nLab Cauchy integral theorem

The Cauchy integral theorem

The Cauchy integral theorem


Cauchy’s integral theorem states that contour integrals of holomorphic functions in the complex plane \mathbb{C} are invariant under homotopy of paths. In particular, if a function is holomorphic on a simply connected subspace of \mathbb{C}, then its contour integral on a path depends only on the beginning and ending points of the path, and indeed can be given by subtracting the values there of an antiderivative (in accordance with the second Fundamental Theorem of Calculus).


Let DD be an open subset of the complex plane \mathbb{C}, let aa and bb be two points in DD, let γ 1\gamma_1 and γ 2\gamma_2 be two curves in DD from aa to bb, let the region between them also lie entirely within DD, and let ff be a holomorphic function on DD. Then we have

γ 1f(z)dz= γ 2f(z)dz. \int_{\gamma_1} f(z) \,\mathrm{d}z = \int_{\gamma_2} f(z) \,\mathrm{d}z .

In particular we have

γ 1f(z)dz=0 \int_{\gamma_1} f(z) \,\mathrm{d}z = 0

if a=ba = b (because then γ 2\gamma_2 may be taken to be a constant); in other words, the contour integral of a holomorphic function is zero around any loop whose inside lies entirely within the function's domain.


category: analysis

Last revised on January 29, 2024 at 00:04:56. See the history of this page for a list of all contributions to it.