An analytic function of a complex variable is a function that is differentiable at every point in its domain. In other words, the function must have a well-defined derivative at every point in the complex plane.
The concept of analytic functions is closely related to the complex plane because it is defined as a function of a complex variable. This means that the function can be represented as a mapping from the complex plane to itself.
The main properties of analytic functions include being differentiable at every point in the complex plane, having a well-defined derivative, and satisfying the Cauchy-Riemann equations. These properties allow for the use of powerful tools such as the Cauchy integral theorem and the Cauchy integral formula.
Analytic functions are used in a wide range of mathematical and scientific fields, including complex analysis, differential equations, physics, and engineering. They are particularly useful in solving problems involving complex numbers and in the study of functions with complex inputs and outputs.
To determine if a function is analytic, one can check if it satisfies the Cauchy-Riemann equations, which are necessary conditions for a function to be analytic. Additionally, if a function is expressible as a power series, then it is also analytic. Other methods include using the Cauchy-Riemann integral theorem or the Cauchy-Riemann integral formula.