Analytic Function

We all know what a function is in math and we also know what its types are but we might not know what an analytic function is? So, here we are on our way to knowing 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 at every point in their domain. So what 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 another value in a domain which converges the series to one point.

Types of Analytic Functions

Analytic Functions can be classified into two different categories. These categories have different distinguishing properties but are similar in some ways. The two types of analytic functions are:

• Real Analytic Function

• Complex Analytic Function

Real Analytic Function

Any function can be referred to as a real analytic function on the open set C in the real line provided that it fulfills the following condition:

• for any x0 ∈ C, then we can write that the coefficients a0, a1, a2, … are the real numbers. Moreover, the series should be convergent to the function f(x) for x in the neighborhood of x0.

This means that any real analytic function is an infinitely differentiable function and the collection of all the real analytic functions on a given set C can be represented by Cω (C).

Complex Analytic Function

A function is said to be a complex analytic function if and only if it is holomorphic which implies that the function should be complex and differentiable.

Conditions that Make a Complex Function Analytic

Let us look at what makes complex functions analytic:

• Let us assume that f(x, y) = u(x, y) + iv(x, y) is  a complex function. Since \[x = \frac{(z + z)}{2}\] and \[y = \frac{(z − z)}{2i}\], substituting for x and y ends up yielding f(z, z) = u(x, y) + iv(x, y).

• f(z, z) is analytic if \[\frac{∂f}{∂z}\] = 0

• For f = u + iv to be analytic, f should depend only on z. In terms of the real and imaginary parts u, v off is equivalent to \[\frac{∂u}{∂x} = \frac{∂v}{∂y}\]. Thus, \[\frac{∂u}{∂y} = − \frac{∂v}{∂x}\]

These are known as the Cauchy-Riemann equations. They are a requisite condition for f = u + iv to be termed analytic.  If f(z) = u(x,y) + iv(x,y) is analytic in a region R of the z-plane then, we can infer that:

• ux, uy, vx , vy exist 

• ux = vy and uy = -vx at every point in this given region.

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 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) is 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, \[\frac{f(z)−f(z_0)}{z – z_0}\] 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 \[W (C, z_0) = f(z_0) = \frac{1}{2\pi i} \int C \frac{f(z)−f(z_0)}{z – z_0} dz\], where W(C, z₀) is the winding number of C around z.

Solved Examples

Question 1: Explain why the function f(z)=2z2−3−e−z is entire?  

Solution 1: Proof: Since all polynomials are entire, 2z2−3 is also entire. Since -z and e−z are both entire, their product −ze−z is also entire. Since -z and ez are entire, their composition e−z is also entire. Lastly,f(z) is the sum of 2z2−3, -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 y2 - 2w0y + 1 = 0 contains a complex root y0. Now, we can’t have y0 = 0 since O2 - 2w0 . 0 + 1 ≠ 0. Therefore, y0 ≠ 0 and there is z₀ ∈ C so that ez0  = y0. Therefore,  

\[cosh(z_0) = \frac{e^{z0} + e^{-z0}}{2} = \frac{y_0^2 + 1}{2y_0} = \frac{2w_0y_0}{2y_0} = w_0  \]