A holomorphic function is a complex-valued function defined on an open subset of the complex plane that is differentiable at every point in its domain. This differentiability implies that the function is infinitely differentiable and can be represented by a convergent power series within its domain, making it an essential object of study in complex analysis.

Holomorphic Function

holomorphic function

complex-valued function

open subset of the complex plane

differentiable at every point

An infinitely differentiable function is one that has derivatives of all orders, meaning it can be differentiated an infinite number of times without encountering discontinuities or undefined behavior. Such functions are smooth and continuous, often used in theoretical mathematics to explore properties of calculus and analysis.

infinitely differentiable

A convergent power series is a series of the form ∑aₙ(x-c)ⁿ that converges to a finite value within a certain radius of convergence around the center c. The convergence is determined by the ratio or root test, and the series represents a function that is analytic within this radius.

convergent power series

essential object of study

Complex analysis is a branch of mathematics that studies functions of complex numbers and their properties, such as differentiability and integrability, which often lead to elegant and powerful results not seen in real analysis. It plays a crucial role in various fields, including engineering, physics, and number theory, due to its ability to simplify problems and provide deep insights into the nature of mathematical structures.

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