\[\operatorname{dc}\left(z,k\right)-\frac{\pi}{2K}\sec\zeta=\frac{2\pi}{K}\sum_{%
n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos\left((2n+1)\zeta\right)}{1-q^{2n+1}},\]