Elliptic function

In complex analysis, an elliptic function is, roughly speaking , a function defined on the complex plane which is periodic in two directions. The elliptic functions can be seen as analogs of the trigonometric functions (which have a single period only). 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.

Related Topics:
Complex analysis - Function - Complex plane - Periodic - Trigonometric function - Elliptic integral - Arc length - Ellipse

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Formally, an elliptic function is a meromorphic function f defined on C for which there exist two non-zero complex numbers a and b such that

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:f(z + a) = f(z + b) = f(z)   for all z in C

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and such that a/b is not real. From this it follows that

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:f(z + ma + nb) = f(z)   for all z in C and all integers m and n.

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In developments of the theory of elliptic functions, 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. Weierstrass became interested in these functions as a student of Christoph Gudermann, a student of Carl Friedrich Gauss. The elliptic functions introduced by Carl Jacobi, and the auxiliary theta functions (not doubly-periodic), are more complex; but important both for the history and for general theory. The primary difference between these two theories is that the Weierstrass functions have high-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.

Related Topics:
Karl Weierstrass - Weierstrass's elliptic functions - Christoph Gudermann - Carl Friedrich Gauss - Elliptic functions - Carl Jacobi - Theta function - Poles - Lattice

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More generally, the study of elliptic functions is closely related to the study of modular functions and modular forms, examples of which include the j-invariant, the Eisenstein series and the Dedekind eta function.

Related Topics:
Modular function - Modular forms - J-invariant - Eisenstein series - Dedekind eta function

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