\[-2iKkk^{\prime}\operatorname{sd}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}%
\frac{(-1)^{n}\pi}{\sin\left(\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)\right)}=%
\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t+%
\frac{1}{2}-m-(n+\frac{1}{2})\tau}\right),\]