Problem 12 - Complex Analysis

Question

If \(D \subset \mathbb{C}\) is a bounded domain, and \(f: \bar{D} \rightarrow \mathbb{C}\) is a continuous function on the closure of \(D\) which is analytic on \(D\), then \(|f|\) takes its maximum on the boundary of \(D\) Using the example of the strip $$ S=\left\\{z \in \mathbb{C} ; \quad|\operatorname{Im} z|<\frac{\pi}{2}\right\\} $$ and the function \(f(z)=\exp (\exp (z))\) show the necessity of the boundedness of \(D\)

Step-by-Step Solution

Answer

The example shows that boundedness is necessary for the Maximum Modulus Principle as \(|f(z)|\) can become unbounded in unbounded domains like \(S\).

Step 1: Understand the Maximum Modulus Principle

The Maximum Modulus Principle states that if a function is analytic and non-constant on a bounded domain, the maximum modulus of the function occurs on the boundary of that domain. This exercise aims to show that if the domain is unbounded, like the strip \( S \), then the function \( |f| \) can achieve its maximum inside the domain.

Step 2: Analyze the Given Strip and Function

The strip \( S = \left\{ z \in \mathbb{C} ; \quad |\operatorname{Im} z| < \frac{\pi}{2} \right\} \) is unbounded in the real direction. The function \( f(z) = \exp (\exp (z)) \) is analytic across the complex plane including \( \bar{S} \). The goal is to show that there could be scenarios where \( |f| \) doesn't achieve its maximum on the boundary of \( D \) due to unboundedness.

Step 3: Evaluate the Behavior of \( f \) on \( S \)

For \( f(z) = \exp(\exp(z)) \), consider the property of the exponential function that it maps large real parts of \( z \) to very large values. Since \( S \) is unbounded in the real part as \( \text{Re}(z) \rightarrow \infty \), \( \exp(z) \rightarrow \infty \) and hence \( f(z) \rightarrow \infty \). Therefore, \(|f(z)|\) can increase indefinitely within \( S \).

Step 4: Show the Necessity of Boundedness

Since \( \text{Re}(z) \) is unbounded in \( S \), \( |f(z)| \) can become arbitrarily large as discussed in Step 3. This violates the condition that the maximum modulus should occur on the boundary of a bounded domain. Hence, this example shows that boundedness of \( D \) is necessary for the Maximum Modulus Principle to hold.

Key Concepts

Bounded Domain

In complex analysis, a bounded domain is an important concept when discussing the behavior of analytic functions. A domain, in this context, refers to an open and connected subset of the complex plane, denoted as \( D \). It is "bounded" if there exists a real number \( M \) such that the distance from the origin to any point in \( D \) is less than \( M \).

• A typical way to visualize this is thinking of a domain that can fit within a large circle in the complex plane.
• The boundaries of such a domain play a key role when applying the Maximum Modulus Principle.

The Maximum Modulus Principle asserts that for a bounded domain, the maximum value of \(|f(z)|\), where \( f \) is an analytic function, occurs on the boundary of the domain. This exercise illustrates why the boundedness of \( D \) is crucial by using a counterexample where the domain is unbounded.

Analytic Functions

Analytic functions are central to complex analysis. These are functions that are differentiable at every point in a domain in the complex plane. Even more interestingly, they are infinitely differentiable and can be represented as power series.

• Analytic functions have powerful properties, like the Maximum Modulus Principle.
• For a function to be analytic everywhere in a domain, it must satisfy Cauchy-Riemann equations and must be continuous.

In this exercise, the function \( f(z) = \exp (\exp (z)) \) is analytic, meaning it behaves nicely across its domain - in this case, even an unbounded strip \( S \). Despite being analytic, the behavior of this function differs depending on whether the domain has boundaries, demonstrating analyticity is not the only determinant for the Maximum Modulus Principle.

Unbounded Domain

An unbounded domain, unlike a bounded one, extends infinitely in at least one direction. This means there is no real number \( M \) limiting the distance of points in the domain from the origin.

• An example of an unbounded domain is the strip \( S = \{ z \in \mathbb{C} \, | \, |\operatorname{Im} z| < \frac{\pi}{2} \} \), which extends infinitely in the real direction.
• Such domains do not have a fixed "boundary" where the function achieves its maximum, as the limit at infinity might surpass any boundary.

The importantly differing outcome in the Maximum Modulus Principle in unbounded domains is that \(|f|\) might achieve values larger than any potential boundary point. In the exercise's example, \( |f(z)| \) increases indefinitely within the strip, which is why the boundedness of \( D \) is essential.