# Cauchy Theorem

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## Cauchy Theorem

a theorem concerned with the expansion of an analytic function into a power series. Suppose f(z) is a function that is single-valued and analytic in a region G, Z0 is an arbitrary (finite) point of G, and ρ is the distance from z0 to the boundary of this region. Then there exists a power series in z – z0 that converges to the function in the interior of the circle ǀz – z0ǀ = ρ:

If the boundary of G reduces to the point at infinity, then ρ is infinite. This theorem was established by A. Cauchy (1831), who based it on his representation of an analytic function in the form of the so-called Cauchy integral.

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5, using Cauchy's theorem, we have, for all T [member of] C
From Cauchy's theorem, for T [member of] C, we have
He continues with Abel's theorem, the gamma function, universal covering spaces, Cauchy's theorem for non-holomorphic functions and harmonic conjugates.
The integral can be evaluated by shifting vertically the integration line and making use of Cauchy's Theorem.

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