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.