§23.11 Integral Representations

Let $\tau ={\omega }_3/{\omega }_1$ and

23.11.1		$f_1(s,\tau )$	$={\displaystyle \frac{{\cosh }^2\left(\frac{1}{2}\tau s\right)}{1-2{\mathrm{e}}^{-s}\cosh \left(\tau s\right)+{\mathrm{e}}^{-2s}}},$
		$f_2(s,\tau )$	$={\displaystyle \frac{{\cos }^2\left(\frac{1}{2}s\right)}{1-2{\mathrm{e}}^{\mathrm{i}\tau s}\cos s+{\mathrm{e}}^{2\mathrm{i}\tau s}}}.$
ⓘ Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\tau$: complex variable and $f_j(s,\tau )$: functions Permalink: http://dlmf.nist.gov/23.11.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §23.11 and Ch.23

Then

23.11.2		$$\mathrm{\wp }\left(z\right)=\frac{1}{z^2}+8{\int }_0^{\mathrm{\infty }}s\left({\mathrm{e}}^{-s}{\sinh }^2\left(\frac{1}{2}zs\right)f_1(s,\tau )+{\mathrm{e}}^{\mathrm{i}\tau s}{\sin }^2\left(\frac{1}{2}zs\right)f_2(s,\tau )\right)ds,$$
ⓘ Symbols: $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $𝕃$: lattice, $z$: complex, $\tau$: complex variable and $f_j(s,\tau )$: functions Permalink: http://dlmf.nist.gov/23.11.E2 Encodings: TeX, pMML, png See also: Annotations for §23.11 and Ch.23

and

23.11.3		$$\zeta \left(z\right)=\frac{1}{z}+{\int }_0^{\mathrm{\infty }}\left({\mathrm{e}}^{-s}\left(zs-\sinh \left(zs\right)\right)f_1(s,\tau )-{\mathrm{e}}^{\mathrm{i}\tau s}\left(zs-\sin \left(zs\right)\right)f_2(s,\tau )\right)ds,$$
ⓘ Symbols: $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $𝕃$: lattice, $z$: complex, $\tau$: complex variable and $f_j(s,\tau )$: functions Permalink: http://dlmf.nist.gov/23.11.E3 Encodings: TeX, pMML, png See also: Annotations for §23.11 and Ch.23

provided that and .