Holomorfní

Holomorfní, in Czech often written as holomorfní, is a term from complex analysis describing a function that is complex differentiable on an open domain. A function f: U → C, defined on an open set U in the complex plane, is holomorfní if the complex derivative f′(z) exists for every z in U. This property is stronger than mere real differentiability and has strong consequences in the theory.

One key aspect of holomorphic functions is their local representability by power series. At every point z0 in U, there exists a neighborhood on which f admits a convergent Taylor series, making holomorphic functions analytic. Equivalently, holomorfní functions satisfy the Cauchy-Riemann equations in every coordinate representation, linking partial derivatives of the real and imaginary parts.

Holomorfní functions exhibit many important properties. They are infinitely differentiable and satisfy the Cauchy integral formula, which leads to results such as Liouville’s theorem (every bounded entire holomorphic function is constant) and the identity theorem (coincidence on a set with a limit point implies equality everywhere on the domain). They also obey the maximum modulus principle and the open mapping theorem, reflecting their rigid structure.

Examples include polynomials, the exponential function e^z, sine and cosine, and any holomorphic function on the whole plane (entire functions). Functions like 1/z are holomorfní on C without the origin but not on domains containing 0. Meromorphic functions generalize holomorfní by allowing isolated poles.

In Czech mathematical usage, holomorfní and holomorfnost describe the same concept as holomorphic functions in English, typically within the context of domains in the complex plane or complex manifolds.