The respective complete elliptic integrals are obtained by setting the amplitude, Φ, the upper limit of the integrals, to π/2.
The Legendre form of an elliptic curve is given by y2 = x(x-1)(x-λ)
Numerical evaluation The classic method of evaluation is by means of Landen's transformations. Descending Landen transformation decreases the modulus k k towards zero, while increasing the amplitude Φ. Conversely, ascending transformation increases the modulus towards unity, while decreasing the amplitude. In either limit of k, zero or one, the integral is readily evaluated.
Most modern authors recommend evaluation in terms of the Carlson symmetric forms, for which there exist efficient, robust and relatively simple algorithms. This approach has been adopted by Boost C++ Libraries, GNU Scientific Library and Numerical Recipes.[3]
