# §22.11 Fourier and Hyperbolic Series

Throughout this section $q$ and $\zeta$ are defined as in §22.2.

If $q\exp\left(2|\Im\zeta|\right)<1$, then

 22.11.1 $\displaystyle\operatorname{sn}\left(z,k\right)$ $\displaystyle=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\sin% \left((2n+1)\zeta\right)}{1-q^{2n+1}},$ 22.11.2 $\displaystyle\operatorname{cn}\left(z,k\right)$ $\displaystyle=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\cos% \left((2n+1)\zeta\right)}{1+q^{2n+1}},$ 22.11.3 $\displaystyle\operatorname{dn}\left(z,k\right)$ $\displaystyle=\frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{n}\cos% \left(2n\zeta\right)}{1+q^{2n}}.$
 22.11.4 $\displaystyle\operatorname{cd}\left(z,k\right)$ $\displaystyle=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}% }\cos\left((2n+1)\zeta\right)}{1-q^{2n+1}},$ 22.11.5 $\displaystyle\operatorname{sd}\left(z,k\right)$ $\displaystyle=\frac{2\pi}{Kkk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+% \frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1+q^{2n+1}},$ 22.11.6 $\displaystyle\operatorname{nd}\left(z,k\right)$ $\displaystyle=\frac{\pi}{2Kk^{\prime}}+\frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{% \infty}\frac{(-1)^{n}q^{n}\cos\left(2n\zeta\right)}{1+q^{2n}}.$

Next, if $q\exp\left(|\Im\zeta|\right)<1$, then

 22.11.7 $\displaystyle\operatorname{ns}\left(z,k\right)-\frac{\pi}{2K}\csc\zeta$ $\displaystyle=\frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin\left((2n+1)% \zeta\right)}{1-q^{2n+1}},$ 22.11.8 $\displaystyle\operatorname{ds}\left(z,k\right)-\frac{\pi}{2K}\csc\zeta$ $\displaystyle=-\frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin\left((2n+1)% \zeta\right)}{1+q^{2n+1}},$ 22.11.9 $\displaystyle\operatorname{cs}\left(z,k\right)-\frac{\pi}{2K}\cot\zeta$ $\displaystyle=-\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{2n}\sin\left(2n\zeta% \right)}{1+q^{2n}},$
 22.11.10 $\operatorname{dc}\left(z,k\right)-\frac{\pi}{2K}\sec\zeta=\frac{2\pi}{K}\sum_{% n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos\left((2n+1)\zeta\right)}{1-q^{2n+1}},$
 22.11.11 $\operatorname{nc}\left(z,k\right)-\frac{\pi}{2Kk^{\prime}}\sec\zeta=-\frac{2% \pi}{Kk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos\left((2n+1)% \zeta\right)}{1+q^{2n+1}},$
 22.11.12 $\operatorname{sc}\left(z,k\right)-\frac{\pi}{2Kk^{\prime}}\tan\zeta=\frac{2\pi% }{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{2n}\sin\left(2n\zeta\right)}% {1+q^{2n}}.$

In (22.11.7)–(22.11.12) the left-hand sides are replaced by their limiting values at the poles of the Jacobian functions.

Next, with $E=E\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and $q\exp\left(2|\Im\zeta|\right)<1$,

 22.11.13 ${\operatorname{sn}}^{2}\left(z,k\right)=\frac{1}{k^{2}}\left(1-\frac{E}{K}% \right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{2n}}% \cos\left(2n\zeta\right).$

Similar expansions for ${\operatorname{cn}}^{2}\left(z,k\right)$ and ${\operatorname{dn}}^{2}\left(z,k\right)$ follow immediately from (22.6.1).

For further Fourier series see Oberhettinger (1973, pp. 23–27).

A related hyperbolic series is

 22.11.14 $k^{2}{\operatorname{sn}}^{2}\left(z,k\right)=\frac{{E^{\prime}}}{{K^{\prime}}}% -\left(\frac{\pi}{2{K^{\prime}}}\right)^{2}\sum_{n=-\infty}^{\infty}\left({% \operatorname{sech}}^{2}\left(\frac{\pi}{2{K^{\prime}}}(z-2nK)\right)\right),$

where ${E^{\prime}}={E^{\prime}}\left(k\right)$ is defined by §19.2.9. Again, similar expansions for ${\operatorname{cn}}^{2}\left(z,k\right)$ and ${\operatorname{dn}}^{2}\left(z,k\right)$ may be derived via (22.6.1). See Dunne and Rao (2000).