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| <div class="titlepage"><div><div><h3 class="title"> |
| <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 |
| Functions</a> |
| </h3></div></div></div> |
| <p> |
| There are twelve Jacobi Elliptic functions, of which the three copolar functions |
| <span class="emphasis"><em>sn</em></span>, <span class="emphasis"><em>cn</em></span> and <span class="emphasis"><em>dn</em></span> |
| are the most important as the other nine can be computed from these three |
| <a href="#ftn.math_toolkit.jacobi.jac_over.f0" class="footnote"><sup class="footnote"><a name="math_toolkit.jacobi.jac_over.f0"></a>[2]</sup></a> <a href="#ftn.math_toolkit.jacobi.jac_over.f1" class="footnote"><sup class="footnote"><a name="math_toolkit.jacobi.jac_over.f1"></a>[3]</sup></a> <a href="#ftn.math_toolkit.jacobi.jac_over.f2" class="footnote"><sup class="footnote"><a name="math_toolkit.jacobi.jac_over.f2"></a>[4]</sup></a>. |
| </p> |
| <p> |
| These functions each take two arguments: a parameter, and a variable as described |
| below. |
| </p> |
| <p> |
| Like all elliptic functions these can be parameterised in a number of ways: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| In terms of a parameter <span class="emphasis"><em>m</em></span>. |
| </li> |
| <li class="listitem"> |
| In terms of the elliptic modulus <span class="emphasis"><em>k</em></span> where <span class="emphasis"><em>m |
| = k<sup>2</sup></em></span>. |
| </li> |
| <li class="listitem"> |
| In terms of the modular angle α, where <span class="emphasis"><em>m = sin<sup>2</sup>α</em></span>. |
| </li> |
| </ul></div> |
| <p> |
| In our implementation, these functions all take the elliptic modulus <span class="emphasis"><em>k</em></span> |
| as the parameter. |
| </p> |
| <p> |
| In addition the variable <span class="emphasis"><em>u</em></span> is sometimes expressed as |
| an amplitude φ, in our implementation we always use <span class="emphasis"><em>u</em></span>. |
| </p> |
| <p> |
| Finally note that our functions all take the elliptic modulus as the first |
| argument - this is for alignment with the Elliptic Integrals. |
| </p> |
| <p> |
| There are twenve functions for computing the twelve individual Jacobi elliptic |
| functions: <a class="link" href="jacobi_cd.html" title="Jacobi Elliptic Function cd">jacobi_cd</a>, |
| <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>, |
| <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>, |
| <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>, |
| <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>, |
| <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> |
| and <a class="link" href="jacobi_sn.html" title="Jacobi Elliptic Function sn">jacobi_sn</a>. |
| </p> |
| <p> |
| They are all called as for example: |
| </p> |
| <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> |
| </pre> |
| <p> |
| Note however that these individual functions are all really thin wrappers |
| around the function <a class="link" href="jacobi_elliptic.html" title="Jacobi Elliptic SN, CN and DN">jacobi_elliptic</a> |
| which calculates the three copolar functions <span class="emphasis"><em>sn</em></span>, <span class="emphasis"><em>cn</em></span> |
| and <span class="emphasis"><em>dn</em></span> in a single function call. Thus if you need more |
| than one of these functions for a given set of arguments, it's most efficient |
| to use <a class="link" href="jacobi_elliptic.html" title="Jacobi Elliptic SN, CN and DN">jacobi_elliptic</a>. |
| </p> |
| <div class="footnotes"> |
| <br><hr style="width:100; align:left;"> |
| <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> |
| <a href="http://en.wikipedia.org/wiki/Jacobi_elliptic_functions" target="_top">Wikipedia: |
| Jacobi elliptic functions</a> |
| </p></div> |
| <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> |
| <a href="http://mathworld.wolfram.com/JacobiEllipticFunctions.html" target="_top">Weisstein, |
| Eric W. "Jacobi Elliptic Functions." From MathWorld - A Wolfram |
| Web Resource.</a> |
| </p></div> |
| <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> |
| <a href="http://dlmf.nist.gov/22" target="_top">Digital Library of Mathematical Functions: |
| Jacobian Elliptic Functions</a> |
| </p></div> |
| </div> |
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| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| 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>) |
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