Cauchy's integral formula states that f(z_0) = 1/(2π i) ∮_γ (f(z) d z)/(z - z_0), where the integral is a contour integral along the contour γ enclosing the point z_0. It can be derived by considering the contour integral ∮_γ (f(z) d z)/(z - z_0), defining a path γ_r as an infinitesimal counterclockwise circle around the point z_0, and defining the path γ_0 as an arbitrary loop with a cut line (on which the forward and reverse contributions cancel each other out) so as to go around z_0.

argument principle | Cauchy integral theorem | complex residue | contour integral | Morera's theorem | pole