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.

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.