Definite Integral to Infinity of Sine of a x over Hyperbolic Sine of b x

Theorem

 * $\ds \int_0^\infty \frac {\sin a x} {\sinh b x} \rd x = \frac \pi {2 b} \tanh \frac {a \pi} {2 b}$

where:
 * $a$ and $b$ are positive real numbers
 * $\tanh$ denotes the hyperbolic cotangent function.

Proof
We have, as $b, n > 0$:

We similarly have:

So:

By Mittag-Leffler Expansion for Hyperbolic Tangent Function, we have:


 * $\ds \pi \map {\tanh} {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {4 z^2 + \paren {2 n + 1}^2}$

where $z$ is not a half-integer multiple of $i$.

Setting $z = \dfrac a {2 b}$ gives:


 * $\ds \pi \map {\tanh} {\frac {a \pi} {2 b} } = 8 \sum_{n \mathop = 0}^\infty \frac {\paren {\frac a {2 b} } } {4 \paren {\frac a {2 b} }^2 + \paren {2 n + 1}^2}$

Therefore: