# Jacobi Elliptic Functions built from Jacobi theta functions

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

• Looks like it might be a good book, but $40 and several day waiting for shipping wasn't what i had in mind. I was hoping for some online book or article. Thanks, i might see if any libraries in my area have it in stock tomorrow Jul 6, 2015 at 3:31 • Actually a quick google book search and I found a great preview, Thanks. ^.^ Jul 6, 2015 at 3:39 • @T.Poindexter Glad I could help. That book is a classic and I really like the treatment of elliptic functions done there, it covers up pretty much all you must know about them Jul 6, 2015 at 3:59 • from the Weierstrass$\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\$ Jul 10, 2019 at 4:20