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<html>
<head>
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<title>Elliptic Integrals - Carlson Form</title>
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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.ellint.ellint_carlson"></a><a class="link" href="ellint_carlson.html" title="Elliptic Integrals - Carlson Form">Elliptic Integrals
- Carlson Form</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.ellint.ellint_carlson.h0"></a>
<span class="phrase"><a name="math_toolkit.ellint.ellint_carlson.synopsis"></a></span><a class="link" href="ellint_carlson.html#math_toolkit.ellint.ellint_carlson.synopsis">Synopsis</a>
</h5>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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_rf</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<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>
<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<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_rf</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<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_rf</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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_rd</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<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>
<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<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_rd</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<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_rd</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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_rj</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<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>
<span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T4</span><span class="special">&gt;</span>
<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_rj</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">T4</span> <span class="identifier">p</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T4</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<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_rj</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">T4</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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_rc</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<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>
<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<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_rc</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<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_rc</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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_rg</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<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>
<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<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_rg</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<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_rg</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.ellint.ellint_carlson.h1"></a>
<span class="phrase"><a name="math_toolkit.ellint.ellint_carlson.description"></a></span><a class="link" href="ellint_carlson.html#math_toolkit.ellint.ellint_carlson.description">Description</a>
</h5>
<p>
These functions return Carlson's symmetrical elliptic integrals, the functions
have complicated behavior over all their possible domains, but the following
graph gives an idea of their behavior:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/ellint_carlson.svg" align="middle"></span>
</p></blockquote></div>
<p>
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
type calculation rules</em></span></a> when the arguments are of different
types: otherwise the return is the same type as the arguments.
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<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_rf</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<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_rf</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
</pre>
<p>
Returns Carlson's Elliptic Integral <span class="emphasis"><em>R<sub>F</sub></em></span>:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/ellint9.svg"></span>
</p></blockquote></div>
<p>
Requires that all of the arguments are non-negative, and at most one may
be zero. Otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<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_rd</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<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_rd</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
</pre>
<p>
Returns Carlson's elliptic integral R<sub>D</sub>:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/ellint10.svg"></span>
</p></blockquote></div>
<p>
Requires that x and y are non-negative, with at most one of them zero, and
that z &gt;= 0. Otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T4</span><span class="special">&gt;</span>
<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_rj</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">T4</span> <span class="identifier">p</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T4</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<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_rj</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">T4</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
</pre>
<p>
Returns Carlson's elliptic integral R<sub>J</sub>:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/ellint11.svg"></span>
</p></blockquote></div>
<p>
Requires that x, y and z are non-negative, with at most one of them zero,
and that <span class="emphasis"><em>p != 0</em></span>. Otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
When <span class="emphasis"><em>p &lt; 0</em></span> the function returns the <a href="http://en.wikipedia.org/wiki/Cauchy_principal_value" target="_top">Cauchy
principal value</a> using the relation:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/ellint17.svg"></span>
</p></blockquote></div>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<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_rc</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<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_rc</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
</pre>
<p>
Returns Carlson's elliptic integral R<sub>C</sub>:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/ellint12.svg"></span>
</p></blockquote></div>
<p>
Requires that <span class="emphasis"><em>x &gt; 0</em></span> and that <span class="emphasis"><em>y != 0</em></span>.
Otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
When <span class="emphasis"><em>y &lt; 0</em></span> the function returns the <a href="http://mathworld.wolfram.com/CauchyPrincipalValue.html" target="_top">Cauchy
principal value</a> using the relation:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/ellint18.svg"></span>
</p></blockquote></div>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<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_rg</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<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_rg</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;)</span>
</pre>
<p>
Returns Carlson's elliptic integral <span class="emphasis"><em>R<sub>G</sub>:</em></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/ellint27.svg"></span>
</p></blockquote></div>
<p>
Requires that x and y are non-negative, otherwise returns the result of
<a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<h5>
<a name="math_toolkit.ellint.ellint_carlson.h2"></a>
<span class="phrase"><a name="math_toolkit.ellint.ellint_carlson.testing"></a></span><a class="link" href="ellint_carlson.html#math_toolkit.ellint.ellint_carlson.testing">Testing</a>
</h5>
<p>
There are two sets of tests.
</p>
<p>
Spot tests compare selected values with test data given in:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
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, pp 13-26.
</p></blockquote></div>
<p>
Random test data generated using NTL::RR at 1000-bit precision and our implementation
checks for rounding-errors and/or regressions.
</p>
<p>
There are also sanity checks that use the inter-relations between the integrals
to verify their correctness: see the above Carlson paper for details.
</p>
<h5>
<a name="math_toolkit.ellint.ellint_carlson.h3"></a>
<span class="phrase"><a name="math_toolkit.ellint.ellint_carlson.accuracy"></a></span><a class="link" href="ellint_carlson.html#math_toolkit.ellint.ellint_carlson.accuracy">Accuracy</a>
</h5>
<p>
These functions are computed using only basic arithmetic operations, so there
isn't much variation in accuracy over differing platforms. Note that only
results for the widest floating-point type on the system are given as narrower
types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
zero error</a>. All values are relative errors in units of epsilon.
</p>
<div class="table">
<a name="math_toolkit.ellint.ellint_carlson.table_ellint_rc"></a><p class="title"><b>Table&#160;8.58.&#160;Error rates for ellint_rc</b></p>
<div class="table-contents"><table class="table" summary="Error rates for ellint_rc">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
<tbody><tr>
<td>
<p>
RC: Random data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 2.4&#949; (Mean = 0.624&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.995&#949; (Mean = 0.433&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.962&#949; (Mean = 0.407&#949;)</span>
</p>
</td>
</tr></tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.ellint.ellint_carlson.table_ellint_rd"></a><p class="title"><b>Table&#160;8.59.&#160;Error rates for ellint_rd</b></p>
<div class="table-contents"><table class="table" summary="Error rates for ellint_rd">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
RD: Random data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 2.59&#949; (Mean = 0.878&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.73&#949; (Mean = 0.831&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.16&#949; (Mean = 0.803&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RD: y = z
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.896&#949; (Mean = 0.022&#949;)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.88&#949; (Mean = 0.839&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.65&#949; (Mean = 0.82&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 16.5&#949; (Mean = 0.843&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RD: x = y
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.824&#949; (Mean = 0.0272&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 3.74&#949; (Mean = 0.84&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.85&#949; (Mean = 0.865&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.51&#949; (Mean = 0.816&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RD: x = 0, y = z
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 2&#949; (Mean = 0.656&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.19&#949; (Mean = 0.522&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.16&#949; (Mean = 0.497&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RD: x = y = z
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.03&#949; (Mean = 0.418&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.998&#949; (Mean = 0.387&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.03&#949; (Mean = 0.418&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RD: x = 0
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 2.85&#949; (Mean = 0.781&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.79&#949; (Mean = 0.883&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.64&#949; (Mean = 0.894&#949;)</span>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.ellint.ellint_carlson.table_ellint_rg"></a><p class="title"><b>Table&#160;8.60.&#160;Error rates for ellint_rg</b></p>
<div class="table-contents"><table class="table" summary="Error rates for ellint_rg">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
RG: Random Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.983&#949; (Mean = 0.0172&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0.983&#949; (Mean = 0.0172&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.95&#949; (Mean = 0.951&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.65&#949; (Mean = 0.929&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RG: two values 0
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0&#949; (Mean = 0&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RG: All values the same or zero
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0&#949; (Mean = 0&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.992&#949; (Mean = 0.288&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.06&#949; (Mean = 0.348&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RG: two values the same
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.594&#949; (Mean = 0.0103&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0.594&#949; (Mean = 0.0103&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.51&#949; (Mean = 0.404&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.96&#949; (Mean = 0.374&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RG: one value zero
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0&#949; (Mean = 0&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.14&#949; (Mean = 0.722&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.96&#949; (Mean = 0.674&#949;)</span>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.ellint.ellint_carlson.table_ellint_rf"></a><p class="title"><b>Table&#160;8.61.&#160;Error rates for ellint_rf</b></p>
<div class="table-contents"><table class="table" summary="Error rates for ellint_rf">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
RF: Random data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 2.73&#949; (Mean = 0.804&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.54&#949; (Mean = 0.674&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.02&#949; (Mean = 0.677&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RF: x = y = z
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.999&#949; (Mean = 0.34&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.991&#949; (Mean = 0.345&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.999&#949; (Mean = 0.34&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RF: x = y or y = z or x = z
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.536&#949; (Mean = 0.00658&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.89&#949; (Mean = 0.749&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.95&#949; (Mean = 0.418&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.21&#949; (Mean = 0.394&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RF: x = 0, y = z
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.29&#949; (Mean = 0.527&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.894&#949; (Mean = 0.338&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.999&#949; (Mean = 0.407&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RF: z = 0
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 2.54&#949; (Mean = 0.781&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.7&#949; (Mean = 0.539&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.89&#949; (Mean = 0.587&#949;)</span>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.ellint.ellint_carlson.table_ellint_rj"></a><p class="title"><b>Table&#160;8.62.&#160;Error rates for ellint_rj</b></p>
<div class="table-contents"><table class="table" summary="Error rates for ellint_rj">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
RJ: Random data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.52&#949; (Mean = 0.0184&#949;)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 3.57&#949; (Mean = 0.704&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_ellint_rj_GSL_2_1_RJ_Random_data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 186&#949; (Mean = 6.67&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 215&#949; (Mean = 7.66&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RJ: 4 Equal Values
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.03&#949; (Mean = 0.418&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.998&#949; (Mean = 0.387&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.03&#949; (Mean = 0.418&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RJ: 3 Equal Values
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 3.96&#949; (Mean = 1.06&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 20.8&#949; (Mean = 0.986&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 39.9&#949; (Mean = 1.17&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RJ: 2 Equal Values
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.6&#949; (Mean = 0.0228&#949;)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.57&#949; (Mean = 0.754&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 220&#949; (Mean = 6.64&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 214&#949; (Mean = 5.28&#949;)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
RJ: Equal z and p
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.742&#949; (Mean = 0.0166&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.62&#949; (Mean = 0.699&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 17.2&#949; (Mean = 1.16&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 16.1&#949; (Mean = 1.14&#949;)</span>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><h5>
<a name="math_toolkit.ellint.ellint_carlson.h4"></a>
<span class="phrase"><a name="math_toolkit.ellint.ellint_carlson.implementation"></a></span><a class="link" href="ellint_carlson.html#math_toolkit.ellint.ellint_carlson.implementation">Implementation</a>
</h5>
<p>
The key of Carlson's algorithm [<a class="link" href="ellint_intro.html#ellint_ref_carlson79">Carlson79</a>]
is the duplication theorem:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/ellint13.svg"></span>
</p></blockquote></div>
<p>
By applying it repeatedly, <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>,
<span class="emphasis"><em>z</em></span> get closer and closer. When they are nearly equal,
the special case equation
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/ellint16.svg"></span>
</p></blockquote></div>
<p>
is used. More specifically, <span class="emphasis"><em>[R F]</em></span> is evaluated from
a Taylor series expansion to the fifth order. The calculations of the other
three integrals are analogous, except for R<sub>C</sub> which can be computed from elementary
functions.
</p>
<p>
For <span class="emphasis"><em>p &lt; 0</em></span> in <span class="emphasis"><em>R<sub>J</sub>(x, y, z, p)</em></span>
and <span class="emphasis"><em>y &lt; 0</em></span> in <span class="emphasis"><em>R<sub>C</sub>(x, y)</em></span>, the integrals
are singular and their <a href="http://mathworld.wolfram.com/CauchyPrincipalValue.html" target="_top">Cauchy
principal values</a> are returned via the relations:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/ellint17.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/ellint18.svg"></span>
</p></blockquote></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 &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, 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>
</div></td>
</tr></table>
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