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.