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I believe I understand the general theory of elliptic functions to an extent.

What I can't seem to find is the distinct method which was used to show that a particular combination of Jacobi Theta functions defined any specific elliptic function.

So my question is, how would I go about defining Weierstrass-$\wp$, $\text{sn}, \text{cn}$, or $\text{dn}$ elliptic functions in terms of Jacobi elliptic functions. Reference material or direct answers would help. Thanks

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For a basic treatment of elliptic functions I strongly recommend you the book Theory of Functions of a Complex Variable by A.I. Markushevich. Especially take a look at part III chapters 5 and 6. Chapter 5 is a complete treatment on Weierstrass theory, meanwhile chapter 6 introduces Jacobi's theory and the relation to the Weierstrass one.

Here is a screen shot from what is cover on the book

enter image description here

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  • $\begingroup$ Looks like it might be a good book, but $40 and several day waiting for shipping wasn't what i had in mind. I was hoping for some online book or article. Thanks, i might see if any libraries in my area have it in stock tomorrow $\endgroup$ Jul 6, 2015 at 3:31
  • $\begingroup$ Actually a quick google book search and I found a great preview, Thanks. ^.^ $\endgroup$ Jul 6, 2015 at 3:39
  • $\begingroup$ @T.Poindexter Glad I could help. That book is a classic and I really like the treatment of elliptic functions done there, it covers up pretty much all you must know about them $\endgroup$ Jul 6, 2015 at 3:59
  • $\begingroup$ from the Weierstrass $\sigma$ functions for $\Im(\tau) > 0$ they obtain $S_\tau(z)$ an entire function such that $S_\tau(z+1)= S_\tau(z),e^{-i\pi \tau} S_\tau(z+\tau) =e^{2i \pi z} S_\tau(z)$ and its Fourier series is easily shown to be $S_\tau(z) = a_\tau(0)+a_\tau(0) e^{2i \pi n z}\sum_{n=1}^\infty e^{-i \pi n^2 \tau} e^{2i \pi n z} = \frac{a_\tau(0)}{2} + \frac{a_\tau(0)}{2} \vartheta(-\tau,z)$. The constant term is found from analyticity in $\tau$ $\endgroup$
    – reuns
    Jul 10, 2019 at 4:20

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