Holomorfia

Holomorfia 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 of a complex variable is called holomorphic if it has a complex derivative at every point in some open set. This is a stronger condition than real differentiability for a function of two real variables. A function that is holomorphic on its entire domain is called an entire function.

The Cauchy-Riemann equations are a necessary condition for a function to be holomorphic. If a function $f(x + iy) = u(x, y) + iv(x, y)$ has partial derivatives, these partial derivatives must satisfy $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ for the function to be complex differentiable. If these equations are satisfied and the partial derivatives are continuous, then the function is indeed holomorphic.

Holomorphic functions possess many remarkable properties. They are infinitely differentiable, analytic (meaning they can be represented by a convergent power series in a neighborhood of any point), and satisfy Cauchy's integral theorem and its related formulas. These properties make holomorphic functions a central object of study in complex analysis and have applications in various fields, including physics and engineering. The term "analytic" is often used interchangeably with "holomorphic" for functions of a complex variable.