Elliptic Exponential Function

The elliptic exponential function eexp_(a,b)(u) gives the value of x in the elliptic logarithm


for a and b real such that u=eln_(a,b)(x).

It is implemented in the Wolfram Language as EllipticExp[u, {a, b}], which returns x together with the superfluous parameter y=+/-sqrt(x^3+ax^2+bx) which multiplies the above integral by a factor of sqrt(y^2)/y.


The top plot above shows x(u) (red), y(u) (violet), and sqrt(y^2)/y (blue) for a=b=1. The other plots show x(z) in the complex plane.


The plots above show y(z) in the complex plane for a=b=1.

As can be seen from the plots, the elliptic exponential function is doubly periodic in the complex plane.

See also

Elliptic Logarithm, Weierstrass Elliptic Function

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

Referenced on Wolfram|Alpha

Elliptic Exponential Function

Cite this as:

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

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