Wolfram Language Paclet Repository

Community-contributed installable additions to the Wolfram Language

JanM/ Dixon

The Dixon elliptic functions in Mathematica

Contributed by: Jan Mangaldan

Alfred Cardew Dixon, in a paper from 1890, introduced two elliptic functions, cm(u,α) and sm(u,α), now named after him that parameterize the cubic curve x3+y3-3α x y=1. In the special case α =0, cm and sm, can be visualized as having "period hexagons" in the complex plane. Dixon's paper also introduced some associated functions that appear when the Dixon elliptic functions are differentiated or integrated with respect to their arguments.

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall["JanM/Dixon"]


To load the code after installation, evaluate this code:
Needs["JanM`Dixon`"]

Details

All functions from the Dixon paclet are suitable for symbolic or numeric evaluation.
Version 12.3 or newer of the Wolfram Language is required.

Paclet Guide

Examples

Basic Examples (3) 

Evaluate the Dixon elliptic functions numerically:

In[1]:=
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Out[1]=
In[2]:=
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Out[2]=

Plot the Dixon functions on the real line:

In[3]:=
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Out[3]=

Series expansions for the Dixon elliptic functions:

In[4]:=
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Out[4]=
In[5]:=
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Out[5]=

Scope (4) 

Periods of a Dixon elliptic function corresponding to the modulus 1/3:

In[6]:=
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Out[6]=

Visualize the associated Dixon function in the complex plane:

In[7]:=
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Out[7]=

Plot a unit-speed parameterization of the trefoil curve:

In[8]:=
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Out[8]=

For α =-1, the Dixon functions degenerate to elementary functions:

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Out[9]=

Publisher

Jan Mangaldan

Disclosures

Compatibility

Wolfram Language Version 12.3

Version History

  • 1.0.0 – 28 April 2023

License Information

MIT License

Paclet Source

Source Metadata

  • Citation:
    • Dixon, A. C. (1890). "On the doubly periodic functions arising out of the curve x³ + y³ − 3𝛼xy = 1". Quarterly Journal of Pure and Applied Mathematics. XXIV: 167–233.