23 Weierstrass Elliptic and Modular Functions Weierstrass Elliptic Functions 23.7 Quarter Periods 23.9 Laurent and Other Power Series

§23.8 Trigonometric Series and Products

Permalink:: http://dlmf.nist.gov/23.8

§23.8(i) Fourier Series
§23.8(ii) Series of Cosecants and Cotangents
§23.8(iii) Infinite Products

§23.8(i) Fourier Series

Notes:: See Lawden (1989, §8.6).
Keywords:: Fourier series, Weierstrass elliptic functions
Permalink:: http://dlmf.nist.gov/23.8.i

If $q=e^{{i\pi\omega_{3}/\omega_{1}}}$ , $\imagpart{(z/\omega_{1})}<2\imagpart{(\omega_{3}/\omega_{1})}$ , and $z\notin\mathbb{L}$ , then

23.8.1 $\mathop{\wp\/}\nolimits\!\left(z\right)+\frac{\eta_{1}}{\omega_{1}}-\frac{\pi^% {2}}{4\omega_{1}^{2}}{\mathop{\csc\/}\nolimits^{{2}}}\!\left(\frac{\pi z}{2% \omega_{1}}\right)=-\frac{2\pi^{2}}{\omega_{1}^{2}}\sum_{{n=1}}^{\infty}\frac{% nq^{{2n}}}{1-q^{{2n}}}\mathop{\cos\/}\nolimits\!\left(\frac{n\pi z}{\omega_{1}% }\right),$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathbb{L}$ : lattice, : nome, : integer, : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$
Permalink:: http://dlmf.nist.gov/23.8.E1
Encodings:: TeX, pMML, png

23.8.2 $\mathop{\zeta\/}\nolimits\!\left(z\right)-\frac{\eta_{1}z}{\omega_{1}}-\frac{% \pi}{2\omega_{1}}\mathop{\cot\/}\nolimits\!\left(\frac{\pi z}{2\omega_{1}}% \right)=\frac{2\pi}{\omega_{1}}\sum_{{n=1}}^{\infty}\frac{q^{{2n}}}{1-q^{{2n}}% }\mathop{\sin\/}\nolimits\!\left(\frac{n\pi z}{\omega_{1}}\right).$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathbb{L}$ : lattice, : nome, : integer, : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$
Permalink:: http://dlmf.nist.gov/23.8.E2
Encodings:: TeX, pMML, png

§23.8(ii) Series of Cosecants and Cotangents

Notes:: See Lawden (1989, pp. 183–184).
Keywords:: Eisenstein convention, Weierstrass elliptic functions
Referenced by:: §23.12
Permalink:: http://dlmf.nist.gov/23.8.ii

When $z\notin\mathbb{L}$ ,

23.8.3 $\mathop{\wp\/}\nolimits\!\left(z\right)=-\frac{\eta_{1}}{\omega_{1}}+\frac{\pi% ^{2}}{4\omega_{1}^{2}}\sum_{{n=-\infty}}^{\infty}{\mathop{\csc\/}\nolimits^{{2% }}}\!\left(\frac{\pi(z+2n\omega_{3})}{2\omega_{1}}\right),$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathbb{L}$ : lattice, : integer, : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$
Permalink:: http://dlmf.nist.gov/23.8.E3
Encodings:: TeX, pMML, png

23.8.4 $\mathop{\zeta\/}\nolimits\!\left(z\right)=\frac{\eta_{1}z}{\omega_{1}}+\frac{% \pi}{2\omega_{1}}\sum_{{n=-\infty}}^{\infty}\mathop{\cot\/}\nolimits\!\left(% \frac{\pi(z+2n\omega_{3})}{2\omega_{1}}\right),$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathbb{L}$ : lattice, : integer, : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$
Referenced by:: §23.8(ii)
Permalink:: http://dlmf.nist.gov/23.8.E4
Encodings:: TeX, pMML, png

where in (23.8.4) the terms in and are to be bracketed together (the Eisenstein convention or principal value: see Weil (1999, p. 6) or Walker (1996, p. 3)).

23.8.5 $\eta_{1}=\frac{\pi^{2}}{2\omega_{1}}\left(\frac{1}{6}+\sum_{{n=1}}^{\infty}{% \mathop{\csc\/}\nolimits^{{2}}}\!\left(\frac{n\pi\omega_{3}}{\omega_{1}}\right% )\right),$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, : integer, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$
Permalink:: http://dlmf.nist.gov/23.8.E5
Encodings:: TeX, pMML, png

with similar results for $\eta_{2}$ and $\eta_{3}$ obtainable by use of (23.2.14).

§23.8(iii) Infinite Products

Notes:: See Lawden (1989, §6.5 and p. 183).
Keywords:: Weierstrass elliptic functions, infinite products
Permalink:: http://dlmf.nist.gov/23.8.iii

23.8.6 $\mathop{\sigma\/}\nolimits\!\left(z\right)=\frac{2\omega_{1}}{\pi}\mathop{\exp% \/}\nolimits\!\left(\frac{\eta_{1}z^{2}}{2\omega_{1}}\right)\mathop{\sin\/}% \nolimits\!\left(\frac{\pi z}{2\omega_{1}}\right)\*\prod_{{n=1}}^{\infty}\frac% {1-2q^{{2n}}\mathop{\cos\/}\nolimits\!\left(\pi z/\omega_{1}\right)+q^{{4n}}}{% (1-q^{{2n}})^{2}},$

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathbb{L}$ : lattice, : nome, : integer, : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$
Permalink:: http://dlmf.nist.gov/23.8.E6
Encodings:: TeX, pMML, png

23.8.7 $\mathop{\sigma\/}\nolimits\!\left(z\right)=\frac{2\omega_{1}}{\pi}\mathop{\exp% \/}\nolimits\!\left(\frac{\eta_{1}z^{2}}{2\omega_{1}}\right)\mathop{\sin\/}% \nolimits\!\left(\frac{\pi z}{2\omega_{1}}\right)\prod_{{n=1}}^{\infty}\frac{% \mathop{\sin\/}\nolimits\!\left(\pi(2n\omega_{3}+z)/(2\omega_{1})\right)% \mathop{\sin\/}\nolimits\!\left(\pi(2n\omega_{3}-z)/(2\omega_{1})\right)}{{% \mathop{\sin\/}\nolimits^{{2}}}\!\left(\pi n\omega_{3}/\omega_{1}\right)}.$

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathbb{L}$ : lattice, : integer, : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$
Referenced by:: §23.10(iii)
Permalink:: http://dlmf.nist.gov/23.8.E7
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 23.7 Quarter Periods 23.9 Laurent and Other Power Series

§23.8 Trigonometric Series and Products

Contents

§23.8(i) Fourier Series

§23.8(ii) Series of Cosecants and Cotangents

§23.8(iii) Infinite Products