Weierstrass functions

 

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 ${\displaystyle \Lambda -\{0\}}$  or ${\displaystyle \{\omega _{1},\omega _{2}\}}$  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

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

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).

 

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

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

This article incorporates material from Weierstrass sigma function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.