22 Jacobian Elliptic Functions Properties 22.11 Fourier and Hyperbolic Series 22.13 Derivatives and Differential Equations

§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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Keywords:: Eisenstein series, Jacobian elliptic functions, expansions in doubly-infinite partial fractions, trigonometric series expansions
Notes:: For the first right-hand sides of (22.12.2)–(22.12.4) see Lawden (1989, (8.8.3), (8.8.4), and (8.8.7)). The first right-hand sides of (22.12.5)–(22.12.13) can be obtained from the first right-hand sides of (22.12.2)–(22.12.4) by translations of the variable in accordance with the intersection of the first four rows and first four columns of Table 22.4.3. For example, in the case of (22.12.5) we apply the translation $\operatorname{cd}\left(z,k\right)=\operatorname{sn}\left(z+K,k\right)$ to (22.12.2). The second right-hand sides of (22.12.2)–(22.12.13) can be obtained from the corresponding first right-hand sides by substituting, as appropriate, the expansions $\pi\csc\left(\pi\zeta\right)=\sum_{m=-\infty}^{\infty}(-1)^{m}/(\zeta-m)$ or $\pi\cot\left(\pi\zeta\right)=\lim_{M\to\infty}\sum_{m=-M}^{M}1/(\zeta-m)$ ; compare (4.22.5) and (4.22.3). See also Walker (1996, §5.1) and Weil (1999, pp. 14–47).
Permalink:: http://dlmf.nist.gov/22.12
See also:: Annotations for Ch.22

With $t\in\mathbb{C}$ and

22.12.1		$\tau=i{K^{\prime}}\left(k\right)/K\left(k\right),$
ⓘ Symbols: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $k$ : modulus and $\tau$ : ratio Permalink: http://dlmf.nist.gov/22.12.E1 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22

22.12.2		$2Kk\operatorname{sn}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{% \sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(% \sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m-(n+\frac{1}{2})\tau}\right),$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio Referenced by: §22.12, §22.12 Permalink: http://dlmf.nist.gov/22.12.E2 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22

22.12.3		$2iKk\operatorname{cn}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{(-1)^{n% }\pi}{\sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}% \left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t-m-(n+\frac{1}{2})\tau}% \right),$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio Permalink: http://dlmf.nist.gov/22.12.E3 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22

22.12.4		$2iK\operatorname{dn}\left(2Kt,k\right)=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n% }\frac{\pi}{\tan\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\lim_{N\to\infty}\sum% _{n=-N}^{N}(-1)^{n}\left(\lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t-m-(n+\frac% {1}{2})\tau}\right).$
ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\tan\NVar{z}$ : tangent function, $k$ : modulus and $\tau$ : ratio Referenced by: §22.12, §22.12 Permalink: http://dlmf.nist.gov/22.12.E4 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22

The double sums in (22.12.2)–(22.12.4) are convergent but not absolutely convergent, hence the order of the summations is important. Compare §20.5(iii).

22.12.5	$\displaystyle 2Kk\operatorname{cd}\left(2Kt,k\right)$	$\displaystyle=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(t+\frac{1}{2}-% (n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{% \infty}\frac{(-1)^{m}}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau}\right),$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio Referenced by: §22.12 Permalink: http://dlmf.nist.gov/22.12.E5 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22
22.12.6	$\displaystyle-2iKkk^{\prime}\operatorname{sd}\left(2Kt,k\right)$	$\displaystyle=\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t+% \frac{1}{2}-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{% m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau}% \right),$
ⓘ Symbols: $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $k$ : modulus, $k^{\prime}$ : complementary modulus and $\tau$ : ratio Permalink: http://dlmf.nist.gov/22.12.E6 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22
22.12.7	$\displaystyle 2iKk^{\prime}\operatorname{nd}\left(2Kt,k\right)$	$\displaystyle=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan\left(\pi% (t+\frac{1}{2}-(n+\frac{1}{2})\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-% 1)^{n}\lim_{M\to\infty}\left(\sum_{m=-M}^{M}\frac{1}{t+\frac{1}{2}-m-(n+\frac{% 1}{2})\tau}\right),$
ⓘ Symbols: $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\tan\NVar{z}$ : tangent function, $k$ : modulus, $k^{\prime}$ : complementary modulus and $\tau$ : ratio Permalink: http://dlmf.nist.gov/22.12.E7 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22
22.12.8	$\displaystyle 2K\operatorname{dc}\left(2Kt,k\right)$	$\displaystyle=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(t+\frac{1}{2}-% n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(% -1)^{m}}{t+\frac{1}{2}-m-n\tau}\right),$
ⓘ Symbols: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio Permalink: http://dlmf.nist.gov/22.12.E8 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22
22.12.9	$\displaystyle 2Kk^{\prime}\operatorname{nc}\left(2Kt,k\right)$	$\displaystyle=\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t+% \frac{1}{2}-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{% \infty}\frac{(-1)^{m+n}}{t+\frac{1}{2}-m-n\tau}\right),$
ⓘ Symbols: $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $k$ : modulus, $k^{\prime}$ : complementary modulus and $\tau$ : ratio Permalink: http://dlmf.nist.gov/22.12.E9 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22
22.12.10	$\displaystyle-2Kk^{\prime}\operatorname{sc}\left(2Kt,k\right)$	$\displaystyle=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan\left(\pi% (t+\frac{1}{2}-n\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\left(% \lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t+\frac{1}{2}-m-n\tau}\right),$
ⓘ Symbols: $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\tan\NVar{z}$ : tangent function, $k$ : modulus, $k^{\prime}$ : complementary modulus and $\tau$ : ratio Permalink: http://dlmf.nist.gov/22.12.E10 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22
22.12.11	$\displaystyle 2K\operatorname{ns}\left(2Kt,k\right)$	$\displaystyle=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(t-n\tau)\right% )}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m% -n\tau}\right),$
ⓘ Symbols: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio Permalink: http://dlmf.nist.gov/22.12.E11 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22
22.12.12	$\displaystyle 2K\operatorname{ds}\left(2Kt,k\right)$	$\displaystyle=\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t-n% \tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-% 1)^{m+n}}{t-m-n\tau}\right),$
ⓘ Symbols: $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio Permalink: http://dlmf.nist.gov/22.12.E12 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22
22.12.13	$\displaystyle 2K\operatorname{cs}\left(2Kt,k\right)$	$\displaystyle=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan\left(\pi% (t-n\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\left(\lim_{M\to% \infty}\sum_{m=-M}^{M}\frac{1}{t-m-n\tau}\right).$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\tan\NVar{z}$ : tangent function, $k$ : modulus and $\tau$ : ratio Referenced by: §22.12 Permalink: http://dlmf.nist.gov/22.12.E13 Encodings: TeX, pMML, png See also: Annotations for §22.12 and Ch.22

22.11 Fourier and Hyperbolic Series 22.13 Derivatives and Differential Equations

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§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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