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Weierstraß Zeta Function

The Quasiperiodic Function defined by

\begin{displaymath}
{d\zeta(z)\over dz} \equiv-\wp(z)
\end{displaymath} (1)

with
\begin{displaymath}
\lim_{z\to 0} [\zeta(z)-z^{-1}]=0.
\end{displaymath} (2)

Then
$\displaystyle \zeta(z)-z^{-1}$ $\textstyle =$ $\displaystyle -\int_0^z [\wp(z)-z^{-2}]\,dz$  
  $\textstyle =$ $\displaystyle -\Sigma'\int_0^z [(z-\Omega_{mn})^{-2}-\Omega_{mn}^{-2}]\,dz$ (3)


\begin{displaymath}
\zeta(z)=z^{-1} + \setbox0=\hbox{$\scriptstyle{m,n =-\infty}...
...fty [(z-\Omega_{mn})^{-1}+\Omega_{mn}^{-1} +z\Omega_{mn}^{-2}]
\end{displaymath} (4)

so $\zeta(z)$ is an Odd Function. Integrating $\wp(z+2\omega_1)=\wp(z)$ gives
\begin{displaymath}
\zeta(z+2\omega_1)=\zeta(z)+2\eta_1.
\end{displaymath} (5)

Letting $z=-\omega_1$ gives $\zeta(-\omega_1)+2\eta_1=-\zeta(\omega_1)+2\eta_1$, so $\eta_1=\zeta(\omega_1)$. Similarly, $\eta_2=\zeta(\omega_2)$. From Whittaker and Watson (1990),
\begin{displaymath}
\eta_1\omega_2-\eta_2\omega_1={\textstyle{1\over 2}}\pi i.
\end{displaymath} (6)


If $x+y+z=0$, then

\begin{displaymath}[\zeta(x)+\zeta(y)+\zeta(z)]^2+\zeta'(x)+\zeta'(y)\zeta'(z)=0.
\end{displaymath} (7)

Also,
\begin{displaymath}
2 {\left\vert\matrix{1 & \wp(x) & \wp^2(x)\cr 1 & \wp(y) & \...
...(z)\cr}\right\vert}
= \zeta(x+y+z)-\zeta(x)-\zeta(y)-\zeta(z)
\end{displaymath} (8)

(Whittaker and Watson 1990, p. 446).


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.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.




© 1996-9 Eric W. Weisstein
1999-05-26