# WeierstrassPPrime

WeierstrassPPrime[u,{g2,g3}]

gives the derivative of the Weierstrass elliptic function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• For certain special arguments, WeierstrassPPrime automatically evaluates to exact values.
• WeierstrassPPrime can be evaluated to arbitrary numerical precision.
• WeierstrassPPrime can be used with CenteredInterval objects. »

# 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(30)

### Numerical Evaluation(5)

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:

WeierstrassPPrime can be used with CenteredInterval objects:

### Specific Values(4)

Value at zero:

Find a value of x for which WeierstrassPPrime[x,1/2,1/2]=10:

WeierstrassPPrime automatically evaluates to simpler functions for certain parameters:

Find a few singular points of WeierstrassPPrime[x,{1/2,1/2}]:

### Visualization(2)

Plot the WeierstrassPPrime function for various parameters:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(10)

Real domain of WeierstrassPPrime:

WeierstrassPPrime is an odd function with respect to x:

WeierstrassPPrime threads elementwise over lists in its first argument: is not an analytic function of :

It has both singularities and discontinuities: is neither nondecreasing nor nonincreasing: is not injective: is surjective: is neither non-negative nor non-positive: is neither convex nor concave:

### Differentiation(2)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to :

### Integration(4)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

Definite integral of WeierstrassPPrime[z,{g2,g3}] over a period is 0:

More integrals:

### Series Expansions(3)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

## Applications(5)

Conformal map from a triangle to the upper halfplane:

Map a triangle:

Uniformization of a generic elliptic curve :

The parametrized uniformization:

Check the correctness of the uniformization:

Define the Dixon elliptic functions:

These functions are cubic generalizations of Cos and Sin:

Real and imaginary periods of the Dixon elliptic functions:

Plot the Dixon elliptic functions on the real line:

Visualize the Dixon elliptic functions in the complex plane:

Series expansions of the Dixon elliptic functions:

Plot an elliptic function over a period parallelogram:

Compute the invariants corresponding to the lemniscatic case of the Weierstrass elliptic function, in which the ratio of the periods is :

Parameterization of the ChenGackstatter minimal surface:

## Properties & Relations(2)

Integrate expressions involving WeierstrassPPrime:

WeierstrassPPrime is closely related to EllipticExpPrime:

Evaluate numerically:

Compare with the built-in function value:

## Possible Issues(1)

Machine-precision input is insufficient to give a correct answer:

Use arbitraryprecision arithmetic to obtain a correct result:

## Neat Examples(1)

Weierstrass functions are doubly periodic over the complex plane: