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

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 $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)=\mathop{\mathrm{sn}\/}\nolimits\left(z+K,k\right)$ 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 $\pi\mathop{\csc\/}\nolimits\!\left(\pi\zeta\right)=\sum _{{m=-\infty}}^{{\infty}}(-1)^{m}/(\zeta-m)$ or $\pi\mathop{\cot\/}\nolimits\!\left(\pi\zeta\right)=\lim _{{M\to\infty}}\sum _{{m=-M}}^{M}1/(\zeta-m)$ ; compare (4.22.5) and (4.22.3). See also Walker (1996, §5.1) and Weil (1999, pp. 14–47).
Keywords:: Eisenstein series, Jacobian elliptic functions
Permalink:: http://dlmf.nist.gov/22.12

With $t\in\Complex$ and

22.12.1 $\tau=i\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)/\mathop{K\/}\nolimits\!\left(k\right),$

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $k$ : modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E1
Encodings:: TeX, pMML, png

22.12.2 $2Kk\mathop{\mathrm{sn}\/}\nolimits\left(2Kt,k\right)=\sum _{{n=-\infty}}^{{\infty}}\frac{\pi}{\mathop{\sin\/}\nolimits\!\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum _{{n=-\infty}}^{{\infty}}\left(\sum _{{m=-\infty}}^{{\infty}}\frac{(-1)^{m}}{t-m-(n+\frac{1}{2})\tau}\right),$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\sin\/}\nolimits z$ : 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

22.12.3 $2iKk\mathop{\mathrm{cn}\/}\nolimits\left(2Kt,k\right)=\sum _{{n=-\infty}}^{{\infty}}\frac{(-1)^{n}\pi}{\mathop{\sin\/}\nolimits\!\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum _{{n=-\infty}}^{{\infty}}\left(\sum _{{m=-\infty}}^{{\infty}}\frac{(-1)^{{m+n}}}{t-m-(n+\frac{1}{2})\tau}\right),$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E3
Encodings:: TeX, pMML, png

22.12.4 $2iK\mathop{\mathrm{dn}\/}\nolimits\left(2Kt,k\right)=\lim _{{N\to\infty}}\sum _{{n=-N}}^{N}(-1)^{n}\frac{\pi}{\mathop{\tan\/}\nolimits\!\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\lim _{{N\to\infty}}\sum _{{n=-N}}^{N}(-1)^{n}\left(\lim _{{M\to\infty}}\sum _{{m=-M}}^{M}\frac{1}{t-m-(n+\frac{1}{2})\tau}\right).$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\tan\/}\nolimits z$ : 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

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 $2Kk\mathop{\mathrm{cd}\/}\nolimits\left(2Kt,k\right)=\sum _{{n=-\infty}}^{{\infty}}\frac{\pi}{\mathop{\sin\/}\nolimits\!\left(\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)\right)}=\sum _{{n=-\infty}}^{{\infty}}\left(\sum _{{m=-\infty}}^{{\infty}}\frac{(-1)^{m}}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau}\right),$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : modulus and $\tau$ : ratio
Referenced by:: §22.12
Permalink:: http://dlmf.nist.gov/22.12.E5
Encodings:: TeX, pMML, png

22.12.6 $-2iKkk^{{\prime}}\mathop{\mathrm{sd}\/}\nolimits\left(2Kt,k\right)=\sum _{{n=-\infty}}^{{\infty}}\frac{(-1)^{n}\pi}{\mathop{\sin\/}\nolimits\!\left(\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)\right)}=\sum _{{n=-\infty}}^{{\infty}}\left(\sum _{{m=-\infty}}^{{\infty}}\frac{(-1)^{{m+n}}}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau}\right),$

Symbols:: $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : modulus, $k^{{\prime}}$ : complementary modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E6
Encodings:: TeX, pMML, png

22.12.7 $2iKk^{{\prime}}\mathop{\mathrm{nd}\/}\nolimits\left(2Kt,k\right)=\lim _{{N\to\infty}}\sum _{{n=-N}}^{N}(-1)^{n}\frac{\pi}{\mathop{\tan\/}\nolimits\!\left(\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)\right)}=\lim _{{N\to\infty}}\sum _{{n=-N}}^{N}(-1)^{n}\lim _{{M\to\infty}}\left(\sum _{{m=-M}}^{M}\frac{1}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau}\right),$

Symbols:: $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\tan\/}\nolimits z$ : tangent function, $k$ : modulus, $k^{{\prime}}$ : complementary modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E7
Encodings:: TeX, pMML, png

22.12.8 $2K\mathop{\mathrm{dc}\/}\nolimits\left(2Kt,k\right)=\sum _{{n=-\infty}}^{{\infty}}\frac{\pi}{\mathop{\sin\/}\nolimits\!\left(\pi(t+\frac{1}{2}-n\tau)\right)}=\sum _{{n=-\infty}}^{{\infty}}\left(\sum _{{m=-\infty}}^{{\infty}}\frac{(-1)^{m}}{t+\frac{1}{2}-m-n\tau}\right),$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E8
Encodings:: TeX, pMML, png

22.12.9 $2Kk^{{\prime}}\mathop{\mathrm{nc}\/}\nolimits\left(2Kt,k\right)=\sum _{{n=-\infty}}^{{\infty}}\frac{(-1)^{n}\pi}{\mathop{\sin\/}\nolimits\!\left(\pi(t+\frac{1}{2}-n\tau)\right)}=\sum _{{n=-\infty}}^{{\infty}}\left(\sum _{{m=-\infty}}^{{\infty}}\frac{(-1)^{{m+n}}}{t+\frac{1}{2}-m-n\tau}\right),$

Symbols:: $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : modulus, $k^{{\prime}}$ : complementary modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E9
Encodings:: TeX, pMML, png

22.12.10 $-2Kk^{{\prime}}\mathop{\mathrm{sc}\/}\nolimits\left(2Kt,k\right)=\lim _{{N\to\infty}}\sum _{{n=-N}}^{N}(-1)^{n}\frac{\pi}{\mathop{\tan\/}\nolimits\!\left(\pi(t+\frac{1}{2}-n\tau)\right)}=\lim _{{N\to\infty}}\sum _{{n=-N}}^{N}(-1)^{n}\left(\lim _{{M\to\infty}}\sum _{{m=-M}}^{M}\frac{1}{t+\frac{1}{2}-m-n\tau}\right),$

Symbols:: $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\tan\/}\nolimits z$ : tangent function, $k$ : modulus, $k^{{\prime}}$ : complementary modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E10
Encodings:: TeX, pMML, png

22.12.11 $2K\mathop{\mathrm{ns}\/}\nolimits\left(2Kt,k\right)=\sum _{{n=-\infty}}^{{\infty}}\frac{\pi}{\mathop{\sin\/}\nolimits\!\left(\pi(t-n\tau)\right)}=\sum _{{n=-\infty}}^{{\infty}}\left(\sum _{{m=-\infty}}^{{\infty}}\frac{(-1)^{m}}{t-m-n\tau}\right),$

Symbols:: $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E11
Encodings:: TeX, pMML, png

22.12.12 $2K\mathop{\mathrm{ds}\/}\nolimits\left(2Kt,k\right)=\sum _{{n=-\infty}}^{{\infty}}\frac{(-1)^{n}\pi}{\mathop{\sin\/}\nolimits\!\left(\pi(t-n\tau)\right)}=\sum _{{n=-\infty}}^{{\infty}}\left(\sum _{{m=-\infty}}^{{\infty}}\frac{(-1)^{{m+n}}}{t-m-n\tau}\right),$

Symbols:: $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E12
Encodings:: TeX, pMML, png

22.12.13 $2K\mathop{\mathrm{cs}\/}\nolimits\left(2Kt,k\right)=\lim _{{N\to\infty}}\sum _{{n=-N}}^{N}(-1)^{n}\frac{\pi}{\mathop{\tan\/}\nolimits\!\left(\pi(t-n\tau)\right)}=\lim _{{N\to\infty}}\sum _{{n=-N}}^{N}(-1)^{n}\left(\lim _{{M\to\infty}}\sum _{{m=-M}}^{M}\frac{1}{t-m-n\tau}\right).$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\tan\/}\nolimits z$ : tangent function, $k$ : modulus and $\tau$ : ratio
Referenced by:: §22.12
Permalink:: http://dlmf.nist.gov/22.12.E13
Encodings:: TeX, pMML, png