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Elliptic Exponential Function


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

 eln_(a,b)(x)=1/2int_infty^x(dt)/(sqrt(t^3+at^2+bt))

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.

EllipticExpEllipticExpReIm1EllipticExpContours1

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.

EllipticExpReIm2EllipticExpContours2

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

Related Wolfram sites

http://functions.wolfram.com/EllipticFunctions/EllipticExp/, http://functions.wolfram.com/EllipticFunctions/EllipticExpPrime/

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References

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. https://mathworld.wolfram.com/EllipticExponentialFunction.html

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