Analyticity of holomorphic functions

In complex analysis, a complex-valued function of a complex variable :

  • is said to be holomorphic at a point if it is differentiable at every point within some open disk centered at , and
  • is said to be analytic at if in some open disk centered at 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. 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 of their domains that contains the set , 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 from the center to the nearest non-removable singularity; if there are no singularities (i.e., if 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 on 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

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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   and suppose   is differentiable everywhere within an open neighborhood containing the closure of  . Let   be the positively oriented (i.e., counterclockwise) circle which is the boundary of   and let   be a point in  . Starting with Cauchy's integral formula, we have

 

Interchange of the integral and infinite sum is justified by observing that   is bounded on   by some positive number  , while for all   in  

 

for some positive   as well. We therefore have

 

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

As the factor   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  :

 

with coefficients

 

Remarks

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  • Since power series can be differentiated term-wise, applying the above argument in the reverse direction and the power series expression for   gives   This is a Cauchy integral formula for derivatives. Therefore the power series obtained above is the Taylor series of  .
  • The argument works if   is any point that is closer to the center   than is any singularity of  . Therefore, the radius of convergence of the Taylor series cannot be smaller than the distance from   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 follows from the preceding remark. If two holomorphic functions agree on a (possibly quite small) open neighborhood   of  , then they coincide on the open disk  , where   is the distance from   to the nearest singularity.
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  • "Existence of power series". PlanetMath.