GNU Scientific Library – Reference Manual: Definition of Legendre Forms

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#### 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, 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.