An analytic function is a complex function that is locally given by a convergent power series, meaning it is differentiable at every point in its domain and its derivatives are continuous. These functions are central to complex analysis, as they exhibit properties such as conformality, the ability to be represented by Taylor or Laurent series, and adherence to the Cauchy-Riemann equations.

Analytic Function

analytic function

complex function

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

differentiable at every point

derivatives are continuous

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.

complex analysis

conformality

The Taylor series is a mathematical representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. This powerful tool allows for the approximation of complex functions by polynomials, making it essential in fields like calculus, numerical analysis, and differential equations.

Taylor Series

The Laurent series is a representation of a complex function as an infinite sum of terms, which can include negative powers, and is particularly useful for analyzing functions with singularities. It generalizes the Taylor series and is essential in complex analysis for understanding the behavior of functions near points where they are not analytic.

Laurent series

The Cauchy-Riemann equations are a set of two partial differential equations that provide necessary and sufficient conditions for a complex function to be holomorphic, meaning it is complex differentiable at every point in its domain. These equations link the real and imaginary parts of a complex function, ensuring the function is conformal and preserves angles locally.

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