WeierstrassSigma

WeierstrassSigma[u,{g2,g3}]

gives the Weierstrass sigma function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Related to WeierstrassZeta by the differential equation .
  • WeierstrassSigma is not periodic and is therefore not strictly an elliptic function.
  • For certain special arguments, WeierstrassSigma automatically evaluates to exact values.
  • WeierstrassSigma can be evaluated to arbitrary numerical precision.

Examples

open allclose all

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  (19)

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)

Value at zero:

WeierstrassSigma automatically evaluates to simpler functions for certain parameters:

Find the first positive maximum of WeierstrassSigma[x,1/2,1/2]:

Visualization  (2)

Plot the WeierstrassSigma function for various parameters:

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

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

Function Properties  (5)

Real domain of WeierstrassSigma:

Complex domain of WeierstrassSigma:

Approximate function range of WeierstrassSigma:

WeierstrassSigma is an odd function with respect to x:

WeierstrassSigma 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 :

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  (2)

The system of coupled nonlinear differential equations for a heavy symmetric top:

The solutions can be expressed through Weierstrass sigma and zeta functions:

Numerically check the correctness of the solutions:

Form any elliptic function with given periods, poles and zeros as a rational function of WeierstrassSigma:

Form an elliptic function with a single and a double zero and a triple pole:

Plot the resulting elliptic function:

Properties & Relations  (1)

Derivatives:

Neat Examples  (1)

Plot WeierstrassSigma over the complex plane:

Introduced in 1996
 (3.0)