In complex analysis, a complex-valued function f of a complex variable Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

• is holomorphic at a point a iff it is differentiable at every point within some open disk centered at a, and

• is analytic at a if in some open disk centered at a 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. Among the corollaries of this theorem are Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include Q is necessary and sufficient for P and P... In mathematics, an analytic function is one that is locally given by a convergent power series. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In...

• 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 holomorphic functions are differentiable not just once, but infinitely often, and

• the fact that the radius of convergence is always the distance from the center a to the nearest singularity; if there are no singularities (i.e., if f 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.

In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ... 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. ... In complex analysis, an entire function is a function that is holomorphic everywhere on the whole complex plane. ...

Proof

Suppose f 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. Then, using Cauchy's integral formula, we get In mathematics, Cauchys integral formula, named after Augustin Louis Cauchy, is a central statement in complex analysis. ...

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.

The generalized Cauchy integral formula

The fact that the coefficient is given by

is a generalization of Cauchy's integral formula, since the latter is just the case in which n = 0.

A by-product of the proof

The argument works if z is any point that is closer to the center than is any singularity of f. Therefore the radius of convergence of the power series cannot be smaller than the distance from the center to the nearest singularity (nor can it be larger, since power series have no singularities in the interiors of their circles of convergence). 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. ... In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In...

External links

• Existence of power series on PlanetMath.