Weierstraß Zeta Function

The Quasiperiodic Function defined by

$${d\zeta(z)\over dz} \equiv-\wp(z)$$ (1)

with

$$\lim_{z\to 0} [\zeta(z)-z^{-1}]=0.$$ (2)

Then

$$\zeta(z)-z^{-1} = -\int_0^z [\wp(z)-z^{-2}]\,dz$$

$$= -\Sigma'\int_0^z [(z-\Omega_{mn})^{-2}-\Omega_{mn}^{-2}]\,dz$$ (3)

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

so $\zeta(z)$ is an Odd Function. Integrating $\wp(z+2\omega_1)=\wp(z)$ gives

$$\zeta(z+2\omega_1)=\zeta(z)+2\eta_1.$$ (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),

$$\eta_1\omega_2-\eta_2\omega_1={\textstyle{1\over 2}}\pi i.$$ (6)

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

$$[\zeta(x)+\zeta(y)+\zeta(z)]^2+\zeta'(x)+\zeta'(y)\zeta'(z)=0.$$ (7)

Also,

$$2 {\left\vert\matrix{1 & \wp(x) & \wp^2(x)\cr 1 & \wp(y) & \... ...(z)\cr}\right\vert} = \zeta(x+y+z)-\zeta(x)-\zeta(y)-\zeta(z)$$ (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.