# EllipticExp

EllipticExp[u,{a,b}]

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.

# Examples

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## Basic Examples(2)

Evaluate numerically:

Check relation with the inverse function:

Plot the components of EllipticExp over several real periods:

## Scope(10)

### Numerical Evaluation(4)

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:

### Specific Values(3)

Values at fixed points:

Evaluate symbolically:

Value at zero:

### Visualization(2)

Plot the EllipticExp function for various parameters:

Plot the real part of EllipticExp[z,{1,2}]:

Plot the imaginary part of EllipticExp[z,{1,2}]:

### Differentiation(1)

First derivative with respect to u:

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

## Applications(4)

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:

Visualize the elliptic exponential in the complex plane:

Define multiplication on the elliptic curve :

Use multiplication on the elliptic curve to add rational numbers:

The value of EllipticLog at the product point equals the sum of values of EllipticLog at the corresponding factors:

## Properties & Relations(5)

Differentiation:

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

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

Compare numerical values:

Evaluate the elliptic exponential and its derivative:

EllipticExpPrime can be expressed in terms of the components of EllipticExp:

WeierstrassHalfPeriods can be used to compute the two linearly independent periods of EllipticExp:

Compare numerical evaluations of EllipticExp at congruent points in the complex plane:

## Possible Issues(1)

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

The discrepancy equals two periods of the lattice: