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I need to start with elliptic function and elliptic integrals. I would like to have three or four fundamental references. If possible one easy and introductory with the very basic stuff, two intermediate and finally the "must have" bible on the subject. I don't know if anyone has something to suggest me. If available also videos would be interesting. Thank you in advance

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Some references for Elliptic Functions are:

  1. ("Easy") Complex Analysis (Ch.5, Ch.6) - E. Freitag & R.Busam

  2. (Intermediate) Modular Functions and Dirichlet Series in Number Theory - T. Apostol

  3. (Intermediate) Elliptic Functions - J.V. Armitage & W.F. Eberlein

  4. (Reference) Die elliptischen Funktionen und ihre Anwendungen ( Erster, Zweiter, Dritter Teil ) - Robert Fricke

UPDATE 10/5-'21 (Intermediate) Elements of the Theory of Elliptic Functions - N.I. Akhiezer

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I found the complex analysis text by Frietag very good for getting started on Elliptic Functions.

After the first half of the book on complex analysis you have the necessary information about meromorphic functions, the way residues, zeros and poles match up and also the weierstrass product and mittag-lefler partial fractions methods. This makes the construction of $\wp$ transparent as the search for a natural meromorphic function with poles on lattice points.

Then the theory of Laurent series is applied to get the Eisenstein series and their relationships, as well as finding the elliptic curve equation. There is some abstract algebra next to study the field of elliptic functions on a lattice. The next chapter follows naturally by looking into modular transforms that preserve lattices and theta functions.

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