Are holomorphic and analytic the same thing?

For functions of a complex variable they are equivalent: complex differentiability on an open set, called holomorphy, is exactly the condition that the function is locally a convergent power series, called analyticity.

Why can a holomorphic function not have a local maximum of its size inside a region?

The maximum modulus principle follows from the mean value property of harmonic functions; the modulus can only reach its largest value on the boundary unless the function is constant.

Holomorphic Functions

A holomorphic function is one that is complex differentiable on an open set; this single condition forces the function to be analytic, infinitely differentiable, and locally representable by a convergent power series.

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Definition

A function of a complex variable is holomorphic on an open set if it has a complex derivative at every point of that set; equivalently it is analytic there, meaning it is locally the sum of a convergent power series.

Scope

This topic covers complex differentiability and the Cauchy-Riemann equations, the equivalence of holomorphy and analyticity, power-series representations, the relation to harmonic functions, the identity and maximum modulus principles, entire functions and Liouville's theorem, and the classification of zeros and isolated singularities.

Core questions

Why does the existence of a complex derivative impose the Cauchy-Riemann equations?
Why is every holomorphic function automatically analytic and infinitely differentiable?
How are the real and imaginary parts of a holomorphic function constrained to be harmonic?
What kinds of singularities can a holomorphic function have, and how are they classified?

Key theories

Cauchy-Riemann equations: Complex differentiability is equivalent to the real and imaginary parts satisfying a coupled pair of partial differential equations, which forces each part to be harmonic and links complex analysis to potential theory.
Maximum modulus and identity principles: A non-constant holomorphic function attains no interior maximum of its modulus, and two holomorphic functions agreeing on a set with a limit point agree everywhere on a connected domain, expressing the rigidity of holomorphic functions.
Liouville's theorem: A bounded entire function is constant, a consequence of the Cauchy estimates that yields a short proof of the fundamental theorem of algebra.

Clinical relevance

Because the real and imaginary parts of a holomorphic function are harmonic, holomorphic functions model two-dimensional steady-state phenomena such as electrostatic potentials and ideal fluid flow, and the rigidity properties make them powerful in number theory, special-function theory, and the analytic continuation of transforms.

History

The defining role of the Cauchy-Riemann equations was recognized by Cauchy and Riemann in the mid-nineteenth century, while Weierstrass developed the equivalent power-series viewpoint. Their combined work established that complex differentiability and analyticity coincide.

Key figures

Augustin-Louis Cauchy
Bernhard Riemann
Karl Weierstrass

Seminal works

ahlfors1979
conway1978

Frequently asked questions

Are holomorphic and analytic the same thing?: For functions of a complex variable they are equivalent: complex differentiability on an open set, called holomorphy, is exactly the condition that the function is locally a convergent power series, called analyticity.
Why can a holomorphic function not have a local maximum of its size inside a region?: The maximum modulus principle follows from the mean value property of harmonic functions; the modulus can only reach its largest value on the boundary unless the function is constant.