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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.jacobi.jac_over"></a><a class="link" href="jac_over.html" title="Overview of the Jacobi Elliptic Functions">Overview of the Jacobi
  28       Elliptic Functions</a>
  29 </h3></div></div></div>
  30 <p>
  31         There are twelve Jacobi Elliptic functions, of which the three copolar functions
  32         <span class="emphasis"><em>sn</em></span>, <span class="emphasis"><em>cn</em></span> and <span class="emphasis"><em>dn</em></span>
  33         are the most important as the other nine can be computed from these three
  34         <a href="#ftn.math_toolkit.jacobi.jac_over.f0" class="footnote" name="math_toolkit.jacobi.jac_over.f0"><sup class="footnote">[2]</sup></a> <a href="#ftn.math_toolkit.jacobi.jac_over.f1" class="footnote" name="math_toolkit.jacobi.jac_over.f1"><sup class="footnote">[3]</sup></a> <a href="#ftn.math_toolkit.jacobi.jac_over.f2" class="footnote" name="math_toolkit.jacobi.jac_over.f2"><sup class="footnote">[4]</sup></a>.
  35       </p>
  36 <p>
  37         These functions each take two arguments: a parameter, and a variable as described
  38         below.
  39       </p>
  40 <p>
  41         Like all elliptic functions these can be parameterised in a number of ways:
  42       </p>
  43 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  44 <li class="listitem">
  45             In terms of a parameter <span class="emphasis"><em>m</em></span>.
  46           </li>
  47 <li class="listitem">
  48             In terms of the elliptic modulus <span class="emphasis"><em>k</em></span> where <span class="emphasis"><em>m
  49             = k<sup>2</sup></em></span>.
  50           </li>
  51 <li class="listitem">
  52             In terms of the modular angle &#945;, where <span class="emphasis"><em>m = sin<sup>2</sup> &#8201;&#945;</em></span>.
  53           </li>
  54 </ul></div>
  55 <p>
  56         In our implementation, these functions all take the elliptic modulus <span class="emphasis"><em>k</em></span>
  57         as the parameter.
  58       </p>
  59 <p>
  60         In addition the variable <span class="emphasis"><em>u</em></span> is sometimes expressed as
  61         an amplitude &#966;, in our implementation we always use <span class="emphasis"><em>u</em></span>.
  62       </p>
  63 <p>
  64         Finally note that our functions all take the elliptic modulus <span class="emphasis"><em>k</em></span>
  65         as the <span class="bold"><strong>first</strong></span> argument - this is for alignment
  66         with the Elliptic Integrals (but is different from other implementations,
  67         for example Mathworks).
  68       </p>
  69 <p>
  70         A simple example comparing use of <a href="http://www.wolframalpha.com/" target="_top">Wolfram
  71         Alpha</a> with Boost.Math (including much higher precision using Boost.Multiprecision)
  72         is <a href="../../../../example/jacobi_zeta_example.cpp" target="_top">jacobi_zeta_example.cpp</a>.
  73       </p>
  74 <p>
  75         There are twelve functions for computing the twelve individual Jacobi elliptic
  76         functions: <a class="link" href="jacobi_cd.html" title="Jacobi Elliptic Function cd">jacobi_cd</a>,
  77         <a class="link" href="jacobi_cn.html" title="Jacobi Elliptic Function cn">jacobi_cn</a>, <a class="link" href="jacobi_cs.html" title="Jacobi Elliptic Function cs">jacobi_cs</a>,
  78         <a class="link" href="jacobi_dc.html" title="Jacobi Elliptic Function dc">jacobi_dc</a>, <a class="link" href="jacobi_dn.html" title="Jacobi Elliptic Function dn">jacobi_dn</a>,
  79         <a class="link" href="jacobi_ds.html" title="Jacobi Elliptic Function ds">jacobi_ds</a>, <a class="link" href="jacobi_nc.html" title="Jacobi Elliptic Function nc">jacobi_nc</a>,
  80         <a class="link" href="jacobi_nd.html" title="Jacobi Elliptic Function nd">jacobi_nd</a>, <a class="link" href="jacobi_ns.html" title="Jacobi Elliptic Function ns">jacobi_ns</a>,
  81         <a class="link" href="jacobi_sc.html" title="Jacobi Elliptic Function sc">jacobi_sc</a>, <a class="link" href="jacobi_sd.html" title="Jacobi Elliptic Function sd">jacobi_sd</a>
  82         and <a class="link" href="jacobi_sn.html" title="Jacobi Elliptic Function sn">jacobi_sn</a>.
  83       </p>
  84 <p>
  85         They are all called as for example:
  86       </p>
  87 <pre class="programlisting"><span class="identifier">jacobi_cs</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">u</span><span class="special">);</span>
  88 </pre>
  89 <p>
  90         Note however that these individual functions are all really thin wrappers
  91         around the function <a class="link" href="jacobi_elliptic.html" title="Jacobi Elliptic SN, CN and DN">jacobi_elliptic</a>
  92         which calculates the three copolar functions <span class="emphasis"><em>sn</em></span>, <span class="emphasis"><em>cn</em></span>
  93         and <span class="emphasis"><em>dn</em></span> in a single function call.
  94       </p>
  95 <div class="tip"><table border="0" summary="Tip">
  96 <tr>
  97 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td>
  98 <th align="left">Tip</th>
  99 </tr>
 100 <tr><td align="left" valign="top"><p>
 101           If you need more than one of these functions for a given set of arguments,
 102           it's most efficient to use <a class="link" href="jacobi_elliptic.html" title="Jacobi Elliptic SN, CN and DN">jacobi_elliptic</a>.
 103         </p></td></tr>
 104 </table></div>
 105 <div class="footnotes">
 106 <br><hr style="width:100; text-align:left;margin-left: 0">
 107 <div id="ftn.math_toolkit.jacobi.jac_over.f0" class="footnote"><p><a href="#math_toolkit.jacobi.jac_over.f0" class="para"><sup class="para">[2] </sup></a>
 108           <a href="http://en.wikipedia.org/wiki/Jacobi_elliptic_functions" target="_top">Wikipedia:
 109           Jacobi elliptic functions</a>
 110         </p></div>
 111 <div id="ftn.math_toolkit.jacobi.jac_over.f1" class="footnote"><p><a href="#math_toolkit.jacobi.jac_over.f1" class="para"><sup class="para">[3] </sup></a>
 112           <a href="http://mathworld.wolfram.com/JacobiEllipticFunctions.html" target="_top">Weisstein,
 113           Eric W. "Jacobi Elliptic Functions." From MathWorld - A Wolfram
 114           Web Resource.</a>
 115         </p></div>
 116 <div id="ftn.math_toolkit.jacobi.jac_over.f2" class="footnote"><p><a href="#math_toolkit.jacobi.jac_over.f2" class="para"><sup class="para">[4] </sup></a>
 117           <a href="http://dlmf.nist.gov/22" target="_top">Digital Library of Mathematical Functions:
 118           Jacobian Elliptic Functions, Reinhardt, W. P., Walker, O. L.</a>
 119         </p></div>
 120 </div>
 121 </div>
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 128       Daryle Walker and Xiaogang Zhang<p>
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