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<title>GNU Scientific Library &ndash; Reference Manual: Definition of Legendre Forms</title>

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<a name="Definition-of-Legendre-Forms"></a>
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Next: <a href="Definition-of-Carlson-Forms.html#Definition-of-Carlson-Forms" accesskey="n" rel="next">Definition of Carlson Forms</a>, Up: <a href="Elliptic-Integrals.html#Elliptic-Integrals" accesskey="u" rel="up">Elliptic Integrals</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Definition-of-Legendre-Forms-1"></a>
<h4 class="subsection">7.13.1 Definition of Legendre Forms</h4>
<a name="index-Legendre-forms-of-elliptic-integrals"></a>
<p>The Legendre forms of elliptic integrals <em>F(\phi,k)</em>,
<em>E(\phi,k)</em> and <em>\Pi(\phi,k,n)</em> are defined by,
</p>
<div class="example">
<pre class="example">  F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))

  E(\phi,k) = \int_0^\phi dt   \sqrt((1 - k^2 \sin^2(t)))

Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))
</pre></div>

<p>The complete Legendre forms are denoted by <em>K(k) = F(\pi/2, k)</em> and
<em>E(k) = E(\pi/2, k)</em>.  
</p>
<p>The notation used here is based on Carlson, <cite>Numerische
Mathematik</cite> 33 (1979) 1 and differs slightly from that used by
Abramowitz &amp; Stegun, where the functions are given in terms of the
parameter <em>m = k^2</em> and <em>n</em> is replaced by <em>-n</em>.
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