Analytic Function

Analytic Functions are defined as per the converging series; that revolves around a particular 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 the expanded Taylor series is an example of this can be given below.

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

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.

\(T_{f} = \sum_{\infty }^{k=0}\frac{(z-c)^{k}}{2\pi i}\int _{\gamma }\frac{f(w)}{(w-c)^{k+1}}dw\)

\(= \frac{1}{2\pi i}\int _{\gamma }\frac{f(w)}{w-c}\sum_{k=0}^{\infty}\left ( \frac{z-c}{w-c} \right )^{k}dw\)

\(= \frac{1}{2\pi i}\int _{\gamma }\frac{f(w)}{w-c}\left ( \frac{1}{1-\frac{z-c}{w-c}} \right )dw\)

\(= \frac{1}{2\pi i}\int _{\gamma }\frac{f(w)}{w-z}dw=f(z)\)

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.

Let \(f:C\rightarrow C\) be an analytic function. For \(z = x + iy\), let \(u, v: R^{2}\) be such that \(u(x, y) = Re f(z)\) and \(v(x, y) = lm\; f(z)\). Which of the following are correct?

\(\frac{\partial ^{2}u}{\partial ^{2}x}+\frac{\partial ^{2}u}{\partial ^{2}y}=0\)

\(\frac{\partial ^{2}v}{\partial ^{2}x}+\frac{\partial ^{2}v}{\partial ^{2}y}=0\)

\(\frac{\partial ^{2}u}{\partial x \partial y}+\frac{\partial ^{2}u}{\partial y \partial x}=0\)

\(\frac{\partial ^{2}v}{\partial x \partial y}+\frac{\partial ^{2}v}{\partial y \partial x}=0\)

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 so are their sumf(z) + g(z) and their product f(z)g(z).
  • 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, ∫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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