$$f(a,x)=\sum_{\tau=-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}$$
How behave this function?  How to  interpolate it with integral?
Can I somehow interpolate it with
$$\int_{-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}d\tau$$?