Built-in Mathematica Symbol

WeierstrassSigma

WeierstrassSigma[u, {g₂, g₃}]
gives the Weierstrass sigma function

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WeierstrassSigma is not periodic and is therefore not strictly an elliptic function.

For certain special arguments, WeierstrassSigma automatically evaluates to exact values.

Evaluate numerically:

Series expansion:

Evaluate numerically:

Series expansion:

Evaluate for complex arguments and invariants:

Evaluate to high precision:

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

WeierstrassSigma threads element-wise over lists in its first argument:

WeierstrassSigma automatically evaluates to simpler functions for certain parameters:

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:

Derivatives:

Plot WeierstrassSigma over the complex plane: