Liouville’s Theorem

Liouville’s Theorem implies that every entire bounded function must be constant. This theorem of complex analysis was given by a French mathematician Joseph Liouville. We can also say that according to Liouville’s theorem, “An entire (that is, holomorphic in the whole complex plane C) function cannot be bounded if it is not constant.”

In this article, applications of Liouville’s Theorem as well as its importance in complex analysis.

What is Liouville’s Theorem?

Liouville’s Theorem is a concept of complex analysis that tells us that if a function is bounded, it must be a constant. This theorem focuses on various kinds of functions. It tells us about the distinct properties of certain functions. A mathematician named Joseph Liouville gave this theorem in 1847. But it is also believed that this theorem was already proved in 1844 by another mathematician named Cauchy.

There are various statements of Liouville’s Theorem and the basic one is,

“A bounded entire function is a constant function.”

Statement of Liouville’s Theorem

Liouville’s Theorem states that,

Every bounded entire function must be a constant function.

To understand its statement profoundly, Let’s take a simple example:

Suppose a function (f) is an entire function.

The given inequality is |f(x)| ≤ P, where P is a constant.

Now, for all values of ‘x’, if the given function satisfies this inequality in a plane C, f is a constant function.

Proof of Liouville’s Theorem

Let’s Proof Liouville’s Theorem:

Consider ƒ an entire function in a plane C.

Suppose a and b are arbitrary points.

Let f’= 0. Then ƒ is a constant function.

To prove that f is a constant function, we need to show that f(a) = f(b) for all a, b ∈ C. Since C is a plane, which is path connected. Then we can choose a curve F: I➝C so that F(0) = a and F(1) = b.

Now,

∫_xf'(x) dx = f(b)- f(a)

Since f’ = 0, f(b) = f(a)

Now,

Suppose |f(x)| ≤ P for all x ∈ C.

To prove ƒ is a constant, we only need to prove f'(z) = 0. Let a ∈ C.

Now, by using Cauchy-Integral Formula, rest of arithmetic operations can be performed.

In this way, you can prove the Liouville’s Theorem.

Corollaries of Liouville’s Theorem

The corollaries of Liouville’s theorem are mentioned below:

• Boundedness of an entire function always depends on its constancy. In simple words, if an entire function is not constant, it can never be a bounded function.

• If f is an entire function that is not constant, the set of values it takes spreads out densely across the complex plane C.

Applications of Liouville’s Theorem

Liouville’s Theorem has many applications across various mathematical concepts, especially in complex analysis. Some of its applications are discussed below:

• Holomorphic Functions: Liouville’s Theorem helps us to understand the constraints on holomorphic functions regarding their boundedness.

• Differential Equations: Liouville’s Theorem is important in determining the properties of a differential equation within an entire function.

• Function Theory: Liouville’s Theorem helps to analyze different functions based on their boundedness.

• Number Theory: Liouville’s Theorem plays an important role in number theory, especially in analyzing the behaviors and properties of transcendental numbers.

Liouville’s Theorem and Fundamental Theorem of Algebra

The differences between Liouville’s Theorem and Fundamental Theorem of Algebra is added in the table below,

Difference Between Liouville’s Theorem and Fundamental Theorem of Algebra
Liouville’s Theorem	Fundamental Theorem of Algebra
Focuses on an entire function in complex analysis.	Focuses on various characteristics of polynomial functions.
Entire bounded functions must be constant.	Roots of non-constant polynomials must be in the complex plane.
Refers to functions defined on the entire complex plane	Specifically, it belongs to the polynomial equations in 1 variable
Restricts the behavior of entire functions	Define the existence of roots for non-constant polynomials
Calls out the behavior related to boundedness	Determines the roots and factorization of polynomials

Read More,

• Mean Value Theorem
• Rolle’s Theorem and Lagrange’s Mean Value Theorem

Examples on Liouville’s Theorem

Some examples verifying Liouville’s Theorem are,

Example 1: Determine whether the given function f(z) = sinz satisfies the conditions of Liouville’s Theorem.

Solution:

Given,

f(z) = sinz

We need to prove whether the function is satisfying all the conditions of Liouville’s Theorem or not.

Liouville’s Theorem has 2 important conditions: 1) Entire Function, 2) Boundedness

• In every complex plane, the sine function is holomorphic therefore the function f(z)= sinz is an entire function.

• In function f(z) = sinz, the value of z varies between -1 and 1. Therefore the sin function is not entirely bounded.

Hence, the function f(z)= sinz does not satisfy both the conditions of Liouville’s Theorem.

Example 2: Prove that the function f(z)= cosz is constant using Liouville’s Theorem.

Solution:

Given,

f(z) = cos z

Let’s understand whether the given function is meeting the criteria or not:

• Function f(z)= cosz is an analytic function and is holomorphic in the entire plane. Hence, it is evident that cos function is an entire function.

• Value of cos function lies between -1 and 1 i.e. -1 ≤ cos(z) ≤ 1. Function cos z always bounds within this range. Therefore, the function f(z)= cosz is entirely bounded in the complex plane.

Hence, it is proved that the function f(z)= cosz is a constant function.

Practice Questions on Liouville’s Theorem

Q1. Prove that the function f(z)= e^z is constant using Liouville’s Theorem.

Q2. Consider a function f(z) ≤ 1 in a complex plane. Determine whether f is entirely bounded or not.

Q3. Consider a function f(z) ≤ 3e^z in a complex plane. Determine whether f is entirely bounded or not.

Frequently Asked Questions on Liouville’s Theorem

What is Liouville’s Theorem in Complex Analysis?

Liouville’s theorem in complex ananlysis states that, “a bounded entire function is a constant function.”

What are Requirements of Liouville’s Theorem?

A function that satisfies Liouville’s Theorem must follow the condition that the function has to be an entire function as well as bounded for all values of z in z-plane.

Can Liouville’s Theorem only be applied to complex numbers?

Yes, Liouville’s Theorem is particularly based on complex analysis and entire functions.

Is Liouville’s Theorem Used for Real-World Applications?

Yes, its applications expand to various fields like physics, engineering, economics etc.

Liouville’s Theorem is named after which Mathematician?

Joseph Liouville a famous french mathematiciam is the one after which Liouville’s theorem is named.