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

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Permalink:: http://dlmf.nist.gov/23.8
See also:: Annotations for Ch.23

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

§23.8(i) Fourier Series

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Keywords:: Fourier series, Weierstrass elliptic functions
Notes:: See Lawden (1989, §8.6).
Permalink:: http://dlmf.nist.gov/23.8.i
See also:: Annotations for §23.8 and Ch.23

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

23.8.1	$\displaystyle\wp\left(z\right)+\frac{\eta_{1}}{\omega_{1}}-\frac{\pi^{2}}{4% \omega_{1}^{2}}{\csc}^{2}\left(\frac{\pi z}{2\omega_{1}}\right)$	$\displaystyle=-\frac{2\pi^{2}}{\omega_{1}^{2}}\sum_{n=1}^{\infty}\frac{nq^{2n}% }{1-q^{2n}}\cos\left(\frac{n\pi z}{\omega_{1}}\right),$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cos\NVar{z}$ : cosine function, $\mathbb{L}$ : lattice, $q$ : nome, $n$ : integer, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number Permalink: http://dlmf.nist.gov/23.8.E1 Encodings: TeX, pMML, png See also: Annotations for §23.8(i), §23.8 and Ch.23
23.8.2	$\displaystyle\zeta\left(z\right)-\frac{\eta_{1}z}{\omega_{1}}-\frac{\pi}{2% \omega_{1}}\cot\left(\frac{\pi z}{2\omega_{1}}\right)$	$\displaystyle=\frac{2\pi}{\omega_{1}}\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q^{2n}% }\sin\left(\frac{n\pi z}{\omega_{1}}\right).$
ⓘ Symbols: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\sin\NVar{z}$ : sine function, $\mathbb{L}$ : lattice, $q$ : nome, $n$ : integer, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number Permalink: http://dlmf.nist.gov/23.8.E2 Encodings: TeX, pMML, png See also: Annotations for §23.8(i), §23.8 and Ch.23

§23.8(ii) Series of Cosecants and Cotangents

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Keywords:: Eisenstein convention, Weierstrass elliptic functions, principal value, series of cosecants or cotangents
Notes:: See Lawden (1989, pp. 183–184).
Referenced by:: §23.12
Permalink:: http://dlmf.nist.gov/23.8.ii
See also:: Annotations for §23.8 and Ch.23

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

23.8.3		$\wp\left(z\right)=-\frac{\eta_{1}}{\omega_{1}}+\frac{\pi^{2}}{4\omega_{1}^{2}}% \sum_{n=-\infty}^{\infty}{\csc}^{2}\left(\frac{\pi(z+2n\omega_{3})}{2\omega_{1% }}\right),$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number Permalink: http://dlmf.nist.gov/23.8.E3 Encodings: TeX, pMML, png See also: Annotations for §23.8(ii), §23.8 and Ch.23

23.8.4		$\zeta\left(z\right)=\frac{\eta_{1}z}{\omega_{1}}+\frac{\pi}{2\omega_{1}}\sum_{% n=-\infty}^{\infty}\cot\left(\frac{\pi(z+2n\omega_{3})}{2\omega_{1}}\right),$
ⓘ Symbols: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number Referenced by: §23.8(ii) Permalink: http://dlmf.nist.gov/23.8.E4 Encodings: TeX, pMML, png See also: Annotations for §23.8(ii), §23.8 and Ch.23

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

23.8.5		$\eta_{1}=\frac{\pi^{2}}{2\omega_{1}}\left(\frac{1}{6}+\sum_{n=1}^{\infty}{\csc% }^{2}\left(\frac{n\pi\omega_{3}}{\omega_{1}}\right)\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $n$ : integer, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number Permalink: http://dlmf.nist.gov/23.8.E5 Encodings: TeX, pMML, png See also: Annotations for §23.8(ii), §23.8 and Ch.23

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

§23.8(iii) Infinite Products

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Keywords:: Weierstrass elliptic functions, infinite products
Notes:: See Lawden (1989, §6.5 and p. 183).
Permalink:: http://dlmf.nist.gov/23.8.iii
See also:: Annotations for §23.8 and Ch.23

23.8.6		$\sigma\left(z\right)=\frac{2\omega_{1}}{\pi}\exp\left(\frac{\eta_{1}z^{2}}{2% \omega_{1}}\right)\sin\left(\frac{\pi z}{2\omega_{1}}\right)\*\prod_{n=1}^{% \infty}\frac{1-2q^{2n}\cos\left(\pi z/\omega_{1}\right)+q^{4n}}{(1-q^{2n})^{2}},$
ⓘ Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\exp\NVar{z}$ : exponential function, $\sin\NVar{z}$ : sine function, $\mathbb{L}$ : lattice, $q$ : nome, $n$ : integer, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number Permalink: http://dlmf.nist.gov/23.8.E6 Encodings: TeX, pMML, png See also: Annotations for §23.8(iii), §23.8 and Ch.23

23.8.7		$\sigma\left(z\right)=\frac{2\omega_{1}}{\pi}\exp\left(\frac{\eta_{1}z^{2}}{2% \omega_{1}}\right)\sin\left(\frac{\pi z}{2\omega_{1}}\right)\prod_{n=1}^{% \infty}\frac{\sin\left(\pi(2n\omega_{3}+z)/(2\omega_{1})\right)\sin\left(\pi(2% n\omega_{3}-z)/(2\omega_{1})\right)}{{\sin}^{2}\left(\pi n\omega_{3}/\omega_{1% }\right)}.$
ⓘ Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\sin\NVar{z}$ : sine function, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number Referenced by: §23.10(iii) Permalink: http://dlmf.nist.gov/23.8.E7 Encodings: TeX, pMML, png See also: Annotations for §23.8(iii), §23.8 and Ch.23

23.7 Quarter Periods 23.9 Laurent and Other Power Series

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§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