23 Weierstrass Elliptic and Modular FunctionsWeierstrass Elliptic Functions23.7 Quarter Periods23.9 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

Contents

1. §23.8(i) Fourier Series
2. §23.8(ii) Series of Cosecants and Cotangents
3. §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 $q={\mathrm{e}}^{\mathrm{i}\pi {\omega }_3/{\omega }_1}$, , and $z\notin 𝕃$, then

23.8.1		$\mathrm{\wp }\left(z\right)+{\displaystyle \frac{{\eta }_1}{{\omega }_1}}-{\displaystyle \frac{{\pi }^2}{4{\omega }_1^2}}{\csc }^2\left({\displaystyle \frac{\pi z}{2{\omega }_1}}\right)$	$=-{\displaystyle \frac{2{\pi }^2}{{\omega }_1^2}}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{n{q}^{2n}}{1-q^{2n}}}\cos \left({\displaystyle \frac{n\pi z}{{\omega }_1}}\right),$
ⓘ Symbols: $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $𝕃$: 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		$\zeta \left(z\right)-{\displaystyle \frac{{\eta }_{1}z}{{\omega }_1}}-{\displaystyle \frac{\pi }{2{\omega }_1}}\cot \left({\displaystyle \frac{\pi z}{2{\omega }_1}}\right)$	$={\displaystyle \frac{2\pi }{{\omega }_1}}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{q^{2n}}{1-q^{2n}}}\sin \left({\displaystyle \frac{n\pi z}{{\omega }_1}}\right).$
ⓘ Symbols: $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\sin z$: sine function, $𝕃$: 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 $z\notin 𝕃$,

23.8.3		$$\mathrm{\wp }\left(z\right)=-\frac{{\eta }_1}{{\omega }_1}+\frac{{\pi }^2}{4{\omega }_1^2}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\csc }^2\left(\frac{\pi (z+2n{\omega }_3)}{2{\omega }_1}\right),$$
ⓘ Symbols: $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $𝕃$: 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		$$\zeta \left(z\right)=\frac{{\eta }_{1}z}{{\omega }_1}+\frac{\pi }{2{\omega }_1}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\cot \left(\frac{\pi (z+2n{\omega }_3)}{2{\omega }_1}\right),$$
ⓘ Symbols: $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $𝕃$: 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		$${\eta }_1=\frac{{\pi }^2}{2{\omega }_1}\left(\frac{1}{6}+\sum _{n=1}^{\mathrm{\infty }}{\csc }^2\left(\frac{n\pi {\omega }_3}{{\omega }_1}\right)\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: 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		$$\sigma \left(z\right)=\frac{2{\omega }_1}{\pi }\exp \left(\frac{{\eta }_1{z}^2}{2{\omega }_1}\right)\sin \left(\frac{\pi z}{2{\omega }_1}\right)\prod _{n=1}^{\mathrm{\infty }}\frac{1-2q^{2n}\cos \left(\pi z/{\omega }_1\right)+q^{4n}}{{(1-q^{2n})}^2},$$
ⓘ Symbols: $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\exp z$: exponential function, $\sin z$: sine function, $𝕃$: 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		$$\sigma \left(z\right)=\begin{array}{l}\frac{2{\omega }_1}{\pi }\exp \left(\frac{{\eta }_1{z}^2}{2{\omega }_1}\right)\sin \left(\frac{\pi z}{2{\omega }_1}\right)\\ \times \prod _{n=1}^{\mathrm{\infty }}\frac{\sin \left(\pi (2n{\omega }_3+z)/(2{\omega }_1)\right)\sin \left(\pi (2n{\omega }_3-z)/(2{\omega }_1)\right)}{{\sin }^2\left(\pi n{\omega }_3/{\omega }_1\right)}.\end{array}$$
ⓘ Symbols: $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\sin z$: sine function, $𝕃$: 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 Periods23.9 Laurent and Other Power Series

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