Mathematical functions related to Weierstrass's elliptic 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.
Weierstrass sigma function
The Weierstrass sigma function associated to a two-dimensional lattice
is defined to be the product
![{\displaystyle {\begin{aligned}\operatorname {\sigma } {(z;\Lambda )}&=z\prod _{w\in \Lambda ^{*}}\left(1-{\frac {z}{w}}\right)e^{z/w+{\frac {1}{2}}(z/w)^{2}}\\&=z\prod _{\begin{smallmatrix}m,n=-\infty \\\{m,n\}\neq 0\end{smallmatrix}}^{\infty }\left(1-{\frac {z}{m\omega _{1}+n\omega _{2}}}\right)e^{z/(m\omega _{1}+n\omega _{2})+{\frac {1}{2}}(z/(m\omega _{1}+n\omega _{2}))^{2}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee07c5cbab68c6486039bd2a61f23900f5fb1f62)
where
denotes
or
are a fundamental pair of periods.
Through careful manipulation of the Weierstrass factorization theorem as it relates also to the sine function, another potentially more manageable infinite product definition is
![{\displaystyle \operatorname {\sigma } {(z;\Lambda )}={\frac {\omega _{i}}{\pi }}e^{\eta _{i}z^{2}/\omega _{i}}\sin {\left({\frac {\pi z}{\omega _{i}}}\right)}\prod _{n=1}^{\infty }\left(1-{\frac {\sin ^{2}{\left({\tfrac {\pi z}{\omega _{i}}}\right)}}{\sin ^{2}{\left({\tfrac {n\pi \omega _{j}}{\omega _{i}}}\right)}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/435ea5396bf6a6a2fd79139f9c78509cee1e2038)
for any
with
and where we have used the notation
(see zeta function below).
Weierstrass zeta function
The Weierstrass zeta function is defined by the sum
![{\displaystyle \operatorname {\zeta } {(z;\Lambda )}={\frac {\sigma '(z;\Lambda )}{\sigma (z;\Lambda )}}={\frac {1}{z}}+\sum _{w\in \Lambda ^{*}}\left({\frac {1}{z-w}}+{\frac {1}{w}}+{\frac {z}{w^{2}}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c535bd841c88b1bb4c738b518b96ed7f3dbc65)
The Weierstrass zeta function is the logarithmic derivative of the sigma-function. The zeta function can be rewritten as:
![{\displaystyle \operatorname {\zeta } {(z;\Lambda )}={\frac {1}{z}}-\sum _{k=1}^{\infty }{\mathcal {G}}_{2k+2}(\Lambda )z^{2k+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d47b2f8a308f5dab775c7712dedc9770723c38c)
where
is the Eisenstein series of weight 2k + 2.
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.
Weierstrass eta function
The Weierstrass eta function is defined to be
and any w in the lattice ![{\displaystyle \Lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733)
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.
Weierstrass ℘-function
The Weierstrass p-function is related to the zeta function by
![{\displaystyle \operatorname {\wp } {(z;\Lambda )}=-\operatorname {\zeta '} {(z;\Lambda )},{\mbox{ for any }}z\in \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/589cc0cce69e02608b3786f43e75e5d9e97c758d)
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:
![{\displaystyle \operatorname {\sigma } {(z;\Lambda )}=2e^{z^{2}/24}\sin {\left({\tfrac {z}{2}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd9648e0f78cacc0363d415d5bd20bdfd8f53cf0)
A generalization for other sine-like functions on other doubly-periodic lattices is
![{\displaystyle f(z)={\frac {\pi }{\omega _{1}}}e^{-(4\eta _{1}/\omega _{1})z^{2}}\operatorname {\sigma } {(2z;\Lambda )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75b5462454c420a945448672d8437907e3f45d61)
This article incorporates material from Weierstrass sigma function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.