What sort of integral problem is that? Why's there a semicircle?

Billyaabob said:

What sort of integral problem is that? Why's there a semicircle?

Contour Integrals.

You can use them to solve definite integrals that would otherwise be incredibly messy to approach. Since that works with complex numbers, however, you need to perform analytic continuation to translate your real function into a complex one.

See also here