Maybe artificial, but a nice example (*I think*) demonstrating *analytic continuation* (***NOT*** just the usual $\mathrm{Re}(e^{i \theta})$ method!) I don't know any reasonable way of doing this by real methods.


As a fun exercise, calculate 
$$
I(\omega) = \int_0^\infty e^{-x} \cos (\omega x) \frac{dx}{\sqrt{x}}, \qquad \omega \in \mathbb{R}
$$
from the real part of $F(1+i \omega)$, where 
$$
F(k) = \int_0^\infty e^{-kx} \frac{dx}{\sqrt{x}}, \qquad \mathrm{Re}(k)>0
$$
(*which is easily obtained for $k>0$ by a real substitution*) and using analytic continuation to justify the same formula with $k=1+i \omega$.

You need care with *square roots*, *branch cuts*, etc.; but this can be avoided by considering $F(k)^2$, $I(\omega)^2$.



Of course all the standard integrals provide endless fun examples! (But the books don't have many requiring *genuine* analytic continuation like this!)