# §22.11 Fourier and Hyperbolic Series

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

If $q\mathop{\exp\/}\nolimits\!\left(2|\Im{\zeta}|\right)<1$, then

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

Next, if $q\mathop{\exp\/}\nolimits\!\left(|\Im{\zeta}|\right)<1$, then

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

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 $\mathop{E\/}\nolimits=\mathop{E\/}\nolimits\!\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and $q\mathop{\exp\/}\nolimits\!\left(2|\Im{\zeta}|\right)<1$,

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

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

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

A related hyperbolic series is

 22.11.14 $k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{2}}\left(z,k\right)=\frac{\mathop{{E^{% \prime}}\/}\nolimits}{\mathop{{K^{\prime}}\/}\nolimits}-\left(\frac{\pi}{2\!% \mathop{{K^{\prime}}\/}\nolimits}\right)^{2}\sum_{n=-\infty}^{\infty}\left({% \mathop{\mathrm{sech}\/}\nolimits^{2}}\!\left(\frac{\pi}{2\!\mathop{{K^{\prime% }}\/}\nolimits}(z-2n\!\mathop{K\/}\nolimits)\right)\right),$

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