Mathematical functions related to Weierstrass's elliptic function
For the fractal continuous function without a defined derivative, see Weierstrass function.
In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass. The relation between the sigma, zeta, and functions is analogous to that between the sine, cotangent, and squared cosecant functions: the logarithmic derivative of the sine is the cotangent, whose derivative is negative the squared cosecant.
Through careful manipulation of the Weierstrass factorization theorem as it relates also to the sine function, another potentially more manageable infinite product definition is
for any with and where we have used the notation (see zeta function below).
The Weierstrass zeta function should not be confused with the Riemann zeta function in number theory.
Weierstrass eta function[]
The Weierstrass eta function is defined to be
and any w in the lattice
This is well-defined, i.e. only depends on the lattice vector w. The Weierstrass eta function should not be confused with either the Dedekind eta function or the Dirichlet eta function.
The Weierstrass p-function is related to the zeta function by
The Weierstrass ℘-function is an even elliptic function of order N=2 with a double pole at each lattice point and no other poles.
Degenerate case[]
Consider the situation where one period is real, which we can scale to be and the other is taken to the limit of so that the functions are only singly-periodic. The corresponding invariants are of discriminant . Then we have and thus from the above infinite product definition the following equality:
A generalization for other sine-like functions on other doubly-periodic lattices is
This article incorporates material from Weierstrass sigma function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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