# InverseWeierstrassP

InverseWeierstrassP[p,{g2,g3}]

gives a value of u for which the Weierstrass function is equal to p.

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The value of u returned always lies in the fundamental period parallelogram defined by the complex halfperiods and .
• InverseWeierstrassP[{p,q},{g2,g3}] finds the unique value of u for which and . For such a value to exist, p and q must be related by .
• InverseWeierstrassP can be evaluated to arbitrary numerical precision.

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

## Scope(20)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

### Specific Values(4)

Values at fixed points:

Value at zero:

Find a value of x for which InverseWeierstrassP[x,{1,2}]=2:

### Visualization(2)

Plot the InverseWeierstrassP function for various parameters:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(4)

InverseWeierstrassP has both singularities and discontinuities: is injective: is neither non-negative nor non-positive:

It is complex-valued over part of the real axis is neither convex nor concave:

It is complex-valued over part of the real axis:

### Differentiation(2)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to when and :

### Integration(2)

Compute the indefinite integral using Integrate:

Verify the antiderivative:

Definite integral:

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

## Generalizations & Extensions(1)

Evaluate the generalized form numerically:

These are the inverse relationships with WeierstrassP and WeierstrassPPrime:

## Applications(2)

Plot the real and imaginary part of InverseWeierstrassP:

Form derivatives:

## Properties & Relations(1)

InverseWeierstrassP is closely related to EllipticLog function:

Evaluate numerically:

Compare with the value of the built-in function:

## Possible Issues(2)

If the first argument does not represent a pair of values of Weierstrass functions, InverseWeierstrassP stays unevaluated:

InverseWeierstrassP evaluates to a vectorvalued first argument: