Elliptic Exponential Function

The elliptic exponential function  gives the value of  in the elliptic logarithm

for  and  real such that .

It is implemented in the Wolfram Language as EllipticExp[u, a, b], which returns  together with the superfluous parameter  which multiplies the above integral by a factor of .

The top plot above shows  (red),  (violet), and  (blue) for . The other plots show  in the complex plane.

The plots above show  in the complex plane for .

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

REFERENCES:

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