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<title>Definition of Legendre Forms - GNU Scientific Library -- Reference Manual</title>
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<h4 class="subsection">7.13.1 Definition of Legendre Forms</h4>
<p><a name="index-Legendre-forms-of-elliptic-integrals-442"></a>The Legendre forms of elliptic integrals F(\phi,k),
E(\phi,k) and \Pi(\phi,k,n) are defined by,
<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>
<p class="noindent">The complete Legendre forms are denoted by K(k) = F(\pi/2, k) and
E(k) = E(\pi/2, k).
<p>The notation used here is based on Carlson, <cite>Numerische
Mathematik</cite> 33 (1979) 1 and differs slightly from that used by
Abramowitz & Stegun, where the functions are given in terms of the
parameter m = k^2 and n is replaced by -n.
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