Content-Length: 57875 | pFad | https://dlmf.nist.gov/./../././././././././././bib/.././23.11#info

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

About the Project

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

© 2010–2024 NIST / Disclaimer / Feedback; Version 1.2.3; Release date 2024-12-15.

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov