Problem 10 - Complex Analysis for Mathematics and Engineering

Question

Let \(f\) be an entire function with the property that \(|f(z)| \geq 1\) for all \(z\). Show that \(f\) is constant.

Step-by-Step Solution

Answer

By Liouville's Theorem, \(f(z)\) is constant because \(\frac{1}{f(z)}\) is a bounded entire function.

Step 1: State Liouville's Theorem

Recall that Liouville's Theorem states that any bounded entire function is constant. A function is considered entire if it is holomorphic (complex differentiable) at all points in the complex plane.

Step 2: Analyze the given condition

The function \(f\) is given as an entire function, and it satisfies the inequality \(|f(z)| \geq 1\) for all complex numbers \(z\). This means \(f(z)\) never takes a value less than 1 in magnitude.

Step 3: Modify the condition to use Liouville's Theorem

Consider the function \(g(z) = \frac{1}{f(z)}\). Since \(|f(z)| \geq 1\), it follows that \(|g(z)| = \left|\frac{1}{f(z)}\right| \leq 1\). The function \(g(z)\) is well defined and entire because \(f(z)\) never equals zero.

Step 4: Apply Liouville's Theorem to \(g(z)\)

Since \(g(z)\) is bounded by 1 and entire, by Liouville's Theorem, \(g(z)\) must be constant.

Step 5: Conclude \(f(z)\) is constant

If \(g(z) = \frac{1}{f(z)}\) is constant, then \(f(z)\) must also be constant, specifically a non-zero constant, because otherwise \(g(z)\) would not be defined everywhere on the complex plane.

Key Concepts

Entire Function

An entire function is a special type of complex function that is defined and complex differentiable everywhere in the complex plane. Being complex differentiable, also known as holomorphic, is somewhat akin to differentiability for real functions but in a complex setting. In essence, it means that the function has a derivative at every point in the complex plane.

• An entire function is smooth with no jumps or corners anywhere on the complex plane.
• Examples of entire functions include polynomials, the exponential function, and the sine and cosine functions.

Entire functions are important in complex analysis because their properties are well-behaved and allow us to use powerful theorems like Liouville's Theorem. Since the function is holomorphic everywhere, we can study its global properties, paving the way for understanding behavior and constraints across the entire plane.

Bounded Function

A bounded function in the context of complex analysis is one where the magnitude of the function's output remains within some fixed value, no matter what input is used. More formally, a function \(f(z)\) is bounded if there exists a positive real number \(M\) such that for all complex numbers \(z\), \(|f(z)| \leq M\).
This definition leads us to an important result in complex analysis, namely Liouville's Theorem. Liouville's Theorem states that if a function is both entire and bounded, it must be a constant function. This is because if an entire function's output can never grow beyond a set value, the only possibility is that it does not grow at all, meaning it stays the same for all \(z\).

• Examples of bounded functions include constant functions because their magnitude is always the same.
• When applied to complex functions, boundedness in conjunction with being entire is a strong condition.

This strong condition helps in demonstrating why some entire functions, like \(g(z) = \frac{1}{f(z)}\), must be constant, as seen in applying Liouville's Theorem.

Complex Differentiable

Complex differentiability is a core concept in complex analysis and is essential for understanding entire functions. A complex function \(f(z)\) is said to be complex differentiable at a point \(z_0\) if the limit:\[ \lim_{\Delta z \to 0} \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z} \]exists.
This limit, if it exists, is called the derivative of the function at that point. It is similar to the derivative concept in real analysis, but it considers complex inputs and outputs.
The condition of being complex differentiable at each point in the complex plane is what defines an entire function. This means your function doesn't encounter sudden changes or have undefined points in terms of derivatives.

• Complex differentiability is a stricter condition than real differentiability, providing more structure across the plane.
• It is what allows powerful results and theorems, like Liouville's, to hold true in the realm of complex numbers.

With complex differentiability, the behavior and properties of a function can be studied thoroughly using the theorems available in complex analysis.