Complex Methods | Part IB, 2001

State Jordan's Lemma.

Consider the integral

I=Cdzzsin(xz)(a2+z2)sinπzI=\oint_{C} d z \frac{z \sin (x z)}{\left(a^{2}+z^{2}\right) \sin \pi z}

for real xx and aa. The rectangular contour CC runs from ++iϵ+\infty+i \epsilon to +iϵ-\infty+i \epsilon, to iϵ-\infty-i \epsilon, to +iϵ+\infty-i \epsilon and back to ++iϵ+\infty+i \epsilon, where ϵ\epsilon is infinitesimal and positive. Perform the integral in two ways to show that

n=(1)nnsinnxa2+n2=πsinhaxsinhaπ\sum_{n=-\infty}^{\infty}(-1)^{n} \frac{n \sin n x}{a^{2}+n^{2}}=-\pi \frac{\sinh a x}{\sinh a \pi}

for x<π|x|<\pi.

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