### Definition of Legendre Forms

The Legendre forms of elliptic integrals F(\phi,k), E(\phi,k) and P(\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)))

    P(\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). Further information on the Legendre forms of elliptic integrals can be found in Abramowitz & Stegun, Chapter 17. The notation used here is based on Carlson, Numerische Mathematik 33 (1979) 1 and differs slightly from that used by Abramowitz & Stegun.