Analyticity Of Holomorphic Functions
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In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
-valued
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
f of a complex variable z: *is said to be
holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
at a point a if it is
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
at every point within some
open disk In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius, r, an open disk is usu ...
centered at a, and * is said to be analytic at a if in some open disk centered at a it can be expanded as a convergent
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
f(z)=\sum_^\infty c_n(z-a)^n (this implies that the
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
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 In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions ''f'' and ''g'' analytic on a domain ''D'' (open and connected subset of \mathbb or \mathbb), if ''f'' = ''g'' on so ...
that two holomorphic functions that agree at every point of an infinite set S 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 S, 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 a to the nearest non-removable singularity; if there are no singularities (i.e., if f is an
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
), 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 In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump f ...
on the complex plane can be entire. In particular, on any
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
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 mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0, ...
. 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 In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary ...
and the power series expansion of the expression : \frac 1 . Let D be an open disk centered at a and suppose f is differentiable everywhere within an open neighborhood containing the closure of D. Let C be the positively oriented (i.e., counterclockwise) circle which is the boundary of D and let z be a point in D. Starting with Cauchy's integral formula, we have : \beginf(z) &= \int_C \,\mathrmw \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
&= \int_C \,\mathrmw \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
&=\int_C \cdotf(w)\,\mathrmw \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
&=\int_C \cdot f(w)\,\mathrmw \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
&=\sum_^\infty\int_C f(w)\,\mathrmw.\end Interchange of the integral and infinite sum is justified by observing that f(w)/(w-a) is bounded on C by some positive number M, while for all w in C : \left, \frac\\leq r < 1 for some positive r as well. We therefore have : \left, f(w) \ \le Mr^n, on C, and as the
Weierstrass M-test In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous t ...
shows the series converges uniformly over C, the sum and the integral may be interchanged. As the factor (z-a)^n does not depend on the variable of integration w, it may be factored out to yield : f(z)=\sum_^\infty (z-a)^n \int_C \,\mathrmw, which has the desired form of a power series in z: : f(z)=\sum_^\infty c_n(z-a)^n with coefficients : c_n=\int_C \,\mathrmw.


Remarks

* Since power series can be differentiated term-wise, applying the above argument in the reverse direction and the power series expression for \frac 1 gives f^(a) = \int_C \, dw. This is a Cauchy integral formula for derivatives. Therefore the power series obtained above is the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
of f. * The argument works if z is any point that is closer to the center a than is any singularity of f. Therefore, the radius of convergence of the Taylor series cannot be smaller than the distance from a 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 In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions ''f'' and ''g'' analytic on a domain ''D'' (open and connected subset of \mathbb or \mathbb), if ''f'' = ''g'' on so ...
follows from the preceding remark. If two holomorphic functions agree on a (possibly quite small) open neighborhood U of a, then they coincide on the open disk B_d(a), where d is the distance from a to the nearest singularity.


External links

* {{planetmath reference, urlname=ExistenceOfPowerSeries, title=Existence of power series
holomorphic functions In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
Theorems in complex analysis Article proofs