$$f(a,x)=\sum_{\tau=-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}$$ How does this function behave? How to interpolate it with an integral? Can I somehow interpolate it with $$\int_{-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}d\tau$$ where $a\in(0,0.5)$, p is a natural number, and $x$ is a real number

$$f(a,x)=\sum_{\tau=-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}$$

Can I apply Euler-Maclauren formula to this sum?

where $a\in(0,0.5)$, p is a natural number, and $x$ is a real number

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

where $a\in(0,0.5)$  , p is a natural number, $x$ is a real number

$$f(a,x)=\sum_{\tau=-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}$$ How does this function behave? How to interpolate it with an integral? Can I somehow interpolate it with $$\int_{-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}d\tau$$ where $a\in(0,0.5)$, p is a natural number, and $x$ is a real number

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

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

where $a\in(0,0.5)$ , p is a natural number, $x$ is a real number