What is holomorphism?

Holomorphism is a concept in complex analysis that describes functions that are differentiable and analytic throughout a complex plane. A function that is holomorphic at a point is also said to be a complex differentiable at that point. A holomorphic function is essentially a function that can be represented by a convergent power series over any arbitrary region in the complex plane.

The concept of holomorphism is closely related to the Cauchy-Riemann equations, a set of conditions that describe the necessary and sufficient conditions for a function to be holomorphic. In particular, a function that satisfies the Cauchy-Riemann equations is holomorphic.

Holomorphism plays a central role in many different areas of mathematics, including complex geometry, complex analysis, and the theory of functions of a complex variable. It is also used in various applications, including in physics and engineering, where it is used to model phenomena that involve complex numbers, such as the propagation of electromagnetic waves or quantum mechanics.