Need better resources and explanation to the Weierstrass functions

Question

I have been looking at Wikipedia's description of Weierstrass functions (see https://en.wikipedia.org/wiki/Weierstrass_functions) and they come across quite brief or terse.

I would like to see some examples of Weierstrass Zeta function (along with Eisenstein series) and Weierstrass Sigma function.

• https://en.wikipedia.org/wiki/Weierstrass_functions#Weierstrass_zeta_function
• https://en.wikipedia.org/wiki/Eisenstein_series
• https://en.wikipedia.org/wiki/Weierstrass_functions#Weierstrass_sigma_function

Had a look at Wolfram, they seem to go into even more details but not working out an example.

Resources like programs (with source) or graphs or animations explaining them will certainly help.

Answer 1

There are several books now on the topic of elliptic functions, but 100 years ago almost all of them were not in English. A notable exception is Harris Hancock's Lectures on the Theory of Elliptic Functions published in 1910, but reprinted by Dover Publications. It gives a detailed explanation of both the Jacobi and Weierstrass versions of Elliptic Functions with the relations between them.

Online handbook sources include Abramowitz and Stegun Handbook of Mathematical Functions, also reprinted by Dover Publications, with a chapter on Weierstrass elliptic functions. Its modern version, the NIST Digital Library of Mathematical Functions (DLMF) chapter 23 is comparable.

Answer 2

Thanks @somos for your detailed reply.

At the moment I'm looking for clarification on the following:

• what does the inverted V symbol (capital letter Lambda) symbolise, how do you represent it in terms of numbers, series? pairs? matrix of numbers? (these two questions applied to both the Weierstrass Sigma and Zeta functions)
• what is the output of this function/formula, again is it a single number or a series of them, a matrix of numbers?
• with regards to the zeta function, I would like to know how to apply this function, particular not sure how to generate Eisenstein series and the notations in https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd16a21a70d1bc62b04c8ac8b0c0b101b49d71a - I'm not familiar with them: G2k+2 * (Lambda) * z raise to 2k+2

G2k+2 - is this just a notation to express Eisenstein series of weight 2k+2? and takes input from the Lambda set? What does this set contain, and which range of numbers? I was reading its a lattice / pair of values - would like to see some examples of this.

Any help with explaining or clarifying what the math notations mean will be also helping in starting to understand these two functions, and also Eisenstein series.

Answer 3

I have actually partly been able to answer my questions from above, after finding these links:

• https://en.m.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions
• https://www.maplesoft.com/support/help/errors/view.aspx?path=Weierstrass

Maplesoft has a library of math functions and semi-okay documentation on how to use them, which helped me clarify about inputs and outputs (some examples on the docs page).

The Elliptic functions page also helped understand the Sigma & Zeta functions better. This page & the Maplesoft docs helped understand about Eisenstein series a bit more.

So I have been able to find out how to solve one of the two functions. I have still a couple of questions about the explanation on the Maplesoft docs, I'll take it up with them, but would be great if someone else could help me answer them in case they don't get back in time.