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
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