- 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.
Examplesopen allclose all
Basic Examples (4)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Numerical Evaluation (4)
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]:
Plot the WeierstrassSigma function for various parameters:
Plot the real part of :
Plot the imaginary part of :
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:
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)