Could someone please help me compact the summation as in the left side of the equation below. I tried many ways but it seems to be not right, the result I did is as the right side of the equation below
The summation is written as follows \begin{equation} \sum_{n=1}^{\infty}e^{i\sqrt{k^2-\big(\frac{n\pi}{b}\big)^2}c} = e^{ikc} \end{equation} The condition of the variables is as follows \begin{equation} b,c >0 \end{equation}
Thank you very much!