Analytic Function

Analytic Function Introduction

We all know what a function is in maths and we also know what are its types but we might not know what an analytic function is? So, here we are on our way to know about the analytic function and everything related to it. Okay, so we can define Analytic Functions as per the converging series; one that twirls around a particular variable x for which the series has been extended. Almost every function that we obtained from the basic operations in algebraic and arithmetic and the elementary transcendental functions can be referred to as analytic in every point on their domain. So what actually is an analytic function? And what are its properties? Let us learn about them in detail.


Meaning of Analytic Function

Analytic Function is usually defined as an infinite differential function, covering a variable called x in such a way that the extended Taylor series can be represented as given below.

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

This demonstrates the extended Taylor overvalue Xo; therefore, this function can be called an analytic function as the value x in its domain there is actually another value in a domain which is that converges the series to one point.


Properties of Analytic Function

Given below are a few basic properties of analytic functions:

  • The limit of consistently convergent sequences of analytic functions is also an analytic function

  • If f(z) & g(z) are the two analytic functions on U, then the sum of f(z) + g(z) & the product of f(z).g(z) will also be analytic

  • If f(z) & 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)) will also be an analytic function.

  • The function f(z) = 1/z (z≠0) is usually analytic.

  • Bounded entire functions are called constant functions.

  • Every non-constant polynomial p(z) consists of a root. In other words, there exists some z₀ such that p(z₀) = 0.

  • If f(z) is regarded as an analytic function, that is defined on U, then its modulus of the function |f(z)| will not be able to attain its maximum in U.

  • The zeros of an analytic function, say f(z) are basically the isolated points until and unless f(z) is identically zero.

  • If F(z) is an analytic function & if C is a curve that connects the two points z₀ & z₁ in the domain of f(z), then ∫C F’(z) = F(z₁) – F(z₀)

  • If f(z) is an analytic function that is defined on a disk D, then there will be an analytic function F(z) defined on D so that F′(z) = f(z), known as 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] will be analytic on the U tool.

  • If f(z) is regarded as an analytic function on a disk D, z₀ is the point in the interior of D, C is a closed curve that cannot pass 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.


Solved Examples

Question 1) Explain why the function \[f(z) = 2z^{2} - 3 - e^{-z}\] is entire?  


Solution 1) Proof: Since all polynomials are entire, \[2z^{2} - 3\] is also entire. Since -z and \[e^{z}\] are both entire, their product \[-ze^{z}\] is also entire. Since -z and \[e^{z}\] are entire, their composition \[e^{-z}\] is also entire. Lastly,f(z) is the sum of \[2z^{3} - 3, -ze^{z}\] and \[e^{-z}\] are entire. 


Question 2) Show that the entire function cosh (z) takes each value in C infinitely many times.


Solution 2) Proof: For each w₀ ∈ C, the quadratic equation \[y^{2} - 2w_{0}y + 1 = 0\] contains a complex root \[y_{0}\]. Now, we can’t have \[y_{0} = 0\] since \[o^{2} - 2w_{0} . 0 + 1 \neq 0\]. Therefore, \[y_{0} \neq 0\] and there is z₀ ∈ C so that \[e^{z_{0}} = y_{0}\].


Therefore, 

\[cosh(z_{0}) = \frac{e^{z_{0}} + e^{-z_{0}}}{2} = \frac{y_{0}^{2} + 1}{2y_{0}} = \frac{2w_{0}y_{0}}{2y_{0}} = w_{0}\]

FAQ (Frequently Asked Questions)

Question 1) What is a Real Analytic Function?

Answer 1) A real analytic function can be called a function if a series matches with the Taylor series and also has a derivative of different order on each of its domain points. 


Tf = Σk = 0 [(z - c)k/2πi] ∫_γ [f(w)/(w - c)k + 1] dw


= 1/2πi ∫γ [f(w)/(w - c) Σk = 0 [{(z - c)/(w - c)}k] dw


= 1/2πi ∫γ [f(w)/(w - c) {1/1- [(z-c)/(w - c)]} dw


= 1/2πi ∫γ [f(w)/(w - z) dw = f(z)

Question 2) What is a Complex Analytic Function?

Answer 2) A complex function is referred to as analytic in the area T of complex plane x if, f(x) holds a derivative at each and every point of x. Also, f(x) has some unique values that it follows one to one function.


Here is an example that explains the analytic function on the complex plane.

let f : C ➝ C be the analytic function. For z = x + iy, let u, v : R² be such that u(x, y) = Ref(z) and v(x, y) = lm f(z). Which of the following are correct?


∂²u/∂²x + ∂²u/∂²y = 0


∂²v/∂²x + ∂²v/∂²y = 0


∂²u/∂x∂y + ∂²u/∂y∂x = 0


∂²v/∂x∂y + ∂²v/∂y∂x = 0