Weierstrass functions

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 ${\displaystyle \wp }$ 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.¹

The Weierstrass sigma function associated to a two-dimensional lattice ${\displaystyle \Lambda \subset \mathbb {C} }$ is defined to be the product

${\displaystyle {\begin{aligned}\operatorname {\sigma } {(z;\Lambda )}&=z\prod _{w\in \Lambda ^{*}}\left(1-{\frac {z}{w}}\right)\exp \left({\frac {z}{w}}+{\frac {1}{2}}\left({\frac {z}{w}}\right)^{2}\right)\\[5mu]&=z\prod _{\begin{smallmatrix}m,n=-\infty \\\{m,n\}\neq 0\end{smallmatrix}}^{\infty }\left(1-{\frac {z}{m\omega _{1}+n\omega _{2}}}\right)\exp {\left({\frac {z}{m\omega _{1}+n\omega _{2}}}+{\frac {1}{2}}\left({\frac {z}{m\omega _{1}+n\omega _{2}}}\right)^{2}\right)}\end{aligned}}}$

where ${\displaystyle \Lambda ^{*}}$ denotes ${\displaystyle \Lambda -\{0\}}$ and ${\displaystyle (\omega _{1},\omega _{2})}$ is 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

${\displaystyle \operatorname {\sigma } {(z;\Lambda )}={\frac {\omega _{i}}{\pi }}\exp {\left({\frac {\eta _{i}z^{2}}{\omega _{i}}}\right)}\sin {\left({\frac {\pi z}{\omega _{i}}}\right)}\prod _{n=1}^{\infty }\left(1-{\frac {\sin ^{2}{\left(\pi z/\omega _{i}\right)}}{\sin ^{2}{\left(n\pi \omega _{j}/\omega _{i}\right)}}}\right)}$

for any ${\displaystyle \{i,j\}\in \{1,2,3\}}$ with ${\displaystyle i\neq j}$ and where we have used the notation ${\displaystyle \eta _{i}=\zeta (\omega _{i}/2;\Lambda )}$ (see zeta function below). Also it is a "quasi-periodic" function, with the following property:

${\displaystyle \sigma (z+2\omega _{i})=-e^{2\eta _{i}(z+\omega _{i})}\sigma (z)}$

The sigma function can be used to represent an elliptic function: ${\displaystyle f(z+\omega _{i})=f(z)\quad i\in \{1,\ldots ,n\}}$ when knowing its zeros and poles that lie in the period parallelogram:

${\displaystyle f(z)=c\prod _{j=1}^{n}{\frac {\sigma (z-a_{j})}{\sigma (z-b_{j})}}}$

Where ${\displaystyle c}$ is a constant in ${\displaystyle \mathbb {C} }$ and ${\displaystyle a_{j}}$ are the zeros in the parallelogram and ${\displaystyle b_{j}}$ are the poles

The Weierstrass zeta function is defined by the sum

${\displaystyle \operatorname {\zeta } {(z;\Lambda )}={\frac {\sigma '(z;\Lambda )}{\sigma (z;\Lambda )}}={\frac {1}{z}}+\sum _{w\in \Lambda ^{*}}\left({\frac {1}{z-w}}+{\frac {1}{w}}+{\frac {z}{w^{2}}}\right).}$

The Weierstrass zeta function is the logarithmic derivative of the sigma-function. The zeta function can be rewritten as:

${\displaystyle \operatorname {\zeta } {(z;\Lambda )}={\frac {1}{z}}-\sum _{k=1}^{\infty }{\mathcal {G}}_{2k+2}(\Lambda )z^{2k+1}}$

where ${\displaystyle {\mathcal {G}}_{2k+2}}$ is the Eisenstein series of weight 2k + 2.

The derivative of the zeta function is ${\displaystyle -\wp (z)}$, where ${\displaystyle \wp (z)}$ is the Weierstrass elliptic function.

The Weierstrass zeta function should not be confused with the Riemann zeta function in number theory.

The Weierstrass eta function is defined to be

${\displaystyle \eta (w;\Lambda )=\zeta (z+w;\Lambda )-\zeta (z;\Lambda ),{\mbox{ for any }}z\in \mathbb {C} }$ and any w in the lattice ${\displaystyle \Lambda }$

This is well-defined, i.e. ${\displaystyle \zeta (z+w;\Lambda )-\zeta (z;\Lambda )}$ 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

${\displaystyle \operatorname {\wp } {(z;\Lambda )}=-\operatorname {\zeta '} {(z;\Lambda )},{\mbox{ for any }}z\in \mathbb {C} }$

The Weierstrass ℘-function is an even elliptic function of order N=2 with a double pole at each lattice point and no other poles.

Consider the situation where one period is real, which we can scale to be ${\displaystyle \omega _{1}=2\pi }$ and the other is taken to the limit of ${\displaystyle \omega _{2}\rightarrow i\infty }$ so that the functions are only singly-periodic. The corresponding invariants are ${\displaystyle \{g_{2},g_{3}\}=\left\{{\tfrac {1}{12}},{\tfrac {1}{216}}\right\}}$ of discriminant ${\displaystyle \Delta =0}$. Then we have ${\displaystyle \eta _{1}={\tfrac {\pi }{12}}}$ and thus from the above infinite product definition the following equality:

${\displaystyle \operatorname {\sigma } {(z;\Lambda )}=2e^{z^{2}/24}\sin {\left({\tfrac {z}{2}}\right)}}$

A generalization for other sine-like functions on other doubly-periodic lattices is

${\displaystyle f(z)={\frac {\pi }{\omega _{1}}}e^{-(4\eta _{1}/\omega _{1})z^{2}}\operatorname {\sigma } {(2z;\Lambda )}}$