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The Legendre forms of elliptic integrals *F(\phi,k)*,
*E(\phi,k)* and *\Pi(\phi,k,n)* are defined by,

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)))

The complete Legendre forms are denoted by *K(k) = F(\pi/2, k)* and
*E(k) = E(\pi/2, k)*.

The notation used here is based on Carlson, Numerische
Mathematik 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*.