The Legendre forms of elliptic integrals @math{F(\phi,k)}, @math{E(\phi,k)} and @math{P(\phi,k,n)} are defined by,
The complete Legendre forms are denoted by @math{K(k) = F(\pi/2, k)} and @math{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.