# Jacobi zeta function

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##### 2: 25.1 Special Notation
The main related functions are the Hurwitz zeta function $\zeta\left(s,a\right)$, the dilogarithm $\operatorname{Li}_{2}\left(z\right)$, the polylogarithm $\operatorname{Li}_{s}\left(z\right)$ (also known as Jonquière’s function $\phi\left(z,s\right)$), Lerch’s transcendent $\Phi\left(z,s,a\right)$, and the Dirichlet $L$-functions $L\left(s,\chi\right)$.
##### 4: 22.1 Special Notation
The functions treated in this chapter are the three principal Jacobian elliptic functions $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, $\operatorname{dn}\left(z,k\right)$; the nine subsidiary Jacobian elliptic functions $\operatorname{cd}\left(z,k\right)$, $\operatorname{sd}\left(z,k\right)$, $\operatorname{nd}\left(z,k\right)$, $\operatorname{dc}\left(z,k\right)$, $\operatorname{nc}\left(z,k\right)$, $\operatorname{sc}\left(z,k\right)$, $\operatorname{ns}\left(z,k\right)$, $\operatorname{ds}\left(z,k\right)$, $\operatorname{cs}\left(z,k\right)$; the amplitude function $\operatorname{am}\left(x,k\right)$; Jacobi’s epsilon and zeta functions $\mathcal{E}\left(x,k\right)$ and $\mathrm{Z}\left(x|k\right)$. …
##### 5: 22.20 Methods of Computation
Jacobi’s zeta function can then be found by use of (22.16.32). …
##### 6: 27.1 Special Notation
(For other notation see Notation for the Special Functions.)
 $d,k,m,n$ positive integers (unless otherwise indicated). … Riemann zeta function; see §25.2(i). Jacobi symbol; see §27.9. …
##### 7: 20.10 Integrals
20.10.1 $\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|ix^{2}\right)\,\mathrm{d}x=2^% {s}(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$ $\Re s>1$,
20.10.2 $\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\,\mathrm{d}% x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$ $\Re s>1$,
20.10.3 $\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle|ix^{2}\right))\,\mathrm{d}% x=(1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$ $\Re s>0$.
##### 8: 31.7 Relations to Other Functions
###### §31.7(ii) Relations to Lamé Functions
With $z={\operatorname{sn}}^{2}\left(\zeta,k\right)$ and …equation (31.2.1) becomes Lamé’s equation with independent variable $\zeta$; compare (29.2.1) and (31.2.8). …
##### 9: 22.2 Definitions
22.2.4 $\operatorname{sn}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{ns}\left(z,k\right)},$
22.2.5 $\operatorname{cn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{nc}\left(z,k\right)},$
22.2.6 $\operatorname{dn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{3}% \left(0,q\right)}\frac{\theta_{3}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{nd}\left(z,k\right)},$
22.2.7 $\operatorname{sd}\left(z,k\right)=\frac{{\theta_{3}}^{2}\left(0,q\right)}{% \theta_{2}\left(0,q\right)\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(% \zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}=\frac{1}{\operatorname{ds}% \left(z,k\right)},$
22.2.8 $\operatorname{cd}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q% \right)}=\frac{1}{\operatorname{dc}\left(z,k\right)},$
##### 10: 22.11 Fourier and Hyperbolic Series
22.11.1 $\operatorname{sn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n% +\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1-q^{2n+1}},$
22.11.7 $\operatorname{ns}\left(z,k\right)-\frac{\pi}{2K}\csc\zeta=\frac{2\pi}{K}\sum_{% n=0}^{\infty}\frac{q^{2n+1}\sin\left((2n+1)\zeta\right)}{1-q^{2n+1}},$
22.11.8 $\operatorname{ds}\left(z,k\right)-\frac{\pi}{2K}\csc\zeta=-\frac{2\pi}{K}\sum_% {n=0}^{\infty}\frac{q^{2n+1}\sin\left((2n+1)\zeta\right)}{1+q^{2n+1}},$
22.11.9 $\operatorname{cs}\left(z,k\right)-\frac{\pi}{2K}\cot\zeta=-\frac{2\pi}{K}\sum_% {n=1}^{\infty}\frac{q^{2n}\sin\left(2n\zeta\right)}{1+q^{2n}},$
22.11.10 $\operatorname{dc}\left(z,k\right)-\frac{\pi}{2K}\sec\zeta=\frac{2\pi}{K}\sum_{% n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos\left((2n+1)\zeta\right)}{1-q^{2n+1}},$