Analytic Function

In Mathematics, Analytic Functions is defined as a function that is locally given by the convergent power series. The analytic function is classified into two different types, such as real analytic function and complex analytic function. Both the real and complex analytic functions are infinitely differentiable. Generally, the complex analytic function holds some properties that do not generally hold for real analytic function. In this article, we are going to learn the definition of an analytic function, real and complex analytic functions and some properties in detail.

What is Analytic Function?

A function is said to be an analytic function if and only if its Taylor series about x0 converges to the function in some neighbourhood for every x0 in its domain.

Types of Analytic Function

Analytic Functions can be categorised into two different types, which are similar in some ways, but it has some different characteristics. The two types of analytic functions are:

  • Real Analytic Function
  • Complex Analytic Function

Real Analytic Function

A function “f” is said to be a real analytic function on the open set D in the real line if for any x0 ∈ D, then we can write:

\(f(x)=\sum_{n=0}^{\infty }a_{n}(x-x_{0})^{n}=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+a_{3}(x-x_{0})^{3}+…\)

 

where the coeffienets a0, a1, a2, … are the real numbers and also the series is convergent to the function f(x) for x in the neighbourhood of x0.

In other words, the real analytic function is defined as an infinitely differentiable function, such that the Taylor series at any point x0 in its domain converges to the function f(x) for x in a neighbourhood of x0 pointwise.

\(T(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!}(x-x_{0})^{n}\)

The collection of all the real analytic function on a given set D is represented by Cω (D).

Complex Analytic Function

A function is said to be a complex analytic function if and only if it is holomorphic. It means that the function is complex differentiable.

Properties of Analytic Function

The basic properties of analytic functions are as follows:

  • The limit of a uniformly convergent sequence of analytic functions is also an analytic function
  • If f(z) and g(z) are analytic functions on U, then their sum f(z) + g(z) and product f(z).g(z) are also analytic
  • If f(z) and g(z) are the two analytic functions and f(z) is in the domain of g for all z, then their composite g(f(z)) is also an analytic function.
  • The function f(z) = 1/z (z≠0) is analytic
  • Bounded entire functions are constant functions
  • Every nonconstant polynomial p(z) has a root. That is, there exists some z0 such that p(z0) = 0.
  • If f(z) is an analytic function, which is defined on U, then its modulus of the function |f(z)| cannot attains its maximum in U.
  • The zeros of an analytic function, say f(z) are the isolated points unless f(z) is identically zero
  • If F(z) is an analytic function and if C is a curve connecting two points z0 and z1 in the domain of f(z), then ∫C F’(z) = F(z1) – F(z0)
  • If f(z) is an analytic function defined on a disk D, then there is an analytic function F(z) defined on D such that F′(z) = f(z), called a primitive of f(z), and, as a consequence, ∫f(z) dz =0; for any closed curve C in D.
  • If f(z) is an analytic function and if z0 is any point in the domain U of f(z), then the function, [f(z)-f(z0)]/[z – z0] is analytic on U as well.
  • If f(z) is an analytic function on a disk D, z0 is a point in the interior of D, C is a closed curve not passing through z0, then W = (C, z0)f(z0) = (1/2π i)∫[f(z)]/[z – z0]dz, where W(C; z0) is the winding number of C around z

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