In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Weierstrass functions are
special function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by ...
s of a
complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
that are auxiliary to the
Weierstrass elliptic function
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the ...
. They are named for
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
. 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.
Weierstrass sigma function
![Weierstrass sigma function](https://upload.wikimedia.org/wikipedia/commons/8/8b/Weierstrass_sigma_function.png)
The Weierstrass sigma function associated to a two-dimensional
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
is defined to be the product
:
where
denotes
or
are a ''
fundamental 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.
Definition
...
''.
Through careful manipulation of the
Weierstrass factorization theorem In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an e ...
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).
Weierstrass zeta function
![Weierstrass zeta function](https://upload.wikimedia.org/wikipedia/commons/e/e8/Weierstrass_zeta_function.png)
The Weierstrass zeta function is defined by the sum
:
The Weierstrass zeta function is the
logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula
\frac
where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f''; ...
of the sigma-function. The zeta function can be rewritten as:
:
where
is the
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...
of weight 2''k'' + 2.
The derivative of the zeta function is
, where
is the
Weierstrass elliptic function
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the ...
The Weierstrass zeta function should not be confused with the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
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
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string t ...
or the
Dirichlet eta function
In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0:
\eta(s) = \sum_^ = \frac - \frac + \frac - \frac + \cdo ...
.
Weierstrass ℘-function
![Weierstrass p function](https://upload.wikimedia.org/wikipedia/commons/6/61/Weierstrass_p_function.png)
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
:
{{PlanetMath attribution, id=4650, title=Weierstrass sigma function
Elliptic functions
Analytic functions