Final answer:
Liouville's theorem states that any bounded entire function must be constant. For instance, the exponential function is entire but not bounded and thus does not satisfy the theorem.
Step-by-step explanation:
The example of Liouville's theorem in complex analysis refers to a significant result, which states that any bounded entire function must be constant. An entire function is a complex function that is holomorphic (complex differentiable) at every point in the complex plane. Liouville's theorem is a powerful tool in complex analysis because it gives us a quick way to determine when a function that is complex differentiable everywhere has to take on constant values.
To provide an example, let's consider the exponential function f(z) = ez. This function is entire because it is differentiable at every point in the complex plane. However, it is not bounded because as the real part of z goes to infinity, the value of ez also goes to infinity. Therefore, the exponential function does not comply with the conditions of Liouville's theorem and hence does not prove the theorem itself. However, if we were to consider a function like f(z) = 1, which is constant, this function is both entire and bounded, and thus it satisfies Liouville's theorem.
Liouville's theorem is often used to prove fundamental results in complex analysis, such as the Fundamental Theorem of Algebra, which states that every non-constant polynomial with complex coefficients has at least one complex root. This theorem uses the concept of bounded entire functions to arrive at a contradiction if a polynomial were to lack roots in the complex plane.