>
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">Π(n, φ \ α)</span>
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">Π(n, φ | m)</span>
+ </p></blockquote></div>
<p>
To further complicate matters, some texts refer to the <span class="emphasis"><em>complement
of the parameter m</em></span>, or 1 - m, where:
</p>
-<p>
- 1 - m = 1 - k<sup>2</sup> = cos<sup>2</sup>α
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">1 - m = 1 - k<sup>2</sup> = cos<sup>2</sup>α</span>
+ </p></blockquote></div>
<p>
This implementation uses <span class="emphasis"><em>k</em></span> throughout: this matches
the requirements of the <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf" target="_top">Technical
- Report on C++ Library Extensions</a>. However, you should be extra
- careful when using these functions!
+ Report on C++ Library Extensions</a>.<br>
+ </p>
+<p>
+ So you should be extra careful when using these functions!
</p>
</td></tr>
</table></div>
+<div class="warning"><table border="0" summary="Warning">
+<tr>
+<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../doc/src/images/warning.png"></td>
+<th align="left">Warning</th>
+</tr>
+<tr><td align="left" valign="top"><p>
+ Boost.Math order of arguments differs from other implementations: <span class="emphasis"><em>k</em></span>
+ is always the <span class="bold"><strong>first</strong></span> argument.
+ </p></td></tr>
+</table></div>
+<p>
+ A simple example comparing use of <a href="http://www.wolframalpha.com/" target="_top">Wolfram
+ Alpha</a> with Boost.Math (including much higher precision using <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>)
+ is <a href="../../../../example/jacobi_zeta_example.cpp" target="_top">jacobi_zeta_example.cpp</a>.
+ </p>
<p>
When <span class="emphasis"><em>φ</em></span> = <span class="emphasis"><em>π</em></span> / 2, the elliptic integrals
are called <span class="emphasis"><em>complete</em></span>.
<p>
Complete Elliptic Integral of the First Kind (Legendre form)
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint6.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint6.svg"></span>
+
+ </p></blockquote></div>
<p>
Complete Elliptic Integral of the Second Kind (Legendre form)
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint7.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint7.svg"></span>
+
+ </p></blockquote></div>
<p>
Complete Elliptic Integral of the Third Kind (Legendre form)
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint8.svg"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint8.svg"></span>
- </p>
-<p>
- Legendre also defined a forth integral D(φ,k) which is a combination of the
- other three:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span>
+ Legendre also defined a fourth integral /D(φ,k)/ which is a combination of
+ the other three:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span>
+
+ </p></blockquote></div>
<p>
Like the other Legendre integrals this comes in both complete and incomplete
forms.
<p>
Carlson's Elliptic Integral of the First Kind
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint9.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint9.svg"></span>
+
+ </p></blockquote></div>
<p>
where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, <span class="emphasis"><em>z</em></span>
are nonnegative and at most one of them may be zero.
<p>
Carlson's Elliptic Integral of the Second Kind
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint10.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint10.svg"></span>
+
+ </p></blockquote></div>
<p>
where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span> are nonnegative, at
most one of them may be zero, and <span class="emphasis"><em>z</em></span> must be positive.
<p>
Carlson's Elliptic Integral of the Third Kind
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint11.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint11.svg"></span>
+
+ </p></blockquote></div>
<p>
where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, <span class="emphasis"><em>z</em></span>
are nonnegative, at most one of them may be zero, and <span class="emphasis"><em>p</em></span>
<p>
Carlson's Degenerate Elliptic Integral
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint12.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint12.svg"></span>
+
+ </p></blockquote></div>
<p>
where <span class="emphasis"><em>x</em></span> is nonnegative and <span class="emphasis"><em>y</em></span> is
nonzero.
<p>
Carlson's Symmetric Integral
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint27.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint27.svg"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.ellint.ellint_intro.h3"></a>
<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
Carlson proved in [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>]
that
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint13.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint13.svg"></span>
+
+ </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
<p>
The Legendre form and Carlson form of elliptic integrals are related by equations:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint14.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint14.svg"></span>
+
+ </p></blockquote></div>
<p>
In particular,
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/ellint15.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/ellint15.svg"></span>
+
+ </p></blockquote></div>
<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
There are two functions related to the elliptic integrals which otherwise
defy categorisation, these are the Jacobi Zeta function:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/jacobi_zeta.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/jacobi_zeta.svg"></span>
+
+ </p></blockquote></div>
<p>
and the Heuman Lambda function:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>
+
+ </p></blockquote></div>
<p>
Both of these functions are easily implemented in terms of Carlson's integrals,
and are provided in this library as <a class="link" href="jacobi_zeta.html" title="Jacobi Zeta Function">jacobi_zeta</a>
The conventional methods for computing elliptic integrals are Gauss and Landen
transformations, which converge quadratically and work well for elliptic
integrals of the first and second kinds. Unfortunately they suffer from loss
- of significant digits for the third kind. Carlson's algorithm [<a class="link" href="ellint_intro.html#ellint_ref_carlson79">Carlson79</a>]
+ of significant digits for the third kind.
+ </p>
+<p>
+ 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.
Special mention goes to:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
- A. M. Legendre, <span class="emphasis"><em>Traitd des Fonctions Elliptiques et des Integrales
+ 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>
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).
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