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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1Maybe start here https://en.wikipedia.org/wiki/Jacobi_elliptic_functions and stop there https://dlmf.nist.gov/22. – Jul 14 '19 at 15:24
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Ok, now can you give 4 books in between? – Dac0 Jul 14 '19 at 20:46
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@Daco. Sorry, no. – Jul 14 '19 at 20:48
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I was writing an answer when I realized there is a very similar question already. Is there anything that is not answered in this question: https://math.stackexchange.com/questions/405383 ? Maybe this is also relevant: https://math.stackexchange.com/questions/181503 – Hrodelbert Jan 03 '20 at 15:59
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See the last part of my answer. – Paramanand Singh Aug 23 '20 at 03:19
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Some references for Elliptic Functions are:
("Easy") Complex Analysis (Ch.5, Ch.6) - E. Freitag & R.Busam
(Intermediate) Modular Functions and Dirichlet Series in Number Theory - T. Apostol
(Intermediate) Elliptic Functions - J.V. Armitage & W.F. Eberlein
(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
nilo de roock
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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.