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