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

With $t\in ℂ$ and

22.12.1		$$\tau =\mathrm{i}{K}^{\prime }\left(k\right)/K\left(k\right),$$
ⓘ Symbols: $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: 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		$$2Kk\mathrm{sn}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t-(n+\frac{1}{2})\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t-m-(n+\frac{1}{2})\tau }\right),$$
ⓘ Symbols: $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: 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		$$2\mathrm{i}Kk\mathrm{cn}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^n\pi }{\sin \left(\pi (t-(n+\frac{1}{2})\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^{m+n}}{t-m-(n+\frac{1}{2})\tau }\right),$$
ⓘ Symbols: $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $\sin z$: 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		$$\begin{array}{l}2\mathrm{i}K\mathrm{dn}(2Kt,k)\\ =\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-(n+\frac{1}{2})\tau )\right)}\\ =\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-(n+\frac{1}{2})\tau }\right).\end{array}$$
ⓘ Symbols: $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $\tan z$: 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		$2Kk\mathrm{cd}(2Kt,k)$	$={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{\pi }{\sin \left(\pi (t+\frac{1}{2}-(n+\frac{1}{2})\tau )\right)}}={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}\left({\displaystyle \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^m}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau }}\right),$
ⓘ Symbols: $\mathrm{cd}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: 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		$-2\mathrm{i}Kk{k}^{\prime }\mathrm{sd}(2Kt,k)$	$={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n\pi }{\sin \left(\pi (t+\frac{1}{2}-(n+\frac{1}{2})\tau )\right)}}={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}\left({\displaystyle \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^{m+n}}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau }}\right),$
ⓘ Symbols: $\mathrm{sd}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $\sin z$: 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		$2\mathrm{i}K{k}^{\prime }\mathrm{nd}(2Kt,k)$	$=\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \sum _{n=-N}^N}{(-1)}^n{\displaystyle \frac{\pi }{\tan \left(\pi (t+\frac{1}{2}-(n+\frac{1}{2})\tau )\right)}}$
			$=\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \sum _{n=-N}^N}{(-1)}^n\underset{M\to \mathrm{\infty }}{lim}\left({\displaystyle \sum _{m=-M}^M}{\displaystyle \frac{1}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau }}\right),$
ⓘ Symbols: $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $\tan z$: 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		$2K\mathrm{dc}(2Kt,k)$	$={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{\pi }{\sin \left(\pi (t+\frac{1}{2}-n\tau )\right)}}={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}\left({\displaystyle \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^m}{t+\frac{1}{2}-m-n\tau }}\right),$
ⓘ Symbols: $\mathrm{dc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: 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		$2K{k}^{\prime }\mathrm{nc}(2Kt,k)$	$={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n\pi }{\sin \left(\pi (t+\frac{1}{2}-n\tau )\right)}}={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}\left({\displaystyle \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^{m+n}}{t+\frac{1}{2}-m-n\tau }}\right),$
ⓘ Symbols: $\mathrm{nc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: 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		$-2K{k}^{\prime }\mathrm{sc}(2Kt,k)$	$=\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \sum _{n=-N}^N}{(-1)}^n{\displaystyle \frac{\pi }{\tan \left(\pi (t+\frac{1}{2}-n\tau )\right)}}=\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \sum _{n=-N}^N}{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}{\displaystyle \sum _{m=-M}^M}{\displaystyle \frac{1}{t+\frac{1}{2}-m-n\tau }}\right),$
ⓘ Symbols: $\mathrm{sc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\tan z$: 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		$2K\mathrm{ns}(2Kt,k)$	$={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{\pi }{\sin \left(\pi (t-n\tau )\right)}}={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}\left({\displaystyle \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^m}{t-m-n\tau }}\right),$
ⓘ Symbols: $\mathrm{ns}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: 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		$2K\mathrm{ds}(2Kt,k)$	$={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n\pi }{\sin \left(\pi (t-n\tau )\right)}}={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}\left({\displaystyle \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^{m+n}}{t-m-n\tau }}\right),$
ⓘ Symbols: $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: 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		$2K\mathrm{cs}(2Kt,k)$	$=\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \sum _{n=-N}^N}{(-1)}^n{\displaystyle \frac{\pi }{\tan \left(\pi (t-n\tau )\right)}}=\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \sum _{n=-N}^N}{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}{\displaystyle \sum _{m=-M}^M}{\displaystyle \frac{1}{t-m-n\tau }}\right).$
ⓘ Symbols: $\mathrm{cs}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\tan z$: 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