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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="Overvew of the Jacobi Elliptic Functions">Overvew of the Jacobi Elliptic
  28       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>&#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 as the first
  65         argument - this is for alignment with the Elliptic Integrals.
  66       </p>
  67 <p>
  68         There are twenve functions for computing the twelve individual Jacobi elliptic
  69         functions: <a class="link" href="jacobi_cd.html" title="Jacobi Elliptic Function cd">jacobi_cd</a>,
  70         <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>,
  71         <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>,
  72         <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>,
  73         <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>,
  74         <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>
  75         and <a class="link" href="jacobi_sn.html" title="Jacobi Elliptic Function sn">jacobi_sn</a>.
  76       </p>
  77 <p>
  78         They are all called as for example:
  79       </p>
  80 <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>
  81 </pre>
  82 <p>
  83         Note however that these individual functions are all really thin wrappers
  84         around the function <a class="link" href="jacobi_elliptic.html" title="Jacobi Elliptic SN, CN and DN">jacobi_elliptic</a>
  85         which calculates the three copolar functions <span class="emphasis"><em>sn</em></span>, <span class="emphasis"><em>cn</em></span>
  86         and <span class="emphasis"><em>dn</em></span> in a single function call. Thus if you need more
  87         than one of these functions for a given set of arguments, it's most efficient
  88         to use <a class="link" href="jacobi_elliptic.html" title="Jacobi Elliptic SN, CN and DN">jacobi_elliptic</a>.
  89       </p>
  90 <div class="footnotes">
  91 <br><hr style="width:100; text-align:left;margin-left: 0">
  92 <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>
  93           <a href="http://en.wikipedia.org/wiki/Jacobi_elliptic_functions" target="_top">Wikipedia:
  94           Jacobi elliptic functions</a>
  95         </p></div>
  96 <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>
  97           <a href="http://mathworld.wolfram.com/JacobiEllipticFunctions.html" target="_top">Weisstein,
  98           Eric W. "Jacobi Elliptic Functions." From MathWorld - A Wolfram
  99           Web Resource.</a>
 100         </p></div>
 101 <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>
 102           <a href="http://dlmf.nist.gov/22" target="_top">Digital Library of Mathematical Functions:
 103           Jacobian Elliptic Functions</a>
 104         </p></div>
 105 </div>
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