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<title>Elliptic Integrals of the Third Kind - Legendre Form</title>
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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_3"></a><a class="link" href="ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form">Elliptic Integrals of the
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Third Kind - Legendre Form</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.ellint.ellint_3.h0"></a>
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<span class="phrase"><a name="math_toolkit.ellint.ellint_3.synopsis"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.synopsis">Synopsis</a>
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</h5>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">ellint_3</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h5>
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<a name="math_toolkit.ellint.ellint_3.h1"></a>
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<span class="phrase"><a name="math_toolkit.ellint.ellint_3.description"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.description">Description</a>
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</h5>
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<p>
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These two functions evaluate the incomplete elliptic integral of the third
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kind <span class="emphasis"><em>Π(n, φ, k)</em></span> and its complete counterpart <span class="emphasis"><em>Π(n,
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k) = E(n, π/2, k)</em></span>.
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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="../../../graphs/ellint_3.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
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type calculation rules</em></span></a> when the arguments are of different
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types: when they are the same type then the result is the same type as the
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arguments.
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</p>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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</pre>
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<p>
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Returns the incomplete elliptic integral of the third kind <span class="emphasis"><em>Π(n,
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φ, k)</em></span>:
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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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Requires <span class="emphasis"><em>k<sup>2</sup>sin<sup>2</sup>(phi) < 1</em></span> and <span class="emphasis"><em>n < 1/sin<sup>2</sup>(φ)</em></span>,
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otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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(outside this range the result would be complex).
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
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documentation for more details</a>.
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</p>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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</pre>
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<p>
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Returns the complete elliptic integral of the first kind <span class="emphasis"><em>Π(n, k)</em></span>:
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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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Requires <span class="emphasis"><em>|k| < 1</em></span> and <span class="emphasis"><em>n < 1</em></span>,
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otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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(outside this range the result would be complex).
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
|
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be used to control the behaviour of the function: how it handles errors,
|
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what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
|
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documentation for more details</a>.
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</p>
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<h5>
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<a name="math_toolkit.ellint.ellint_3.h2"></a>
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<span class="phrase"><a name="math_toolkit.ellint.ellint_3.accuracy"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.accuracy">Accuracy</a>
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</h5>
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<p>
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These functions are computed using only basic arithmetic operations, so there
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isn't much variation in accuracy over differing platforms. Note that only
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results for the widest floating point type on the system are given as narrower
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types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
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zero error</a>. All values are relative errors in units of epsilon.
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</p>
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<div class="table">
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<a name="math_toolkit.ellint.ellint_3.table_ellint_3"></a><p class="title"><b>Table 8.65. Error rates for ellint_3</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for ellint_3">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5150<br> Sun Solaris<br> long double
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</p>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 14.1<br> Win32<br> double
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Elliptic Integral PI: Mathworld Data
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 475ε (Mean = 86.3ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> <span class="red">Max = +INFε (Mean
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= +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_3__cmath__Elliptic_Integral_PI_Mathworld_Data">And
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other failures.</a>)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 1.48e+05ε (Mean = 2.54e+04ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_ellint_3_GSL_2_1_Elliptic_Integral_PI_Mathworld_Data">And
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other failures.</a>)
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 475ε (Mean = 86.3ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 565ε (Mean = 102ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Elliptic Integral PI: Random Data
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 4.54ε (Mean = 0.895ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> <span class="red">Max = 3.37e+20ε (Mean
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= 3.47e+19ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_3__cmath__Elliptic_Integral_PI_Random_Data">And
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other failures.</a>)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 633ε (Mean = 50.1ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 4.49ε (Mean = 0.885ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 8.33ε (Mean = 0.971ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Elliptic Integral PI: Large Random Data
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 3.7ε (Mean = 0.893ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> <span class="red">Max = 2.52e+18ε (Mean
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= 4.83e+17ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_ellint_3__cmath__Elliptic_Integral_PI_Large_Random_Data">And
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other failures.</a>)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.557ε (Mean = 0.0389ε)</span><br>
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<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 40.1ε (Mean = 7.77ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 3.7ε (Mean = 0.892ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2.86ε (Mean = 0.944ε)</span>
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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</div>
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<br class="table-break"><h5>
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<a name="math_toolkit.ellint.ellint_3.h3"></a>
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<span class="phrase"><a name="math_toolkit.ellint.ellint_3.testing"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.testing">Testing</a>
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</h5>
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<p>
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The tests use a mixture of spot test values calculated using the online calculator
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at <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>,
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and random test data generated using NTL::RR at 1000-bit precision and this
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implementation.
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</p>
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<h5>
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<a name="math_toolkit.ellint.ellint_3.h4"></a>
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<span class="phrase"><a name="math_toolkit.ellint.ellint_3.implementation"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.implementation">Implementation</a>
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</h5>
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<p>
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The implementation for Π(n, φ, k) first siphons off the special cases:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic"><span class="emphasis"><em>Π(0, φ, k) = F(φ, k)</em></span></span>
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic"><span class="emphasis"><em>Π(n, π/2, k) = Π(n, k)</em></span></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
and
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/ellint23.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
Then if n < 0 the relations (A&S 17.7.15/16):
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/ellint24.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
are used to shift <span class="emphasis"><em>n</em></span> to the range [0, 1].
|
|
</p>
|
|
<p>
|
|
Then the relations:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic"><span class="emphasis"><em>Π(n, -φ, k) = -Π(n, φ, k)</em></span></span>
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic"><span class="emphasis"><em>Π(n, φ+mπ, k) = Π(n, φ, k) + 2mΠ(n, k)
|
|
; n <= 1</em></span></span>
|
|
</p></blockquote></div>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic"><span class="emphasis"><em>Π(n, φ+mπ, k) = Π(n, φ, k) ; n > 1</em></span>
|
|
       
|
|
<a href="#ftn.math_toolkit.ellint.ellint_3.f0" class="footnote" name="math_toolkit.ellint.ellint_3.f0"><sup class="footnote">[1]</sup></a></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
are used to move φ to the range [0, π/2].
|
|
</p>
|
|
<p>
|
|
The functions are then implemented in terms of Carlson's integrals using
|
|
the relations:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/ellint25.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
and
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/ellint26.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<div class="footnotes">
|
|
<br><hr style="width:100; text-align:left;margin-left: 0">
|
|
<div id="ftn.math_toolkit.ellint.ellint_3.f0" class="footnote"><p><a href="#math_toolkit.ellint.ellint_3.f0" class="para"><sup class="para">[1] </sup></a>
|
|
I haven't been able to find a literature reference for this relation,
|
|
but it appears to be the convention used by Mathematica. Intuitively
|
|
the first <span class="emphasis"><em>2 * m * Π(n, k)</em></span> terms cancel out as the
|
|
derivative alternates between +∞ and -∞.
|
|
</p></div>
|
|
</div>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
|
Daryle Walker and Xiaogang Zhang<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
|
</p>
|
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</div></td>
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</tr></table>
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