Weierstraß Sigma Function

The Quasiperiodic Function defined by

$\begin{displaymath} {d\over dz} \ln\sigma(z)=\zeta(z), \end{displaymath}$

(1)

where $\zeta(z)$ is the Weierstraß Zeta Function and

$\begin{displaymath} \lim_{z\to 0} {\sigma(z)\over z}=1. \end{displaymath}$

(2)

Then

$\begin{displaymath} \sigma(z)=z\prod_{mn}'\left[{\left({1-{z\over\Omega_{mn}}}\r... ...{{z\over\Omega_{mn}}+{z^2\over 2\Omega_{mn}^2}}\right)}\right] \end{displaymath}$

(3)

$\begin{displaymath} \sigma(z+2\omega_1)=-e^{2\eta_1(z+\omega_1)}\sigma(z) \end{displaymath}$

(4)

$\begin{displaymath} \sigma(z+2\omega_2)=-e^{2\eta_2(z+\omega_2)}\sigma(z) \end{displaymath}$

(5)

$\begin{displaymath} \sigma_r(z) = {e^{-\eta_rz}\sigma(z+\omega_r)\over \sigma(\omega_r)} \end{displaymath}$

(6)

for

, 2, 3.

$\begin{displaymath} \sigma(z\vert\omega_1, \omega_2) = {2\omega_1\over \pi \vart... ...1\left({\nu \left\vert{\omega_2\over \omega_1}\right.}\right), \end{displaymath}$

(7)

where $\nu\equiv \pi z/(2\omega_1)$ , and

$\begin{displaymath} \eta_1 = -{\pi^2\vartheta_1'''\over 12\omega_1\vartheta_1'} \end{displaymath}$

(8)

$\begin{displaymath} \eta_2 = -{\pi^2\omega_2\vartheta_1'''\over 12{\omega_1}^2 \vartheta_1'} -{\pi i\over 2\omega_1}. \end{displaymath}$

(9)

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Weierstrass Elliptic and Related Functions.'' Ch. 18 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 627-671, 1972.