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   4 <title>Jacobi Zeta Function</title>
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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.ellint.jacobi_zeta"></a><a class="link" href="jacobi_zeta.html" title="Jacobi Zeta Function">Jacobi Zeta Function</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.ellint.jacobi_zeta.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.ellint.jacobi_zeta.synopsis"></a></span><a class="link" href="jacobi_zeta.html#math_toolkit.ellint.jacobi_zeta.synopsis">Synopsis</a>
  32       </h5>
  33 <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">jacobi_zeta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34 </pre>
  35 <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>
  36
  37 <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>
  38 <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">jacobi_zeta</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">phi</span><span class="special">);</span>
  39
  40 <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>
  41 <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">jacobi_zeta</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">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>
  42
  43 <span class="special">}}</span> <span class="comment">// namespaces</span>
  44 </pre>
  45 <h5>
  46 <a name="math_toolkit.ellint.jacobi_zeta.h1"></a>
  47         <span class="phrase"><a name="math_toolkit.ellint.jacobi_zeta.description"></a></span><a class="link" href="jacobi_zeta.html#math_toolkit.ellint.jacobi_zeta.description">Description</a>
  48       </h5>
  49 <p>
  50         This function evaluates the Jacobi Zeta Function <span class="emphasis"><em>Z(&#966;, k)</em></span>
  51       </p>
  52 <div class="blockquote"><blockquote class="blockquote"><p>
  53           <span class="inlinemediaobject"><img src="../../../equations/jacobi_zeta.svg"></span>
  54
  55         </p></blockquote></div>
  56 <p>
  57         Please note the use of &#966;, and <span class="emphasis"><em>k</em></span> as the parameters, the
  58         function is often defined as <span class="emphasis"><em>Z(&#966;, m)</em></span> with <span class="emphasis"><em>m
  59         = k<sup>2</sup></em></span>, see for example <a href="http://mathworld.wolfram.com/JacobiZetaFunction.html" target="_top">Weisstein,
  60         Eric W. "Jacobi Zeta Function." From MathWorld--A Wolfram Web Resource.</a>
  61         Or else as <a href="https://dlmf.nist.gov/22.16#E32" target="_top"><span class="emphasis"><em>Z(x, k)</em></span></a>
  62         with <span class="emphasis"><em>&#966; = am(x, k)</em></span>, where <span class="emphasis"><em>am</em></span> is the
  63         <a href="https://dlmf.nist.gov/22.16#E1" target="_top">Jacobi amplitude function</a>
  64         which is equivalent to <span class="emphasis"><em>asin(jacobi_elliptic(k, x))</em></span>.
  65       </p>
  66 <p>
  67         The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
  68         type calculation rules</em></span></a> when the arguments are of different
  69         types: when they are the same type then the result is the same type as the
  70         arguments.
  71       </p>
  72 <p>
  73         Requires <span class="emphasis"><em>-1 &lt;= k &lt;= 1</em></span>, otherwise returns the result
  74         of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  75         (outside this range the result would be complex).
  76       </p>
  77 <p>
  78         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
  79         be used to control the behaviour of the function: how it handles errors,
  80         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
  81         documentation for more details</a>.
  82       </p>
  83 <p>
  84         Note that there is no complete analogue of this function (where &#966; = &#960; / 2) as
  85         this takes the value 0 for all <span class="emphasis"><em>k</em></span>.
  86       </p>
  87 <h5>
  88 <a name="math_toolkit.ellint.jacobi_zeta.h2"></a>
  89         <span class="phrase"><a name="math_toolkit.ellint.jacobi_zeta.accuracy"></a></span><a class="link" href="jacobi_zeta.html#math_toolkit.ellint.jacobi_zeta.accuracy">Accuracy</a>
  90       </h5>
  91 <p>
  92         These functions are trivially computed in terms of other elliptic integrals
  93         and generally have very low error rates (a few epsilon) unless parameter
  94         &#966;
  95 is very large, in which case the usual trigonometric function argument-reduction
  96         issues apply.
  97       </p>
  98 <div class="table">
  99 <a name="math_toolkit.ellint.jacobi_zeta.table_jacobi_zeta"></a><p class="title"><b>Table&#160;8.68.&#160;Error rates for jacobi_zeta</b></p>
 100 <div class="table-contents"><table class="table" summary="Error rates for jacobi_zeta">
 101 <colgroup>
 102 <col>
 103 <col>
 104 <col>
 105 <col>
 106 <col>
 107 </colgroup>
 108 <thead><tr>
 109 <th>
 110               </th>
 111 <th>
 112                 <p>
 113                   GNU C++ version 7.1.0<br> linux<br> double
 114                 </p>
 115               </th>
 116 <th>
 117                 <p>
 118                   GNU C++ version 7.1.0<br> linux<br> long double
 119                 </p>
 120               </th>
 121 <th>
 122                 <p>
 123                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 124                 </p>
 125               </th>
 126 <th>
 127                 <p>
 128                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 129                 </p>
 130               </th>
 131 </tr></thead>
 132 <tbody>
 133 <tr>
 134 <td>
 135                 <p>
 136                   Elliptic Integral Jacobi Zeta: Mathworld Data
 137                 </p>
 138               </td>
 139 <td>
 140                 <p>
 141                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 142                 </p>
 143               </td>
 144 <td>
 145                 <p>
 146                   <span class="blue">Max = 1.66&#949; (Mean = 0.48&#949;)</span>
 147                 </p>
 148               </td>
 149 <td>
 150                 <p>
 151                   <span class="blue">Max = 1.66&#949; (Mean = 0.48&#949;)</span>
 152                 </p>
 153               </td>
 154 <td>
 155                 <p>
 156                   <span class="blue">Max = 1.52&#949; (Mean = 0.357&#949;)</span>
 157                 </p>
 158               </td>
 159 </tr>
 160 <tr>
 161 <td>
 162                 <p>
 163                   Elliptic Integral Jacobi Zeta: Random Data
 164                 </p>
 165               </td>
 166 <td>
 167                 <p>
 168                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 169                 </p>
 170               </td>
 171 <td>
 172                 <p>
 173                   <span class="blue">Max = 2.99&#949; (Mean = 0.824&#949;)</span>
 174                 </p>
 175               </td>
 176 <td>
 177                 <p>
 178                   <span class="blue">Max = 3.96&#949; (Mean = 1.06&#949;)</span>
 179                 </p>
 180               </td>
 181 <td>
 182                 <p>
 183                   <span class="blue">Max = 3.89&#949; (Mean = 0.824&#949;)</span>
 184                 </p>
 185               </td>
 186 </tr>
 187 <tr>
 188 <td>
 189                 <p>
 190                   Elliptic Integral Jacobi Zeta: Large Phi Values
 191                 </p>
 192               </td>
 193 <td>
 194                 <p>
 195                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 196                 </p>
 197               </td>
 198 <td>
 199                 <p>
 200                   <span class="blue">Max = 2.92&#949; (Mean = 0.951&#949;)</span>
 201                 </p>
 202               </td>
 203 <td>
 204                 <p>
 205                   <span class="blue">Max = 3.05&#949; (Mean = 1.13&#949;)</span>
 206                 </p>
 207               </td>
 208 <td>
 209                 <p>
 210                   <span class="blue">Max = 2.52&#949; (Mean = 0.977&#949;)</span>
 211                 </p>
 212               </td>
 213 </tr>
 214 </tbody>
 215 </table></div>
 216 </div>
 217 <br class="table-break"><h5>
 218 <a name="math_toolkit.ellint.jacobi_zeta.h3"></a>
 219         <span class="phrase"><a name="math_toolkit.ellint.jacobi_zeta.testing"></a></span><a class="link" href="jacobi_zeta.html#math_toolkit.ellint.jacobi_zeta.testing">Testing</a>
 220       </h5>
 221 <p>
 222         The tests use a mixture of spot test values calculated using values calculated
 223         at <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, and random
 224         test data generated using MPFR at 1000-bit precision and a deliberately naive
 225         implementation in terms of the Legendre integrals.
 226       </p>
 227 <h5>
 228 <a name="math_toolkit.ellint.jacobi_zeta.h4"></a>
 229         <span class="phrase"><a name="math_toolkit.ellint.jacobi_zeta.implementation"></a></span><a class="link" href="jacobi_zeta.html#math_toolkit.ellint.jacobi_zeta.implementation">Implementation</a>
 230       </h5>
 231 <p>
 232         The implementation for Z(&#966;, k) first makes the argument &#966; positive using:
 233       </p>
 234 <div class="blockquote"><blockquote class="blockquote"><p>
 235           <span class="serif_italic"><span class="emphasis"><em>Z(-&#966;, k) = -Z(&#966;, k)</em></span></span>
 236         </p></blockquote></div>
 237 <p>
 238         The function is then implemented in terms of Carlson's integral R<sub>J</sub>
 239 using the
 240         relation:
 241       </p>
 242 <div class="blockquote"><blockquote class="blockquote"><p>
 243           <span class="inlinemediaobject"><img src="../../../equations/jacobi_zeta.svg"></span>
 244
 245         </p></blockquote></div>
 246 <p>
 247         There is one special case where the above relation fails: when <span class="emphasis"><em>k
 248         = 1</em></span>, in that case the function simplifies to
 249       </p>
 250 <div class="blockquote"><blockquote class="blockquote"><p>
 251           <span class="serif_italic"><span class="emphasis"><em>Z(&#966;, 1) = sign(cos(&#966;)) sin(&#966;)</em></span></span>
 252         </p></blockquote></div>
 253 <h6>
 254 <a name="math_toolkit.ellint.jacobi_zeta.h5"></a>
 255         <span class="phrase"><a name="math_toolkit.ellint.jacobi_zeta.jacobi_zeta_example"></a></span><a class="link" href="jacobi_zeta.html#math_toolkit.ellint.jacobi_zeta.jacobi_zeta_example">Example</a>
 256       </h6>
 257 <p>
 258         A simple example comparing use of <a href="http://www.wolframalpha.com/" target="_top">Wolfram
 259         Alpha</a> with Boost.Math (including much higher precision using Boost.Multiprecision)
 260         is <a href="../../../../example/jacobi_zeta_example.cpp" target="_top">jacobi_zeta_example.cpp</a>.
 261       </p>
 262 </div>
 263 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 264 <td align="left"></td>
 265 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 266       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 267       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 268       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 269       Daryle Walker and Xiaogang Zhang<p>
 270         Distributed under the Boost Software License, Version 1.0. (See accompanying
 271         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>)
 272       </p>
 273 </div></td>
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