State 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.
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.
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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.
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.
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