One I can't do
Many years ago someone told me that Littlewood found a way to use the Residue Theorem (or perhaps complex methods in general) to find $\int_{-\infty}^\infty e^{-t^2/2}\,dt$. I remain stumped - any clues?
One I can do
Teaching baby complex, it's unfortunate that the students get the idea that $\int_{-\infty}^\infty f(t)\,dt$ is sort of automatically $2\pi i$ times the sum of the residues in the upper half plane, without needing to worry about showing that the "other" part of the contour integral tends to $0$. I noticed a simple example regarding that:
Exercise. Use the Residue Theorem to find $\lim_{R\to\infty}\int_{-R}^R\frac{dt}{t+i}$.
It's a straightforward argument; turns out that the integral is half of $2\pi i$ times the sum of the residues in the upper half-plane...