Weierstrass sigma function
Encyclopedia
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
.
is defined to be the product
where
denotes 
.
Note that the Weierstrass zeta-function is basically the logarithmic derivative
of the sigma-function. The zeta-function can be rewritten as:
where
is the Eisenstein series
of weight
.
Also note that the derivative of the zeta-function is
, where 
is the Weierstrass elliptic function
The Weierstrass zeta-function should not be confused with the Riemann zeta-function in number theory.
It can be proved that this is well-defined, i.e.
only depends on w. The Weierstrass eta-function should not be confused with the Dedekind eta-function
.
The Weierstrass p-function is an even elliptic function of order N=2 with a double pole at each lattice and no others.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
.
Weierstrass sigma-function
The Weierstrass sigma-function associated to a two-dimensional latticeFundamental pair of periods
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined....
is defined to be the productwhere
denotes .Weierstrass zeta-function
The Weierstrass zeta-function is defined by the sumNote that the Weierstrass zeta-function is basically the logarithmic derivative
Logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formulawhere f ′ is the derivative of f....
of the sigma-function. The zeta-function can be rewritten as:
where
is the Eisenstein seriesEisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly...
of weight
.Also note that the derivative of the zeta-function is
, where is the Weierstrass elliptic functionThe 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 beIt can be proved that this is well-defined, i.e.
only depends on w. The Weierstrass eta-function should not be confused with the Dedekind eta-functionDedekind eta function
The Dedekind eta function, named after Richard Dedekind, is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive...
.
Weierstrass p-function
The Weierstrass p-function is defined to beThe Weierstrass p-function is an even elliptic function of order N=2 with a double pole at each lattice and no others.