Problem 33 - A First Course in Complex Analysis with Applications

Question

Consider the function \(f(z)=\frac{1}{(z-5)^{3}}\). What is the Laurent series expansion of \(f\) about \(z_{0}=5\) that is valid on the annulus \(0<|z-5|<\infty\) ?

Step-by-Step Solution

Answer

The Laurent series of \( f(z) \) is \( (z-5)^{-3} \) for \( 0 < |z-5| < \infty \).

Step 1: Identify the Type of Series Required

Given the function \( f(z) = \frac{1}{(z-5)^3} \), we need to find its Laurent series about \( z_0 = 5 \). The function is already in a form that suggests a Laurent expansion, specifically because it has terms that involve negative powers of \( z-5 \).

Step 2: Express Function in Terms of Negative Powers

The function \( f(z) = \frac{1}{(z-5)^3} \) is already expressed in terms suitable for a Laurent series around \( z_0 = 5 \), because it is given by \( (z-5)^{-3} \).

Step 3: Write Out the General Form of a Laurent Series

A Laurent series around \( z_0 \) has the form:\[ f(z) = \sum_{n=-\infty}^{\infty} a_n (z-z_0)^n\]For this function, terms of positive powers do not appear as it is fully expressed with a single term with a negative exponent.

Step 4: Identify the Coefficients of the Laurent Series

For \( f(z) = (z-5)^{-3} \): The function is a single term with \( a_{-3} = 1 \) and \( a_n = 0 \) for all other \( n eq -3 \). Hence, the series is simply:\[ f(z) = 1 \cdot (z-5)^{-3}\]

Step 5: Confirm the Region of Convergence

The problem asks for the Laurent series valid on the annulus \( 0 < |z-5| < \infty \). The series \( (z-5)^{-3} \) naturally converges in this region because it is defined everywhere except at \( z = 5 \).

Key Concepts

Complex Analysis

Complex Analysis is a branch of mathematics that deals with functions of complex numbers. These functions involve a real and an imaginary part, typically represented as \( z = x + iy \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).
Understanding complex analysis requires us to comprehend complex functions, which can often behave in intriguing ways, such as having poles, essential singularities, or being entire (meaning they are differentiable everywhere in the complex plane).
For example, the function \( f(z) = \frac{1}{(z-5)^3} \) examined here illustrates a function with a pole of order 3 at \( z = 5 \). This means the function behaves like \((z-5)^{-3}\) near \( z = 5 \).

• Complex analysis has applications in various fields such as engineering, physics, number theory, and applied mathematics.
• It provides a more comprehensive understanding of differentiable functions.
• Understanding complex planes and the behavior of functions helps in resolving theoretical and practical problems.

Series Expansion

Series expansion is a mathematical technique used to represent a function as an infinite sum of terms. In complex analysis, this can be particularly useful for studying the behavior of complex functions near singular points.
The two main types of series expansions are Taylor and Laurent series. A Taylor series expansion represents a function with only non-negative powers of \( (z-z_0) \), and is used when the function is analytic at \( z_0 \).
A Laurent series, on the other hand, might include negative exponents and is used to represent functions near poles or in annular regions (rings in the complex plane). For instance, the given function \( f(z) = \frac{1}{(z-5)^3} \) can be expanded as a Laurent series with a single negative exponent term: \( (z-5)^{-3} \).

• Laurent series expansion is crucial in handling functions with singularities.
• Provides insights into the nature of singularities (e.g., poles vs. essential singularities).
• Helps in the calculation of residues, which are important for contour integration.

Region of Convergence

The region of convergence for a series is the set of values for which the series converges to a finite value. Understanding this concept is vital when working with series expansions, as it tells us where the expansion accurately represents the function.
For Laurent series, the region of convergence is typically an annulus centered at \( z_0 \), where the function is analyzed.
In the given problem, the series \( (z-5)^{-3} \) converges for \( 0 < |z-5| < \infty \), which means it is valid everywhere in the complex plane except at \( z = 5 \), where the function has a pole.

• Identifying the region of convergence ensures that the Laurent series is correctly applied.
• A correct understanding of these regions is critical in complex integration and residue calculation.
• Misidentifying the region of convergence can lead to incorrect conclusions and results.