Elliptic Logarithm

The elliptic logarithm is generalization of integrals of the form

for  real, which can be expressed in terms of logarithmic and inverse trigonometric functions, to

for  and  real. This integral can be done analytically, but has a complicated form involving incomplete elliptic integrals of the first kind with complex parameters. The plots above show the special case .

The elliptic logarithm is implemented in the Wolfram Language as EllipticLog[x, y, a, b], where  is an unfortunate and superfluous parameter that must be set to either  or  and which multiplies the above integral by a factor of .

The inverse of the elliptic logarithm is the elliptic exponential function.

REFERENCES:

Wolfram, S. The Mathematica Book, 5th ed. Champaign, IL: Wolfram Media, p. 788, 2003.