## File: Definition-of-Legendre-Forms.html

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 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596  GNU Scientific Library – Reference Manual: Definition of Legendre Forms

7.13.1 Definition of Legendre Forms

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.