Analytical function

Analytical function refers to a type of function in the field of complex analysis, a branch of mathematics that studies functions that operate on complex numbers. An analytical function, also known as a holomorphic function, is a function that is locally given by a convergent power series. This means that at every point in its domain, the function can be represented by a power series that converges to the function's value at that point.

Definition

Formally, a function f is said to be analytical on a domain D in the complex plane if for any point z₀ in D, there exists a neighborhood of z₀ in which f can be expressed as a convergent power series: \[f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n\] where aₙ are complex coefficients and the series converges for z in the neighborhood of z₀.

Properties

Analytical functions possess several important properties that make them a central object of study in complex analysis:

• Complex Differentiability: An analytical function is infinitely differentiable within its domain, and its derivatives are also analytical functions.
• Analytic Continuation: If two analytical functions agree on a non-empty open subset of their common domain, they are identical on the entirety of their common domain.
• Cauchy's Integral Theorem: This theorem states that if f is analytical in a simply connected domain, then the integral of f around any closed contour in the domain is zero.
• Morera's Theorem: This theorem provides a converse to Cauchy's Integral Theorem, stating that if a continuous function f satisfies the condition that the integral of f around every closed contour is zero, then f is analytical.

Examples

• The exponential function, e^z, and the trigonometric functions, such as sin(z) and cos(z), are examples of entire functions, meaning they are analytical over the entire complex plane.
• The logarithm function, log(z), is analytical on the complex plane cut along the negative real axis, which is necessary to make the function single-valued.

Applications

Analytical functions are used in various areas of mathematics and applied sciences, including fluid dynamics, electromagnetic theory, and quantum mechanics. Their properties, such as conformal mapping and the ability to represent functions as power series, make them invaluable tools in both theoretical and applied research.