Let the function $F$ be integrable for all real arguments $x$, such that

$$\int_{-\infty}^{\infty}|F(x)| d x<\infty$$

and assume that the series

$$f(\tau)=\sum_{n=-\infty}^{\infty} F(2 n \pi+\tau)$$

converges uniformly for all $0 \leqslant \tau \leqslant 2 \pi$.

Prove the Poisson summation formula

$$f(\tau)=\frac{1}{2 \pi} \sum_{n=-\infty}^{\infty} \hat{F}(n) e^{i n \tau}$$

where $\hat{F}$ is the Fourier transform of $F$. [Hint: You may show that

$$\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{-i m x} f(x) d x=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{-i m x} F(x) d x$$

or, alternatively, prove that $f$ is periodic and express its Fourier expansion coefficients explicitly in terms of $\hat{F}$.]

Letting $F(x)=e^{-|x|}$, use the Poisson summation formula to evaluate the sum

$$\sum_{n=-\infty}^{\infty} \frac{1}{1+n^{2}}$$