In complex analysis, a complex-valued function
: One of the most important theorems of complex analysis is that holomorphic functions are analytic and vice versa.
Among the corollaries of this theorem are The argument, first given by Cauchy, hinges on Cauchy's integral formula and the power series expansion of the expression Let
be an open disk centered at
is differentiable everywhere within an open neighborhood containing the closure of
be the positively oriented (i.e., counterclockwise) circle which is the boundary of
Starting with Cauchy's integral formula, we have Interchange of the integral and infinite sum is justified by observing that
by some positive number
, and as the Weierstrass M-test shows the series converges uniformly over
, the sum and the integral may be interchanged.
does not depend on the variable of integration
, it may be factored out to yield which has the desired form of a power series in