Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x by Cosecant of p x

Theorem

 * $\displaystyle \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \csc p x} \rd x = \arctan \frac b p - \arctan \frac a p$

where:
 * $a$ and $b$ are non-negative real numbers
 * $p$ is a non-zero real number.

Proof
Fix $p$ and set:


 * $\displaystyle \map I \alpha = \int_0^\infty \frac {e^{-\alpha x} } {x \csc p x} \rd x$

for all $\alpha \ge 0$.

Then:


 * $\displaystyle \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \csc p x} \rd x = \map I a - \map I b$

We have:

so:

for all $\alpha \ge 0$ and constant $C \in \R$.

We therefore have: