Doubly Periodic Functions
Doubly periodic functions are meromorphic functions on the complex plane which are periodic in two different directions. Using classical complex analysis it is shown that the sum of residues and the sum of orders of poles and zeroes vanishes. The Weierstrass ℘-function is introduced which together with its derivative generates the field doubly periodic functions for given periods. Its Laurent expansion features the first “modular” functions: the holomorphic Eisenstein series.
KeywordsComplex Plane Holomorphic Function Periodic Function Meromorphic Function Zeta Function
- [Sil09]Silverman, J.H.: The Arithmetic of Elliptic Curves, 2nd edn. Graduate Texts in Mathematics, vol. 106. Springer, Dordrecht (2009) Google Scholar