Proof of Liouville’s theorem in Complex Analysis

Liouville’s Theorem:

The Liouville’s theorem in Complex Analysis can be considered as an improvement of Picard's little theorem. This theorem states that every bounded and entire function is constant. The theorem is can be interpreted as: every entire function whose image omits two or more complex numbers must be constant. In this article the statement and proof of the theorem is explained.

Statement of the theorem:

If \(f\) is entire and bounded in the complex plane, then \(f(z)\)is constant throughout the plane.

Proof of the theorem:

Cauchy’s inequality states that a function  \(f\) is analytic within and on a circle\(\left| {z - {z_0}} \right| = R\), taken in the positive sense, then

\(\left| {{f^n}({z_0})} \right| \le \frac{{n!{M_R}}}{R}\)

Where, \(n = 0,1,2,...\) and \({M_R}\)be the maximum value of \(f(z)\)in the circle\(\left| {z - {z_0}} \right| = R\).

Here, given that, \(f\) is entire and bounded. Therefore, \(\left| {f(z)} \right| \le M\) for some \(M\)and taking \(n = 1\)in Cauchy’s inequality we get

\(\left| {f'({z_0})} \right| \le \frac{M}{R}\)

Where \({z_0}\)is any fixed point in the plane and \(R\)can be arbitrarily large. In this inequality, \(M\)is independent of the value of \(R\). Therefore, \(R\)can be taken arbitrarily large only when\(\left| {f'({z_0})} \right| = 0\). Also, the choice of \({z_0}\)is arbitrary, which means \(f'(z) = 0\)everywhere in the complex plane. Which implies that \(f(z)\)is constant throughout the complex plane.

This proves the theorem.

The proof of Liouville’s theorem is a simple one, but its applicability in Complex Analysis is very vast. Mostly, this theorem is used in questions where we need to check whether a function is constant or not.