§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, $q$ : nome, $n$ : integer, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
Permalink:: http://dlmf.nist.gov/23.8.E1
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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, $q$ : nome, $n$ : integer, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
Permalink:: http://dlmf.nist.gov/23.8.E2
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§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, $n$ : integer, $z$ : 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, $n$ : integer, $z$ : 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
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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 $\eta _{1}=\frac{\pi^{2}}{2\omega _{1}}\left(\frac{1}{6}+\sum _{{n=1}}^{\infty}{\mathop{\csc\/}\nolimits^{{2}}}\!\left(\frac{n\pi\omega _{3}}{\omega _{1}}\right)\right),$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $n$ : integer, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
Permalink:: http://dlmf.nist.gov/23.8.E5
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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, $q$ : nome, $n$ : integer, $z$ : 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, $n$ : integer, $z$ : 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

§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