In complex analysis, a complex-valued function:

• is said to be holomorphic if it is differentiable at every point within some open disk, and

• is said to be analytic it can be expanded as a convergent power series (this implies that the radius of convergence is positive).

One of the most important theorems of complex analysis is that holomorphic functions are analytic and vice versa. (A holomorphic function at a point is analytic at the point and vice versa.) Among the corollaries of this theorem are

• the identity theorem that two holomorphic functions that agree at every point of an infinite set with an accumulation point inside the intersection of their domains also agree everywhere in every connected open subset, and

• the fact that, since power series are infinitely differentiable, so are holomorphic functions (this is in contrast to the case of real differentiable functions), and

• the fact that the radius of convergence is always the distance to the nearest non-removable singularity is an entire function), then the radius of convergence is infinite. Strictly speaking, this is not a corollary of the theorem but rather a by-product of the proof.

• no bump function supported in a non-empty compact subset of the complex plane can be entire. In particular, on any connected open subset of the complex plane, there can be no bump function defined on that set which is holomorphic on the set. This has important ramifications for the study of complex manifolds, as it precludes the use of partitions of unity. In contrast the partition of unity is a tool which can be used on any real manifold.

Proof

The argument, first given by Cauchy, hinges on Cauchy's integral formula and the power series expansion of the expression

is differentiable everywhere within an open neighborhood. Starting with Cauchy's integral formula, we have

as well. We therefore have

, and as the Weierstrass M-test shows the series converges uniformly, the sum and the integral may be interchanged.

, it may be factored out to yield

:

with coefficients

Remarks

• This is a Cauchy integral formula for derivatives. Therefore the power series obtained above is the Taylor series.

• to the nearest singularity (nor can it be larger, since power series have no singularities in the interiors of their circles of convergence).

• A special case of the identity theorem to the nearest singularity.

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