Analyticity of holomorphic functions information

In complex analysis, a complex-valued function $f$ of a complex variable $z$ :

is said to be holomorphic at a point $a$ if it is differentiable at every point within some open disk 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 $f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}$ (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 $S$ 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), 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.

Analyticity of holomorphic functions information

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Analyticity of holomorphic functions

Holomorphic function

Analytic function

Complex analysis

Value distribution theory of holomorphic functions

Holomorphic functional calculus

Harmonic function

Zeros and poles

Univalent function

Complex geometry

Meromorphic function

Hyperbolic functions

Function of several complex variables

Laurent series

Radius of convergence

List of types of functions

Entire function

Conformal map

Trigonometric functions

Dirac delta function

Antiholomorphic function

Gamma function

Residue theorem

Differentiable function

Complex analysis
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