A function has a pole at \(z_0\) and its Laurent series on \(0<|z-z_0|<r\) is \(\frac{3}{(z-z_0)^2}+\frac{5}{z-z_0}+7+\cdots\). What is \(\oint_{|z-z_0|=r/2}f\,dz\)?
Only the \(\frac{5}{z-z_0}\) term survives integration: the integral is \(2\pi i\cdot a_{-1}=10\pi i\). The \(\frac{3}{(z-z_0)^2}\) term integrates to \(0\).
The residue of a function \(f\) at an isolated singularity \(z_0\) is the coefficient \(a_{-1}\) of \((z-z_0)^{-1}\) in its Laurent series on a punctured disc of radius \(r\).