2 - Elliptic functions

Summary

Elliptic functions are introduced via the method of images following a review of periodic functions, Poisson summation, the unfolding trick, and analytic continuation applied to the Riemann zeta-function. The differential equations and addition formulas obeyed by periodic and elliptic functions are deduced from their Kronecker–Eisenstein series representation. The classic constructions of elliptic functions, in terms of their zeros and poles, are presented in terms of the Weierstrass elliptic function, the Jacobi elliptic functions, and the Jacobi theta-functions. The elliptic function theory developed here is placed in the framework of elliptic curves, Abelian differentials, and Abelian integrals.