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. The most basic statement of Liouville’s Theorem 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.

Liouville’s Theorem Formula

Liouville’s Theorem can be expressed mathematically as follows:

Let Ω(t) be the phase space volume enclosed by a region in phase space at time t. Then, for any such region, the volume Ω(t) satisfies

dΩ/dt = 0

Read More:

• Calculus in Maths
• Integral Calculus
• Differential Calculus

Liouville’s Theorem implies that every entire bounded function must be constant. In other words, 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.” This theorem of complex analysis was given by a French mathematician, Joseph Liouville.

In this article, we discuss what Liouville’s theorem is, its application, and its importance in complex analysis.