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.

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  (23)

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  (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 TemplateBox[{z, 2, 1}, WeierstrassPPrime]:

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

Function Properties  (4)

Real domain of WeierstrassPPrime:

WeierstrassPPrime is an odd function with respect to x:

WeierstrassPPrime threads elementwise over lists in its first argument:

TraditionalForm formatting:

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  (3)

Conformal map from a triangle to the upper half‐plane:

Map a triangle:

Uniformization of a generic elliptic curve :

The parametrized uniformization:

Check the correctness of the uniformization:

Define Dixon trigonometric functions:

These functions are cubic generalizations of Cos and Sin:

Plot the Dixon trigonometric functions:

Series expansions of these functions:

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 arbitrary‐precision arithmetic to obtain a correct result:

Neat Examples  (1)

Weierstrass functions are doubly periodic over the complex plane:

Introduced in 1988
 (1.0)
 |
Updated in 1996
 (3.0)