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