Complex Methods | Part IB, 2001

Let the function FF be integrable for all real arguments xx, such that

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

and assume that the series

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

converges uniformly for all 0τ2π0 \leqslant \tau \leqslant 2 \pi.

Prove the Poisson summation formula

f(τ)=12πn=F^(n)einτf(\tau)=\frac{1}{2 \pi} \sum_{n=-\infty}^{\infty} \hat{F}(n) e^{i n \tau}

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

12π02πeimxf(x)dx=12πeimxF(x)dx\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 ff is periodic and express its Fourier expansion coefficients explicitly in terms of F^\hat{F}.]

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

n=11+n2\sum_{n=-\infty}^{\infty} \frac{1}{1+n^{2}}

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