Problem 10 - A First Course in Complex Analysis with Applications

Question

Expand \(f(z)=\frac{1}{z(z-3)}\) in a Laurent series valid for the indicated annular domain. $$ |z-3|>3 $$

Step-by-Step Solution

Answer

The Laurent series is \( \sum_{n=0}^{\infty} \frac{3^n}{z^{n+1}} \) for \(|z-3| > 3\).

Step 1: Analyze the Domain

The problem requires us to find the Laurent series of the function \( f(z) = \frac{1}{z(z-3)} \) valid for the region \( |z - 3| > 3 \). This region implies that the series must converge outside the circle of radius 3 centered at \( z = 3 \).

Step 2: Rewrite the Function

Rewriting \( f(z) \), we have \( f(z) = \frac{1}{z(z-3)} = \frac{1}{z} \cdot \frac{1}{z-3} \). For \( |z-3| > 3 \), it is useful to express \( \frac{1}{z-3} \) using the geometric series expansion.

Step 3: Express Using Geometric Series

We rewrite \( \frac{1}{z-3} \) as \( \frac{1}{z} \cdot \frac{1}{1 - \frac{3}{z}} \) which we can expand using the geometric series \( \frac{1}{1-w} = \sum_{n=0}^{\infty} w^n \) valid for \( |w| < 1 \). Here, \( w = \frac{3}{z} \) implies \( \left| \frac{3}{z} \right| < 1 \). This is satisfied in the domain \( |z| > 3 \).

Step 4: Obtain the Series

Substituting the series expansion, we get \[ \frac{1}{z(z-3)} = \frac{1}{z} \cdot \sum_{n=0}^{\infty} \left( \frac{3}{z} \right)^n = \sum_{n=0}^{\infty} \frac{3^n}{z^{n+1}}. \] This is the Laurent series representation of \( f(z) \) for \( |z-3| > 3 \).

Step 5: Validate Convergence

The series \( \sum_{n=0}^{\infty} \frac{3^n}{z^{n+1}} \) converges in the region \( |z| > 3 \), as required by the problem, therefore, the derived Laurent series is valid for the given domain.

Key Concepts

Complex Analysis

Complex analysis is a branch of mathematics that focuses on functions of complex numbers. It is essential for understanding various aspects of analysis, especially when dealing with complex domains and functions. Complex analysis combines several core principles:

• Complex Numbers: Functions are defined using complex numbers, which include a real part and an imaginary part.
• Analytic Functions: These are functions defined by a power series that converges to the function in a given domain.
• Singularities: Points at which a function is not defined or not well-behaved.
• Convergence: Refers to the behavior of a series or function as its input or index approaches a limit.

In our problem, we are asked to find the Laurent series for a function across a specific region, which involves understanding and applying the concepts of complex analysis to derive the function behavior over a complex domain. This requires a solid understanding of regions within the complex plane, the domain of convergence, and series expansion.

Geometric Series

The geometric series is a fundamental concept in mathematics, providing a simple way to represent a function as an infinite sum. In its standard form, a geometric series is written as:\[rac{1}{1-w} = \\sum_{n=0}^{\infty} w^n,\]valid for \(|w| < 1\). This series is crucial for expanding functions in cases where direct computation is challenging.
In the exercise, the geometric series is used to rewrite the function \(\frac{1}{z-3}\) as a series. Here, the term \(w = \frac{3}{z}\) is used to ensure convergence over the specified domain \(|z| > 3\). By expressing the function in terms of the geometric series, we simplify the process of obtaining the Laurent series, making the solution more manageable and understandable.

Series Expansion

Series expansion is a method to express a function as an infinite sum of terms. This technique is particularly useful in complex analysis to tackle complex functions and singular points in the complex plane.

• Taylor Series: It is used for functions that are analytic at a point, providing expansion around that point.
• Laurent Series: This is a generalized form of Taylor series that allows for negative powers, crucial for dealing with singularities.

In our context, the Laurent series is employed to find the representation of the function \(f(z)\) suitable for the defined region \(|z-3| > 3\). By breaking down the function into simpler components and employing series expansion, we derive a formula that effectively describes the function's behavior in the given domain. Through expansion, complex calculations transform into manageable sums over simpler terms.

Analytic Functions

Analytic functions are vital in complex analysis, defined by their ability to be written as convergent power series within their domain of definition. For a function to be analytic, it must satisfy certain conditions over a complex region:

• Power Series Representation: The function can be expressed as a series that converges to the function.
• Infinitely Differentiable: The function must be smoothly varying and differentiable infinitely many times.
• Rectifiable Boundary: The boundary condition of the analytic region allows for accurate series representation.

In our problem, we explored the analyticity of \(f(z) = \frac{1}{z(z-3)}\) in an annular domain \(|z-3| > 3\). By constructing a Laurent series, we show the region where the function remains analytic and converges as a series. Understanding and identifying regions where a function is analytic allows us to utilize techniques like series expansion to study these functions closely.