Weierstrass Elliptic Functions in Python, SageMath, and Julia

This repository contains:

• WeierstrassP.ipynb - A Jupyter Notebook demonstrating how to implement Weierstrass elliptic functions in Python.
• weierstrass.py - A Python library containing coded implementations of Weierstrass elliptic functions.

Implementation Details

The Jupyter Notebook illustrates how to compute the following Weierstrass elliptic functions:

• $\wp(z)$ – the Weierstrass $\wp$ function
• $\wp'(z)$ – its first derivative
• $\sigma(z)$ – the Weierstrass sigma function
• $\zeta(z)$ – the Weierstrass zeta function

Additionally, the inverse function $\wp^{-1}$ is introduced, along with essential quantities such as:

• The half-periods $\omega_i$ derived from lattice invariants $g_2$, $g_3$.
• Expressions for $g_2$, $g_3$ in terms of the half-periods.

The functions implemented in this notebook are equivalent to those available in Wolfram Mathematica 14. The code relies on standard mathematical references, including:

• DLMF – Digital Library of Mathematical Functions
• Wolfram Function Repository
• Carlson Symmetric Forms (Wikipedia)

Requirements (Python)

To run the notebook, the following libraries are required:

• mpmath
• numpy

To-Do List

• Code a continous expresion for $\ln{\frac{\sigma(x-y)}{\sigma(x+y)}}$
• Develop a SageMath implementation.
• Develop a Julia implementation.

Feel Free to Contribute!

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Thank you for your interest!