Analytic Functions 

Analytic Functions are defined as per the converging series; that revolves around a particular variable x for which the series has been expanded. Almost all the functions that we obtained from the basic algebraic and arithmetic operations and the elementary transcendental functions are analytic in every point on their domain. In this article, let us discuss what is analytic function, real and complex analytic function, properties in detail.

What is Analytic Function?

Analytic Function is defined as an infinitely differential function, over a variable called x such that the expanded Taylor series can be expressed as given below.

T(x)=∑∞n=0f(n)(x0)n!(x−x0)n

This explains the expanded Taylor  overvalue Xo; this function is said to be an analytic function for the value x in its domain there is another value in a domain which is  that converges the series to one point

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

If a series agrees with Taylor series and has derivative of different order on its every domain point then that series  is said to be real analytic function.

Tf=∑k=0∞(z−c)k2πi∫γf(w)(w−c)k+1dw

=12πi∫γf(w)w−c∑∞k=0(z−cw−c)kdw

=12πi∫γf(w)w−c(11−z−cw−c)dw

=12πi∫γf(w)w−zdw=f(z)

Complex Analytic Function

A function is said to be analytic in the region T of complex plane x if, f(x) has derivative at each and every point of x and f(x) has unique values that are it follows one to one function.’

This example explains the analytic function on the complex plane.

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 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, ∫_C 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)∫_C [f(z)]/[z – z₀]dz, where W(C; z₀) is the winding number of C around z