Problem 5 - Complex Analysis

Question

Let \(f\) be analytic in an open set \(D\) containing the closed unit disk \(\overline{\mathbb{E}}=\\{z \in\) \(\mathbb{C} ; \quad|z| \leq 1\\} .\) Assume \(|f(z)|<1\) for \(|z|=1\). For any \(n \in \mathbb{N}\) the equation \(f(z)=z^{n}\) has exactly \(n\) solutions in \(\mathbb{E} .\) Especially, \(f\) has exactly one fixed point in \(\mathbb{E}\).

Step-by-Step Solution

Answer

The equation \(f(z)=z^n\) has \(n\) solutions in \(\mathbb{E}\); \(f\) has one fixed point.

Step 1: Understand the Problem Statement

The function \(f\) is analytic in a set \(D\) which includes the closed unit disk. The problem also states that \(|f(z)| < 1\) for all \(|z| = 1\) on the boundary of the unit disk. We want to find the number of solutions for the equation \(f(z) = z^n\) within the open unit disk \(\mathbb{E}\) for \(n \in \mathbb{N}\).

Step 2: Apply Rouche's Theorem

Rouche's Theorem tells us that if two functions, say \(g(z)\) and \(h(z)\), are analytic inside a contour \(C\), and \(|g(z) - h(z)| < |g(z)|\) on \(C\), then \(g(z)\) and \(h(z)\) have the same number of zeros inside \(C\). Here, choose \(g(z) = z^n\) and \(h(z) = -f(z)\). On the boundary \(|z|=1\), we have \(|f(z)| < 1 = |z^n|\). Hence, \(|z^n + f(z)| = |z^n - (-f(z))| = |z^n|\). Therefore, \(f(z)\) and \(z^n\) have the same number of zeros inside \(\mathbb{E}\).

Step 3: Count the Zeros

Since \(z^n\) has exactly \(n\) zeros at \(z=0\) (each with multiplicity 1) inside \(|z| < 1\), by Rouche's Theorem, the function \(f(z) - z^n\) also has \(n\) zeros inside the unit disk. Hence, the equation \(f(z) = z^n\) has \(n\) solutions in \(\mathbb{E}\).

Step 4: Special Case of Fixed Points

A fixed point of \(f\) is a solution to the equation \(f(z) = z\). For \(n=1\), the equation becomes \(f(z) = z^1 = z\). Since we have \(1\) zero as previously determined, \(f\) has exactly one fixed point in \(\mathbb{E}\).

Key Concepts

Analytic Function

An analytic function is a function that is both complex-differentiable and infinitely differentiable in a neighborhood around every point in its domain. This means if you take any point in the domain of an analytic function, you can compute its derivative, and keep doing so infinitely many times. In simple terms, analytic functions behave very nicely everywhere in their domain, allowing them to be expressed as a power series.
Analytic functions are powerful tools in complex analysis. They ensure high precision in calculations and are incredibly useful for conformal mappings and solving differential equations. For the problem at hand, knowing that function \(f\) is analytic guarantees that the properties and rules we apply, including Rouche's Theorem, hold true.

Unit Disk

The unit disk \(\mathbb{E}\) is the set of all points \(z\) in the complex plane such that \(|z| < 1\). Essentially, it includes all points that are "inside" the circle of radius 1 centered at the origin in the complex plane.
In our exercise, the unit disk is the region where we want to find solutions to the equation \(f(z) = z^n\). This open disk is crucial because Rouche's Theorem, which is used to count the number of roots, needs a closed contour like the unit disk boundary \(|z| = 1\) to apply properties about the behavior of analytic functions.

Fixed Point

A fixed point of a function is a point at which the function's value equals the point itself. Simply, for a function \(f\), a fixed point \(z\) is such that \(f(z) = z\). For instance, if \(z_0\) is a fixed point of \(f\), then substituting \(z_0\) into \(f\), you get \(f(z_0) = z_0\).
In this exercise, when \(n = 1\), the equation becomes \(f(z) = z\). We use Rouche's Theorem to determine that \(f\) has exactly one fixed point within the unit disk \(\mathbb{E}\). Finding fixed points is essential for determining the stability of systems and analyzing iterative processes.

Zero of a Function

A zero of a function, also called a root, is a value \(z_0\) for which the function evaluates to zero, i.e., \(f(z_0) = 0\). Zeros are important because they represent where the function crosses the x-axis or mean a value at which the function's output is null.
In our specific case, we were interested in finding zeros for \(f(z) - z^n = 0\), which translates to finding solutions to \(f(z) = z^n\) inside the open unit disk \(\mathbb{E}\). Thanks to Rouche's Theorem, we established that this function has exactly \(n\) zeros in \(\mathbb{E}\), each corresponding to a unique solution to the equation in the given region.