Cauchy's integral theorem

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In mathematics, the Cauchy integral 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 : UC be a holomorphic function, and let \!\,\gamma be a rectifiable path in U whose start point is equal to its end point. Then,

Briefly, the path integral along a Jordan curve of a function holomorphic in the interior of the curve, is zero. Instaed of a single closed path we can consider a linear combination of closed path where the scalars are integers. Such a combination is called a closed chain and one defines integral along the chain as the linear combination of integrals over individual paths. A closed chain is called a 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 to extend Cauchy's theorem to multiply connected regions analytically.

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Discussion

As was shown by Goursat, Cauchy's integral theorem can be proven assuming only that the complex derivative f '(z) exists everywhere in U. This is significant, because one can then prove Cauchy's integral formula for these functions, and from that one can deduce that these functions are in fact infinitely differentiable.

The condition that U be simply connected means that U has no "holes" or, in homotopy terms, that the fundamental group of U is trivial; for instance, every open disk U = {z: | zz0 | < r} qualifies. The condition is crucial; consider

which traces out the unit circle, and then the path integral

is non-zero; the Cauchy integral theorem does not apply here since f(z) = 1 / z is not defined (and certainly not holomorphic) at z = 0.

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 : UC 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

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