Imported Upstream version 1.72.0

[platform/upstream/boost.git] / libs / math / doc / html / math_toolkit / ellint / ellint_3.html

diff --git a/libs/math/doc/html/math_toolkit/ellint/ellint_3.html b/libs/math/doc/html/math_toolkit/ellint/ellint_3.html
index c28dd35..c122b58 100644 (file)
--- a/libs/math/doc/html/math_toolkit/ellint/ellint_3.html
+++ b/libs/math/doc/html/math_toolkit/ellint/ellint_3.html
@@ -4,7 +4,7 @@
 <title>Elliptic Integrals of the Third Kind - Legendre Form</title>
 <link rel="stylesheet" href="../../math.css" type="text/css">
 <meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
-<link rel="home" href="../../index.html" title="Math Toolkit 2.10.0">
+<link rel="home" href="../../index.html" title="Math Toolkit 2.11.0">
 <link rel="up" href="../ellint.html" title="Elliptic Integrals">
 <link rel="prev" href="ellint_2.html" title="Elliptic Integrals of the Second Kind - Legendre Form">
 <link rel="next" href="ellint_d.html" title="Elliptic Integral D - Legendre Form">
@@ -38,14 +38,14 @@
 <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_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">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;19.&#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_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&#160;19.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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_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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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_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">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;19.&#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_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&#160;19.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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_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&#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>
@@ -58,9 +58,10 @@
         kind <span class="emphasis"><em>&#928;(n, &#966;, k)</em></span> and its complete counterpart <span class="emphasis"><em>&#928;(n,
         k) = E(n, &#960;/2, k)</em></span>.
       </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../graphs/ellint_3.svg" align="middle"></span>
-      </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../graphs/ellint_3.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
@@ -70,48 +71,50 @@
 <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_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">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;19.&#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_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&#160;19.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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_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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 </pre>
 <p>
         Returns the incomplete elliptic integral of the third kind <span class="emphasis"><em>&#928;(n,
         &#966;, k)</em></span>:
       </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>
         Requires <span class="emphasis"><em>k<sup>2</sup>sin<sup>2</sup>(phi) &lt; 1</em></span> and <span class="emphasis"><em>n &lt; 1/sin<sup>2</sup>(&#966;)</em></span>,
         otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
         (outside this range the result would be complex).
       </p>
 <p>
-        The final <a class="link" href="../../policy.html" title="Chapter&#160;19.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+        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;19.&#160;Policies: Controlling Precision, Error Handling etc">policy
+        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">&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_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">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;19.&#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_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&#160;19.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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_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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 </pre>
 <p>
         Returns the complete elliptic integral of the first kind <span class="emphasis"><em>&#928;(n, k)</em></span>:
       </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../equations/ellint8.svg"></span>
-      </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/ellint8.svg"></span>
+
+        </p></blockquote></div>
 <p>
         Requires <span class="emphasis"><em>|k| &lt; 1</em></span> and <span class="emphasis"><em>n &lt; 1</em></span>,
         otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
         (outside this range the result would be complex).
       </p>
 <p>
-        The final <a class="link" href="../../policy.html" title="Chapter&#160;19.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+        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;19.&#160;Policies: Controlling Precision, Error Handling etc">policy
+        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>
@@ -126,7 +129,7 @@
         zero error</a>. All values are relative errors in units of epsilon.
       </p>
 <div class="table">
-<a name="math_toolkit.ellint.ellint_3.table_ellint_3"></a><p class="title"><b>Table&#160;7.65.&#160;Error rates for ellint_3</b></p>
+<a name="math_toolkit.ellint.ellint_3.table_ellint_3"></a><p class="title"><b>Table&#160;8.65.&#160;Error rates for ellint_3</b></p>
 <div class="table-contents"><table class="table" summary="Error rates for ellint_3">
 <colgroup>
 <col>
@@ -274,63 +277,70 @@
 <p>
         The implementation for &#928;(n, &#966;, k) first siphons off the special cases:
       </p>
-<p>
-        <span class="emphasis"><em>&#928;(0, &#966;, k) = F(&#966;, k)</em></span>
-      </p>
-<p>
-        <span class="emphasis"><em>&#928;(n, &#960;/2, k) = &#928;(n, k)</em></span>
-      </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="serif_italic"><span class="emphasis"><em>&#928;(0, &#966;, k) = F(&#966;, k)</em></span></span>
+        </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="serif_italic"><span class="emphasis"><em>&#928;(n, &#960;/2, k) = &#928;(n, k)</em></span></span>
+        </p></blockquote></div>
 <p>
         and
       </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../equations/ellint23.svg"></span>
-      </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/ellint23.svg"></span>
+
+        </p></blockquote></div>
 <p>
         Then if n &lt; 0 the relations (A&amp;S 17.7.15/16):
       </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../equations/ellint24.svg"></span>
-      </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>&#928;(n, -&#966;, k) = -&#928;(n, &#966;, k)</em></span></span>
+        </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="serif_italic"><span class="emphasis"><em>&#928;(n, &#966;+m&#960;, k) = &#928;(n, &#966;, k) + 2m&#928;(n, k)
+          ; n &lt;= 1</em></span></span>
+        </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="serif_italic"><span class="emphasis"><em>&#928;(n, &#966;+m&#960;, k) = &#928;(n, &#966;, k) ; n &gt; 1</em></span>
+          &#8198;&#8198;&#8198;  &#8198;&#8198;&#8198;
+<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>
-        <span class="emphasis"><em>&#928;(n, -&#966;, k) = -&#928;(n, &#966;, k)</em></span>
-      </p>
-<p>
-        <span class="emphasis"><em>&#928;(n, &#966;+m&#960;, k) = &#928;(n, &#966;, k) + 2m&#928;(n, k) ; n &lt;= 1</em></span>
-      </p>
-<p>
-        <span class="emphasis"><em>&#928;(n, &#966;+m&#960;, k) = &#928;(n, &#966;, k) ; n &gt; 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>
-      </p>
-<p>
-        are used to move &#966; &#160; to the range [0, &#960;/2].
+        are used to move &#966; to the range [0, &#960;/2].
       </p>
 <p>
         The functions are then implemented in terms of Carlson's integrals using
         the relations:
       </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../equations/ellint25.svg"></span>
-      </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+          <span class="inlinemediaobject"><img src="../../../equations/ellint25.svg"></span>
+
+        </p></blockquote></div>
 <p>
         and
       </p>
-<p>
-        <span class="inlinemediaobject"><img src="../../../equations/ellint26.svg"></span>
-      </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 * &#928;(n, k)</em></span> terms cancel out as the derivative
-          alternates between +&#8734; and -&#8734;.
-        </p></div>
+            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 * &#928;(n, k)</em></span> terms cancel out as the
+            derivative alternates between +&#8734; and -&#8734;.
+          </p></div>
 </div>
 </div>
 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>