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Elliptic function
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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).

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

f(z + a) = f(z + b) = f(z)   for all z in C
and such that a/b is not real. From this it follows that
f(z + ma + nb) = f(z)   for all z in C and all integers m and n.

In developments of the theory of elliptic functions, modern authors mostly follow Karl Weierstrass: the notations of Weierstrass's elliptic functions based on his -function are convenient, and any elliptic function can be expressed in terms of these. 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.

Elliptic functions are the inverse functions of elliptic integrals, which is how they were introduced historically.

Any complex number ω such that f(z + ω) = f(z) for all z in C is called a period of f. If the two periods a and b are such that any other period ω can be written as ω = ma + nb with integers m and n, then a and b are called fundamental periods. Every elliptic function has a pair of fundamental periods, but this pair is not unique.

If a and b are fundamental periods, then any parallelogram with vertices z, z + a, z + b, z + a + b is called a fundamental parallelogram. Shifting such a parallelogram by integral multiples of a and b yields a copy of the parallelogram, and the function f behaves identically on all these copies, because of the periodicity.

The number of poles in any fundamental parallelogram is finite (and the same for all fundamental parallelograms). Unless the elliptic function is constant, any fundamental parallelogram has at least one pole, a consequence of Liouville's theorem.

The sum of the orders of the poles in any fundamental parallelogram is called the order of the elliptic function. The sum of the residues of the poles in any fundamental parallelogram is equal to zero, so in particular no elliptic function can have order one.

The derivative of an elliptic function is again an elliptic function, with the same periods. The set of all elliptic functions with the same fundamental periods form a field.

References