Elliptic Logarithm


The elliptic logarithm is generalization of integrals of the form


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


for a and b 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 a=b=1.

The elliptic logarithm is implemented in the Wolfram Language as EllipticLog[{x, y}, {a, b}], where y is an unfortunate and superfluous parameter that must be set to either y=sqrt(x^3+ax^2+bx) or y=-sqrt(x^3+ax^2+bx) and which multiplies the above integral by a factor of sqrt(y^2)/y.

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

See also

Elliptic Curve, Elliptic Exponential Function, Logarithm

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

Referenced on Wolfram|Alpha

Elliptic Logarithm

Cite this as:

Weisstein, Eric W. "Elliptic Logarithm." From MathWorld--A Wolfram Web Resource.

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