Residues

A residue is the coefficient of \(z^{-1}\) in the Laurent series of a function around an isolated singularity \(z_0\). For a simple pole where \(f(z)=\frac{g(z)}{h(z)}\) with \(g(z_0)\neq0\) and \(h'(z_0)\neq0\), the residue is \(\frac{g(z_0)}{h'(z_0)}\). For a pole of order \(m\), use \(\operatorname{Res}(f,z_0)=\frac{1}{(m-1)!}\lim_{z\to z_0}\frac{d^{m-1}}{dz^{m-1}}\big((z-z_0)^m f(z)\big)\). For real integrals, close a contour in the complex plane and sum residues inside it; the integral equals \(2\pi i\) times that sum. For trigonometric integrals over \([0,2\pi]\), set \(z=e^{i\theta}\), so \(d\theta=\frac{dz}{iz}\) and \(\cos\theta=\frac{z+z^{-1}}{2}\), turning the integral into a contour integral around the unit circle.