Cauchy's integral theorem allows for integration over a complex variable. It deals only with analytic functions, that is, those functions which have only a single value at each point in the region of interest. These functions have the same limits on a point no matter which direction that point is approached. The analyticity of a function can be tested with the Cauchy-Riemann equations, which relate the first derivatives of the real and imaginary components of a function in terms of the real and imaginary components of the variable. If a function is analytic, Cauchy's integral theorem states that the integral between two points is always the same regardless of what path is taken.

To state Cauchy's theorem another way, any closed curve in an analytic region integrates to zero. This means that a function can be integrated around a circle, a square, or a...