Elliptic function

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In complex analysis, a mathematical discipline, an elliptic function is a function defined on the complex plane that is periodic in two directions (a doubly periodic function) and at the same time is meromorphic. Historically, elliptic functions were discovered as inverse functions of elliptic integrals; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives.

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Definition

Formally, an elliptic function is a meromorphic function f defined on C for which there exist two non-zero complex numbers a and b with a/b not real, such that

wherever f(z) is defined. From this it follows that

There are two methods of constructing 'canonical' elliptic functions: those of Jacobi and Weierstrass. In the theory, modern authors mostly follow Karl Weierstrass: the notations of Weierstrass's elliptic functions based on his \wp-function are convenient, and any elliptic function can be expressed in terms of these. However it is the functions of Jacobi that appear most commonly in practical problems, especially the need to avoid complex numbers, having a mapping from real to real, where the imaginary part is unnecessary or physically insignificant. Weierstrass became interested in these functions as a student of Christoph Gudermann, a student of Carl Friedrich Gauss.

The elliptic functions introduced by Jacobi, and the auxiliary theta functions (not doubly periodic), are more complicated but important both for the history and for general theory. The primary difference between these two theories is that the Weierstrass functions have second-order and higher-order poles located at the corners of the periodic lattice, whereas the Jacobi functions have simple poles. The development of the Weierstrass theory is easier to present and understand, having fewer complications.

More generally, the study of elliptic functions is closely related to the study of modular functions and modular forms, a relationship proven by the modularity theorem. Examples of this relationship include the j-invariant, the Eisenstein series and the Dedekind eta function.

Properties

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