In complex analysis

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, a branch of mathematics, a complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

-valued function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 ƒ of a complex variable z.

• is said to be holomorphic
  Holomorphic function
  In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
  
   at a point a if it is differentiable at every point within some open disk centered at a, and

• is said to be analytic
  Analytic function
  In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
  
   at a if in some open disk centered at a it can be expanded as a convergent power series

.

One of the most important theorems of complex analysis is that holomorphic functions are analytic. Among the corollaries of this theorem are

• the fact 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 some open 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 a to the nearest singularity
  Mathematical singularity
  In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
  
  ; if there are no singularities (i.e., if ƒ is an entire function
  Entire function
  In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
  
  ), 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

The argument, first given by Cauchy, hinges on Cauchy's integral formula

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 of the disk, and it provides integral formulas for all...

 and the power series development of the expression

.

Suppose ƒ is differentiable everywhere within some open disk centered at a. Let z be within that open disk. Let C be a positively oriented (i.e., counterclockwise) circle centered at a, lying within that open disk but farther from a than z is. Starting with Cauchy's integral formula, we have

To justify the interchange of the sum and the integral, one must notice that in the intersection of |(z − a)/(w − a)| ≤ r < 1 and some closed domain containing C, ƒ(w)/(w − a) is holomorphic and therefore bounded by some positive number M. So we have

The Weierstrass M-test says the series converges uniformly, and thus the interchange of the sum and the integral is justified.

Since the factor (z − a)ⁿ does not depend on the variable of integration w, it can be pulled out:

And now the integral and the factor of 1/(2πi) do not depend on z, i.e., as a function of z, that whole expression is a constant c_n, so we can write:

and that is the desired power series.

Remarks

• 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

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

 of ƒ.

• The argument works if z is any point that is closer to the center a than is any singularity of ƒ. 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
  Identity theorem
  In complex analysis, a branch of mathematics, the identity theorem for holomorphic functions states: given functions f and g holomorphic on a connected open set D, if f = g on some neighborhood of z that is in D, then f = g on D. Thus a holomorphic function is completely determined by its values on...
  
  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.

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