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DLMF: §23.11 Integral Representations ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

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23 Weierstrass Elliptic and Modular Functions Weierstrass Elliptic Functions 23.10 Addition Theorems and Other Identities 23.12 Asymptotic Approximations

§23.11 Integral Representations

ⓘ

Keywords:: Weierstrass elliptic functions, integral representations
Notes:: See Dienstfrey and Huang (2006).
Permalink:: http://dlmf.nist.gov/23.11
See also:: Annotations for Ch.23

Let $\tau=\ifrac{\omega_{3}}{\omega_{1}}$ and

23.11.1		$\displaystyle f_{1}(s,\tau)$	$\displaystyle=\frac{{\cosh}^{2}\left(\tfrac{1}{2}\tau s\right)}{1-2e^{-s}\cosh% \left(\tau s\right)+e^{-2s}},$
23.11.1		$\displaystyle f_{2}(s,\tau)$	$\displaystyle=\frac{{\cos}^{2}\left(\tfrac{1}{2}s\right)}{1-2e^{i\tau s}\cos s% +e^{2i\tau s}}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : 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		$\wp\left(z\right)=\frac{1}{z^{2}}+8\int_{0}^{\infty}s\left(e^{-s}{\sinh}^{2}% \left(\tfrac{1}{2}zs\right)f_{1}(s,\tau)+e^{i\tau s}{\sin}^{2}\left(\tfrac{1}{% 2}zs\right)f_{2}(s,\tau)\right)\,\mathrm{d}s,$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathbb{L}$ : 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		$\zeta\left(z\right)=\frac{1}{z}+\int_{0}^{\infty}\left(e^{-s}\left(zs-\sinh% \left(zs\right)\right)f_{1}(s,\tau)-e^{i\tau s}\left(zs-\sin\left(zs\right)% \right)f_{2}(s,\tau)\right)\,\mathrm{d}s,$
ⓘ Symbols: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathbb{L}$ : 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 $-1<\Re\left(z+\tau\right)<1$ and $\left|\Im z\right|<\Im\tau$ .

23.10 Addition Theorems and Other Identities 23.12 Asymptotic Approximations

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