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( I have seen this question: How to start with elliptic functions but it has no answer. )

In Mathematica ( this is not a Mathematica question ) I calculated the following sum:

Sum[E^(-\[Pi] n^2 x), {n, 1, Infinity}]

The answer is:

1/2 (-1 + EllipticTheta[3, 0, E^(-\[Pi] x)])

In my Mathematics studies I never came across this EllipticTheta function.

Which standard university mathematics course introduces this function?

What is your suggested book on the subject and in which standard ( graduate ? ) course is it taught?

nilo de roock
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  • I doubt if the classical theory as developed by Abel, Jacobi and Ramanujan is taught anywhere. The theory by Weierstrass should be available (if not exclusively as a separate course) as a part of complex analysis. – Paramanand Singh Aug 25 '20 at 12:24
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    You may have a look at the references mentioned in the last part of this answer. – Paramanand Singh Aug 25 '20 at 12:28
  • Thank you. I have found several sources for the level I am on ( B.Sc./ Beginning Graduate ): Modular forms: A classical and computational introduction by Lloyd Kilford looks promising because it is very hands on ( computing ). Also Apostol's follow-up on Analytic Number Theory: "Modular Forms and Dirichlet Series for NT" seems a good candidate to study. - I am looking at all this from an ANT perspective.- From your profile I noticed you are also a self-study person! – nilo de roock Aug 25 '20 at 17:50
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    Apostol book is great (I have a copy)! – Paramanand Singh Aug 25 '20 at 18:15
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    By the way, theta/elliptic functions is deeply interesting topic which is rather difficult and not widely studied. It took me quite sometime to figure out the fundamental ideas of this theory. – Paramanand Singh Aug 25 '20 at 18:18
  • Are you on facebook or twitter? – nilo de roock Aug 25 '20 at 18:20
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    Sorry, my only online presence is on this site and somewhat less frequently on my blog. I do have a twitter and facebook account but I haven't used them since years. Social media seems boring to me. – Paramanand Singh Aug 25 '20 at 18:23
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    You may also study theta/elliptic functions from my blog: https://paramanands.blogspot.com/p/archives.html?m=0 (search theta elliptic etc) – Paramanand Singh Aug 25 '20 at 18:24
  • I have added your blog to my collection of bookmarks here: https://papaly.com/ndroock1/d2Edj/Mathematics – nilo de roock Aug 25 '20 at 18:28

1 Answers1

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EllipticTheta[1] to EllipticTheta[4] are also called Jacobi theta functions: $$ \vartheta_3(z,q) = \sum _{n=-\infty}^{\infty }{q}^{{n}^{2}}{{\rm e}^{2\,inz}} $$

So Bellman may be OK. The book on Ramanujan theta functions probably assumes you already know about Jacobi theta functions.

Probably Mathematica has documentation on its functions ... Here is the reference that Maple's documentation recommends:

Chapter 16, "Jacobian Elliptic Functions and Theta Functions" of Handbook of Mathematical Functions edited by Abramowitz and Stegun

GEdgar
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  • But this answers none of my 4 questions, with respect, this should have been a comment. That they are also called Jacobi Theta functions is in the books I mentioned. – nilo de roock Aug 18 '20 at 10:48