Elliptic function

An elliptic function is a meromorphic function on the complex plane with two linearly independent periods. Its periodicity makes it descend to a meromorphic function on a complex torus.

Meromorphic function: A function that is holomorphic except at isolated poles. Elliptic functions belong to this class, with poles repeating across their period lattice.
Period lattice: The set of all integer combinations of a function’s periods. It records every translation that leaves an elliptic function unchanged.
Niels Henrik Abel: A Norwegian mathematician whose work established foundational results in algebra and the theory of elliptic functions. His independent development of elliptic-function theory clarified its structure and transformations.
Simple pendulum: A mass oscillating under gravity on a light rod or string of fixed length. At large amplitudes, its exact motion is expressed using Jacobi elliptic functions.
Complex lattice: A discrete subgroup of the complex plane generated by two linearly independent complex numbers. The integer combinations of two periods form the lattice of an elliptic function.
Fundamental parallelogram: A parallelogram whose lattice translates tile the complex plane without overlapping interiors. One such cell contains the complete pattern of an elliptic function’s poles and values.
Carl Gustav Jacob Jacobi: A German mathematician who made major contributions to elliptic functions, number theory, and mechanics. His systematic treatment introduced the theta functions and influential transformation formulas.
Nonlinear Schrödinger equation: A nonlinear partial differential equation describing the evolution of a complex wave envelope. Certain periodic wave solutions are written with Jacobi elliptic functions.
Complex torus: A quotient of the complex plane by a lattice, forming a compact Riemann surface of genus one. Periodicity identifies the plane’s lattice-equivalent points, producing the torus on which the function lives.
Weierstrass elliptic function: The elliptic function defined by a lattice sum with a double pole at each lattice point. It provides a canonical model from which every elliptic function can be expressed rationally.

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