holomorphic

Holomorphic is a term in complex analysis describing a function defined on an open subset of the complex plane that is complex differentiable at every point of its domain. If f: U → C is holomorphic on U, then f is not only differentiable but analytic: around every point a in U there exists a power series that converges to f in some neighborhood of a. The complex differentiability implies many strong regularity properties not shared by real differentiability.

A common criterion is the Cauchy–Riemann equations: writing f = u + iv with u, v real-valued, the partial derivatives exist and satisfy ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x, and f is differentiable as a function of the real variables; equivalently, ∂f/∂z̄ = 0 in Wirtinger calculus.

Holomorphic functions are infinitely differentiable, and on a connected open set, they are determined by their values on any open subset (the identity theorem). Entire functions are holomorphic on the whole complex plane; examples include polynomials, the exponential function, sine and cosine, and more generally many functions obtained by composing holomorphic functions away from their singularities.

Not all complex-valued functions are holomorphic; for instance, the complex conjugation z ↦ z̄ is nowhere holomorphic.

Important theorems about holomorphic functions include Liouville's theorem, which states that a bounded entire function is constant; the maximum modulus principle, which restricts the maximum of the modulus to the boundary in a domain; and the open mapping theorem, which asserts that non-constant holomorphic functions map open sets to open sets.