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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 $r=1$, 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.




© 1996-9 Eric W. Weisstein
1999-05-26