My question is: how to obtain
$$ \frac{1}{x + y + 2\lambda} = \frac{1}{2 \pi} \int^\infty_{-\infty} \left(\frac{1}{x + \lambda + i\mu}\right)\left(\frac{1}{y + \lambda - i\mu}\right) \,\mathrm d\mu \, ? $$
for $x, y, \lambda > 0$. This is supposed to be a case of Laplace transform but I don't really see how to show the equality.