Introduction
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,
\[cosh(z_0) = \frac{e^{z0} + e^{-z0}}{2} = \frac{y_0^2 + 1}{2y_0} = \frac{2w_0y_0}{2y_0} = w_0 \]
FAQs on Analytic Function
1. What is an analytic function in mathematics?
An analytic function is a function that can be locally described by a convergent power series. This means that for any point in its domain, the function's value can be represented by a Taylor series expanded around that point. A key characteristic of analytic functions is that they are infinitely differentiable, meaning you can calculate their derivatives of any order.
2. What are the two main types of analytic functions?
Analytic functions are primarily classified into two types based on their domain:
- Real Analytic Function: A function defined on the real number line that can be represented by a convergent Taylor series at every point in its domain.
- Complex Analytic Function: A function defined on the complex plane. A complex function is analytic if and only if it is holomorphic, meaning it is complex-differentiable in its domain.
3. How can you determine if a complex function is analytic?
A complex function f(z) = u(x, y) + iv(x, y) is determined to be analytic in a region if its real part (u) and imaginary part (v) have continuous first partial derivatives that satisfy the Cauchy-Riemann equations. These equations are a necessary condition for analyticity:
- ∂u/∂x = ∂v/∂y
- ∂u/∂y = -∂v/∂x
If these conditions hold throughout a region, the function is analytic in that region.
4. What are the key properties of analytic functions?
Analytic functions have several important properties that make them central to complex analysis. Some key properties include:
- The sum and product of two analytic functions are also analytic.
- The composition of two analytic functions (g(f(z))) is analytic.
- If a function is analytic, it is infinitely differentiable.
- The limit of a uniformly convergent sequence of analytic functions is also an analytic function.
- An analytic function is constant if its absolute value has a maximum within its domain (Maximum Modulus Principle).
5. Can you give an example of a function that is analytic and one that is not?
Certainly. Here are examples of both:
- Analytic Function Example: Any polynomial function, such as f(z) = z² + 2z - 1, is analytic everywhere in the complex plane. The exponential function, f(z) = ez, is also analytic everywhere. These are also known as 'entire functions'.
- Non-Analytic Function Example: The function f(z) = z̄ (the complex conjugate of z) is not analytic anywhere because it fails to satisfy the Cauchy-Riemann equations. Another common example is f(z) = |z|², which is differentiable only at z = 0 and is therefore not analytic.
6. What is the main difference between a real analytic function and a complex analytic function?
The main difference lies in the conditions they must satisfy. A real analytic function is defined on the real line and must be representable by a Taylor series. In contrast, a complex analytic function is defined on the complex plane, and its analyticity is linked to the much stricter condition of being complex-differentiable everywhere in its domain. This condition, encapsulated by the Cauchy-Riemann equations, implies that if a complex function is differentiable once, it is infinitely differentiable—a property not always true for real functions.
7. Why are the Cauchy-Riemann equations a necessary condition for a function to be analytic?
The Cauchy-Riemann equations are essential because the derivative of a complex function at a point must be unique, regardless of the path of approach to that point. In the complex plane, you can approach a point from infinitely many directions (e.g., along the real axis, the imaginary axis, or any diagonal path). The Cauchy-Riemann equations ensure that the value of the derivative remains the same for all these paths. If they are not satisfied, the limit defining the derivative would yield different results for different paths, meaning a unique derivative does not exist and the function cannot be analytic.
8. How are analytic functions used in real-world applications like physics or engineering?
Analytic functions are fundamental tools in many scientific and engineering fields due to their well-behaved properties. Key applications include:
- Fluid Dynamics: Modeling two-dimensional potential flow, such as the flow of air over an airplane wing.
- Electromagnetism: Solving 2D problems in electrostatics and magnetostatics, like finding the electric potential in a region.
- Control Theory: Analyzing the stability and response of systems using techniques like the Laplace transform, which is rooted in complex analysis.
- Signal Processing: Using Fourier analysis to decompose signals into constituent frequencies, where complex functions play a crucial role.
9. What does it mean for a function to be 'infinitely differentiable,' and why is this important for analyticity?
A function is 'infinitely differentiable' if you can take its derivative an infinite number of times, and a valid derivative always exists. For complex functions, this is a remarkable consequence of being analytic. If a complex function has just one derivative in a region, it is guaranteed to have derivatives of all orders. This is important because the existence of all these derivatives is precisely what allows the function to be perfectly represented by a Taylor series, which is the formal definition of an analytic function.
10. Are all analytic functions also holomorphic?
Yes, in the context of complex functions, the terms analytic and holomorphic are effectively synonymous. A function is defined as 'holomorphic' if it is complex-differentiable in a neighbourhood of every point in its domain. A fundamental theorem in complex analysis proves that any holomorphic function is also analytic (can be expressed as a power series), and conversely, any analytic function is holomorphic. Therefore, the two terms are used interchangeably.
