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<title>GNU Scientific Library &ndash; Reference Manual: Mathieu Functions</title>

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<a name="Mathieu-Functions"></a>
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Next: <a href="Power-Function.html#Power-Function" accesskey="n" rel="next">Power Function</a>, Previous: <a href="Logarithm-and-Related-Functions.html#Logarithm-and-Related-Functions" accesskey="p" rel="previous">Logarithm and Related Functions</a>, Up: <a href="Special-Functions.html#Special-Functions" accesskey="u" rel="up">Special Functions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Mathieu-Functions-1"></a>
<h3 class="section">7.26 Mathieu Functions</h3>
<a name="index-Mathieu-functions"></a>

<p>The routines described in this section compute the angular and radial
Mathieu functions, and their characteristic values.  Mathieu
functions are the solutions of the following two differential
equations:
</p>
<div class="example">
<pre class="example">d^2y/dv^2 + (a - 2q\cos 2v)y = 0
d^2f/du^2 - (a - 2q\cosh 2u)f = 0
</pre></div>

<p>The angular Mathieu functions <em>ce_r(x,q)</em>, <em>se_r(x,q)</em> are
the even and odd periodic solutions of the first equation, which is known as Mathieu&rsquo;s equation. These exist
only for the discrete sequence of  characteristic values <em>a=a_r(q)</em>
(even-periodic) and <em>a=b_r(q)</em> (odd-periodic).
</p>
<p>The radial Mathieu functions <em>Mc^{(j)}_{r}(z,q)</em>, <em>Ms^{(j)}_{r}(z,q)</em> are the solutions of the second equation,
which is referred to as Mathieu&rsquo;s modified equation.  The
radial Mathieu functions of the first, second, third and fourth kind
are denoted by the parameter <em>j</em>, which takes the value 1, 2, 3
or 4.
</p>










<p>For more information on the Mathieu functions, see Abramowitz and
Stegun, Chapter 20.  These functions are defined in the header file
<samp>gsl_sf_mathieu.h</samp>.
</p>
<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top">&bull; <a href="Mathieu-Function-Workspace.html#Mathieu-Function-Workspace" accesskey="1">Mathieu Function Workspace</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Mathieu-Function-Characteristic-Values.html#Mathieu-Function-Characteristic-Values" accesskey="2">Mathieu Function Characteristic Values</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Angular-Mathieu-Functions.html#Angular-Mathieu-Functions" accesskey="3">Angular Mathieu Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Radial-Mathieu-Functions.html#Radial-Mathieu-Functions" accesskey="4">Radial Mathieu Functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
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