Cauchy's integral theorem
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line 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 be a rectifiable path in U whose start point is equal to its end point. Then
A precise (homology) version can be stated using winding numbers. The winding number of a closed curve around a point a not on the curve is the integral of f(z)/(2πi), where f(z) = 1/(z − a) around the curve. It is an integer. Briefly, the path integral along a Jordan curve of a function holomorphic in the interior of the curve, is zero. Instead of a single closed path we can consider a linear combination of closed paths, where the scalars are integers. Such a combination is called a closed chain, and one defines an integral along the chain as a linear combination of integrals over individual paths. A closed chain is called a cycle in a region if it is homologous to zero in the region; that is, the winding number, expressed by the integral of 1/(z − a) over the closed chain, is zero for each point 'a' not in the region. This means that the closed chain does not wind around points outside the region. Then Cauchy's theorem can be stated as the integral of a function holomorphic in an open set taken around any cycle in the open set is zero. An example is furnished by the ring-shaped region. This version is crucial for rigorous derivation of Laurent series and Cauchy's residue formula without involving any physical notions such as cross cuts or deformations. The version enables the extension of Cauchy's theorem to multiply-connected regions analytically.
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