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- #1

- Jan 17, 2013

- 1,667

7.81.

**Prove that**

\(\displaystyle \int^{\infty}_0 \frac{\sin(ax)}{e^{2 \pi x}-1} \,dx = \frac{1}{4} \coth\left( \frac{a}{2} \right) - \frac{1}{2a}\)

I found a solution here but it is not general , I assumed that \(\displaystyle |a| < 2 \pi \) but there seemed no restriction in the wording of the problem .

This thread will be dedicated to prove the result using contour integration , any comments or replies are always welcomed .