# 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 z_{0} to the boundary of this region. Then there exists a power series in z – z_{0} that converges to the function in the interior of the circle ǀz – z_{0}ǀ = 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.