Legendre chose the name elliptic integrals because[1] the second kind gives the arc length of an ellipse of unit semi-major axis and eccentricity
A more detailed treatment of the Legendre forms is given in the main article on elliptic integrals.
The argument order shown above is that of Gradshteyn and Ryzhik[2] as well as Numerical Recipes.
[3] The choice of sign is that of Abramowitz and Stegun[4] as well as Gradshteyn and Ryzhik,[2] but corresponds to the
[3] The respective complete elliptic integrals are obtained by setting the amplitude,
The Legendre form of an elliptic curve is given by The classic method of evaluation is by means of Landen's transformations.
Descending Landen transformation decreases the modulus
Conversely, ascending transformation increases the modulus towards unity, while decreasing the amplitude.
Most modern authors recommend evaluation in terms of the Carlson symmetric forms, for which there exist efficient, robust and relatively simple algorithms.