\[\operatorname{sd}\left(z,k\right)=\frac{2\pi}{Kkk^{\prime}}\sum_{n=0}^{\infty}%
\frac{(-1)^{n}q^{n+\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1+q^{2n+1}},\]