Problem 35 - A First Course in Complex Analysis with Applications

Question

Suppose the analytic function \(f(z)\) has a zero of order \(n\) at \(z=z_{0} .\) Prove that the function \([f(z)]^{m}, m\) a positive integer, has a zero of order \(m n\) at \(z=z_{0}\).

Step-by-Step Solution

Answer

[f(z)]^m has a zero of order mn at z=z_0.

Step 1: Identify the Problem

We need to prove that if an analytic function \(f(z)\) has a zero of order \(n\) at \(z=z_0\), then the function \([f(z)]^m\) has a zero of order \(mn\) at the same point.

Step 2: Express f(z) Near Zero

If \(f(z)\) has a zero of order \(n\) at \(z=z_0\), it can be expressed as \(f(z) = (z-z_0)^n g(z)\), where \(g(z)\) is analytic and \(g(z_0) eq 0\).

Step 3: Substitute into [f(z)]^m

Substitute the expression of \(f(z)\) into \([f(z)]^m\). We have: \([f(z)]^m = [(z-z_0)^n g(z)]^m = (z-z_0)^{mn} [g(z)]^m\).

Step 4: Identify the Order of the Zero

The expression \((z-z_0)^{mn} [g(z)]^m\) demonstrates that \([f(z)]^m\) has a zero of order \(mn\) at \(z=z_0\), since \(g(z_0)\) is non-zero.

Key Concepts

Analytic Functions

In complex analysis, an analytic function, also known as a holomorphic function, is a function that is locally given by a convergent power series. These functions are incredibly important due to their smooth and continuous nature on their domain. They not only have derivatives at every point in their domain but also have derivatives of all orders. Analytic functions can be thought of as an extension of polynomials or Taylor series for complex numbers. A function is defined as analytic on a region if it can be locally expanded into a power series:

• They are smooth on their domain and have derivatives of all orders.
• They can be expressed using power series around any point in their domain.
• Analytic functions include elementary functions like exponentials, logarithms, and polynomials.

Analyticity implies many significant properties, such as the identity theorem, which states that if two analytic functions agree on an infinite set with an accumulation point, they agree everywhere on their domain. This is a powerful concept used to study and understand zero behavior, convergence, and differentiability properties.

Zeros of Functions

Zeros of functions occur when the function value equals zero at a given point. For an analytic function, these zeros are particularly important as they can be tied to many properties of the function. Finding zeros helps in understanding the function's nature and its behavior in certain domains. When talking specifically about zeros, the terms used are:

• Simple Zeros: Where the function value is zero, and around it, the function crosses the x-axis just once.
• Multiple Zeros: Occur when the x-axis is tangent to the function at the point, meaning the zero has more influence over the function's behavior.

Analytic functions characterized by zeros can be expressed in the form of \[ f(z) = (z-z_0)^n \, g(z) \] where \( g(z) \) has no zeros at \( z_0 \) and \(n\) indicates the multiplicity or order of zero. This helps in clearly identifying and understanding the strength and nature of the zero's influence on the function.

Order of a Zero

The order of a zero refers to the number of times the function can be divided by \((z-z_0)\) before getting a non-zero value. This reveals how the function behaves when approaching that point. For a function \( f(z) \) with a zero of order \( n \) at \( z_0 \), we can write: \[ f(z) = (z-z_0)^n \cdot g(z) \] where \( g(z) \) is analytic and \( g(z_0) eq 0 \). Understanding the order of a zero is crucial:

• Higher Order Zeros: Represent points where the derivative and some higher derivatives of the function are also zero.
• Impacts Behavior: The order indicates the behavior of the function at or near that point and affects the function's local topology.

When considering powers of functions, the order of the zero multiplies according to the power, a property used in proofs. For instance, if \( f(z) \) has a zero of order \( n \) at \( z_0 \), then \( [f(z)]^m \) has a zero of order \( m \cdot n \) at \( z_0 \). This understanding is fundamental in exploring deeper topics in complex analysis such as roots and factorization of analytic functions.