Elliptic function construction

Question

I know the methods to construct nonconstant elliptic function was given by Jacobi and Weirstrass .Is there any other method discovered for constructing elliptic functions?If yes, so where can I find? Tell me references .

Answer 1

The following can be seen as a generalization of Jacobi elliptic functions expressed as inverses of integrals, so it may not be what you’re looking for.

For an orthogonal lattice you can map a rectangle to the upper half plane and and extend it by Schwarz’ reflection principle. Note that the elliptic function in this case is the inverse of the (multi-valued) Schwarz-Christoffel mapping that maps the upper half plane onto a rectangle.

This construction can be generalized: Let $p(z)$ be a polynomial of degree $3$ or $4$ with simple roots and $p(0)=0$. Then the multi-valued function $$z \mapsto \int_0^z \frac{\mathrm{d} w}{\sqrt{p(w)}}$$ has an elliptic function as single valued inverse. In this case the period lattice is given by the period integrals $$\int_{\gamma} \frac{\mathrm{d} w}{\sqrt{p(w)}}$$ where $\gamma$ is any closed path winding around an even number of roots of $p$.

Answer 2

The book Development of Elliptic Functions According to Ramanujan by K. Venkatachaliengar edited by Shaun Cooper. The book Elliptic Functions according to Eisenstein and Kronecker by Andre Weil. The Wikipedia article Abel elliptic functions. There are other less well known approaches.