Editing Analyticity of holomorphic functions

{{Short description|Theorem}}
{{Complex_analysis_sidebar}}
In [[complex analysis]], a [[complex number|complex]]-valued [[function (mathematics)|function]] <math>f</math> of a complex variable <math>z</math>:

*is said to be [[holomorphic function|holomorphic]] at a point <math>a</math> if it is [[Differentiable function|differentiable]] at every point within some [[open disk]] centered at <math>a</math>, and
* is said to be [[analytic function|analytic]] at <math>a</math> if in some open disk centered at <math>a</math> it can be expanded as a [[Convergent series|convergent]] [[power series]] <math display="block">f(z)=\sum_{n=0}^\infty c_n(z-a)^n</math> (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]] <math>S</math> with an [[accumulation point]] inside the [[intersection]] of their [[Domain of a function|domains]] also agree everywhere in every connected open [[subset]] of their domains that contains the set <math>S</math>, 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 <math>a</math> to the nearest non-removable [[mathematical singularity|singularity]]; if there are no singularities (i.e., if <math>f</math> 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 set|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 manifold|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

: <math>\frac 1 {w-z} .</math>

Let <math>D</math> be an open disk centered at <math>a</math> and suppose <math>f</math> is differentiable everywhere within an [[open neighborhood]] containing the closure of <math>D</math>.  Let <math>C</math> be the positively oriented (i.e., counterclockwise) circle which is the boundary of <math>D</math> and let <math>z</math> be a point in <math>D</math>.  Starting with Cauchy's integral formula, we have

: <math>\begin{align}f(z) &{}= {1 \over 2\pi i}\int_C {f(w) \over w-z}\,\mathrm{d}w \\[10pt]

&{}= {1 \over 2\pi i}\int_C {f(w) \over (w-a)-(z-a)} \,\mathrm{d}w \\[10pt]
&{}={1 \over 2\pi i}\int_C {1 \over w-a}\cdot{1 \over 1-{z-a \over w-a}}f(w)\,\mathrm{d}w \\[10pt]
&{}={1 \over 2\pi i}\int_C {1 \over w-a}\cdot{\sum_{n=0}^\infty\left({z-a \over w-a}\right)^n} f(w)\,\mathrm{d}w \\[10pt]
&{}=\sum_{n=0}^\infty{1 \over 2\pi i}\int_C {(z-a)^n \over (w-a)^{n+1}} f(w)\,\mathrm{d}w.\end{align}</math>

Interchange of the integral and infinite sum is justified by observing that <math>f(w)/(w-a)</math> is  bounded on <math>C</math> by some positive number <math>M</math>, while for all <math>w</math> in <math>C</math>

: <math>\left|\frac{z-a}{w-a}\right|\leq r < 1 </math>

for some positive <math>r</math> as well.  We therefore have

: <math>\left| {(z-a)^n \over (w-a)^{n+1} }f(w) \right| \le Mr^n,</math>

on <math>C</math>, and as the [[Weierstrass M-test]] shows the series [[converges uniformly]] over <math>C</math>, the sum and the integral may be interchanged.

As the factor <math>(z-a)^n</math> does not depend on the variable of integration <math>w</math>, it may be factored out to yield

: <math>f(z)=\sum_{n=0}^\infty (z-a)^n {1 \over 2\pi i}\int_C {f(w) \over (w-a)^{n+1}} \,\mathrm{d}w,</math>

which has the desired form of a power series in <math>z</math>:

: <math>f(z)=\sum_{n=0}^\infty c_n(z-a)^n</math>

with coefficients

: <math>c_n={1 \over 2\pi i}\int_C {f(w) \over (w-a)^{n+1}} \,\mathrm{d}w.</math>

== Remarks ==
* Since power series can be differentiated term-wise, applying the above argument in the reverse direction and the power series expression for <math display="block"> \frac 1 {(w-z)^{n+1}} </math> gives <math display="block">f^{(n)}(a) = {n! \over 2\pi i} \int_C {f(w) \over (w-a)^{n+1}}\, dw.</math> This is a [[Cauchy's integral formula|Cauchy integral formula]] for derivatives. Therefore the power series obtained above is the [[Taylor series]] of <math>f</math>.
* The argument works if <math>z</math> is any point that is closer to the center <math>a</math> than is any singularity of <math>f</math>.  Therefore, the radius of convergence of the Taylor series cannot be smaller than the distance from <math>a</math> 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 <math>U</math> of <math>a</math>, then they coincide on the open disk <math>B_d(a)</math>, where <math>d</math> is the distance from <math>a</math> to the nearest singularity.

== External links ==
* {{planetmath reference|urlname=ExistenceOfPowerSeries|title=Existence of power series}}

[[Category:Analytic functions|holomorphic functions]]
[[Category:Theorems in complex analysis]]
[[Category:Article proofs]]