EllipticExp

is the inverse for EllipticLog. It produces a list {x,y} such that u==EllipticLog[{x,y},{a,b}].

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
EllipticExp gives the generalized exponential associated with the elliptic curve .
For certain special arguments, EllipticExp automatically evaluates to exact values.
EllipticExp can be evaluated to arbitrary numerical precision.

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Evaluate numerically:

Check relation with the inverse function:

Plot coordinates of the elliptic curve over several real periods of EllipticExp:

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Values at fixed points:

Evaluate symbolically:

Value at zero:

Plot the EllipticExp function for various parameters:

Plot the real part of :

Plot the imaginary part of :

First derivative with respect to u:

Plot the first derivative with respect to u when a=10 and b=1/3:

Define multiplication on the elliptic curve :

Use multiplication on the elliptic curve to add rational numbers:

Compare with EllipticLog:

Map integers on an elliptic curve:

Differentiation:

The point returned by EllipticExp[u,{a,b}] satisfies :

EllipticExp is closely related to the WeierstrassP function and its derivative:

Compare numerical values:

EllipticExp is a doubly periodic complex function, so the inverse relation does not always hold:

The discrepancy equals two periods of the lattice:

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