Holomorfía

Holomorfía is a concept in complex analysis that describes functions which are differentiable in a neighborhood of every point in an open subset of the complex plane. A complex-valued function is called holomorphic if it has a complex derivative at every point in some open set. This is a much stronger condition than differentiability for real-valued functions. If a function is holomorphic on an open set, it is not only differentiable but also infinitely differentiable, analytic, and can be represented by a convergent power series in a neighborhood of any point in that set.

The term "holomorphic" was introduced by Charles Jean de la Vallée Poussin. An alternative term, "analytic," is often used interchangeably with holomorphic, though some mathematicians draw subtle distinctions. A function is holomorphic on a domain if it is holomorphic at every point in that domain. The study of holomorphic functions is central to complex analysis and has applications in various fields of mathematics, physics, and engineering, including fluid dynamics, electromagnetism, and number theory. Key theorems in complex analysis, such as Cauchy's integral theorem and the residue theorem, are fundamental to understanding the properties of holomorphic functions.