The Cauchy integral theorem in complex analysis is an important statement about path integrals for holomorphic functions in the complex plane. Essentially, it says that if two different paths connect the same two points, and a function is holomorphic everywhere "in between" the two paths, then the two path integrals of the function will be the same.
The theorem is usually formulated for closed paths as follows: let U be an open subset of C which is simply connected, let f : U -> C be a holomorphic function, and let γ be a rectifiable path in U whose start point is equal to its end point. Then,
The condition that U be simply connected means that U have no "holes"; for instance, every open disk U = { z : |z - z0| < r } qualifies. The condition is crucial; for example, if
One important consequence of the theorem is that path integrals of holomorphic functions on simply connected domains can be computed in a manner familiar from the fundamental theorem of real calculus: let U be a simply connected open subset of C, let f : U -> C be a holomorphic function, and let γ be a piecewise continuously differentiable path in U with start point a and end point b. If F is a complex antiderivative of ''f'\', then
- ∫γ f(z) dz = F(b) - F(a).
- ∫γ f(z) dz = 0.