Problem 20 - A First Course in Complex Analysis with Applications

Question

Expand \(f(z)=\frac{z}{(z+1)(z-2)}\) in a Laurent series valid for the given annular domain. $$ 0<|z-2|<3 $$

Step-by-Step Solution

Answer

Laurent series: \( \frac{1}{3(z-2)} + \frac{2}{9} \sum_{n=0}^{\infty} \left(-\frac{z-2}{3}\right)^n \).

Step 1: Identifying the region of convergence

The annular domain given is \(0 < |z-2| < 3\). This means the Laurent series should be expressed such that it is valid between the circles \(|z-2|=0\) and \(|z-2|=3\).

Step 2: Express the function in terms of basic fractions

Express \(f(z)=\frac{z}{(z+1)(z-2)}\) in terms of partial fractions: \[ f(z) = \frac{A}{z+1} + \frac{B}{z-2} \]By finding residues at the poles, we solve for constants \(A\) and \(B\). Substitute \(z = -1\) and \(z = 2\) into the equation separately to find \(A\) and \(B\).

Step 3: Calculate partial fraction coefficients

For the residue at \(z=2\), set: \( \frac{z}{(z+1)(z-2)} = \frac{A}{z+1} + \frac{B}{z-2} \). Multiply through by \((z+1)(z-2)\) and set \(z = 2\) to solve for \(A\): \[ A = \frac{2}{2+1} = \frac{2}{3} \].Similarly for \(B\) by setting \(z=-1\):\[ B = \frac{-1}{-1-2} = \frac{1}{3} \].

Step 4: Rewrite function using partial fractions

Substitute back into the partial fractions: \[ f(z) = \frac{2/3}{z+1} + \frac{1/3}{z-2}. \]

Step 5: Expand each term as applicable for given domain

For \(\frac{1/3}{z-2}\) which is naturally valid for \(|z-2| > 0\), no change is needed:\[ \frac{1}{3} \cdot \frac{1}{z-2}. \]For \(\frac{2/3}{z+1}\), we need a series valid for \(|z-2|<3\) where \(z \approx 2\). Rewrite \(\frac{1}{z+1} = \frac{1}{(z-2)+3}\) and use geometric series:\[ \frac{1}{3} \sum_{n=0}^{\infty} \left(\frac{-(z-2)}{3}\right)^n = \frac{2}{3 \times 3}\sum_{n=0}^{\infty}\left(-\frac{z-2}{3}\right)^n. \]

Step 6: Combine series and simplify

Combine the series for all the terms, yielding the Laurent series valid for the domain:\[ f(z) = \frac{1}{3(z-2)} + \frac{2}{9} \sum_{n=0}^{\infty} \left(-\frac{z-2}{3}\right)^n. \]

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a useful method in complex analysis and calculus for breaking down rational functions into simpler fractions. This technique helps in evaluating integrals and in finding series expansions, like the Laurent series. When dealing with a rational function such as \( f(z) = \frac{z}{(z+1)(z-2)} \), we can decompose it into partial fractions. The goal is to express the original function as:

• \( \frac{A}{z+1} + \frac{B}{z-2} \)

To find the constants \(A\) and \(B\), known as residues, we substitute specific values for \(z\) that simplify the expressions. In this case:

• Substituting \(z = -1\) helps to find \(B\),
• Substituting \(z = 2\) helps to find \(A\).

Through these substitutions and solving equations, values are determined as \(A = \frac{2}{3}\) and \(B = \frac{1}{3}\).
Partial fraction decomposition is especially handy for analyzing the behavior of functions and finding more manageable forms suitable for specific mathematical operations.

Annular Domain

An annular domain refers to a ring-shaped region in the complex plane. It is defined by two concentric circles and is expressed in a constraint like \(0 < |z-2| < 3\). This means the region of interest is bounded by the circle of radius 0, excluding the center, and extends to the circle of radius 3, centered at \(z = 2\).

• An annular domain allows for covering regions where a function is not analytic at the center circle but can be expressed in a series within the bounds set by both circles.
• In the context of expanding into a Laurent series, functions are expanded into series which can contain terms of both positive and negative powers of \(z\).

Being aware of where each term in the series converges is essential, as it determines the validity and usefulness of the expansion in different parts of the complex plane. For example, understanding the domain \(0 < |z-2| < 3\) helps ensure the resulting Laurent series faithfully represents the function within that specific region.

Complex Analysis

Complex analysis is a profound area of mathematics focusing on functions of complex numbers. It harnesses concepts like differentiation and integration in the setting of the complex plane. One of the powerful tools in complex analysis is the Laurent series, which generalizes Taylor series:

• Laurent series allows functions with singularities, such as poles, to be expressed as an infinite series.
• Unlike Taylor series that only handles regular points in a function, Laurent series can express a function in terms of positive and negative powers of \(z\).

This capability makes the Laurent series valuable for handling functions with certain types of singularities inside an annular domain. In the process of solving, the partial fraction decomposition helps break down functions, and the series expansion adapts being around a specific singular point, maximizing the utility of functions in either analytic or non-analytic forms.
Complex analysis provides the theoretical foundation that makes handling these advanced concepts feasible, enabling deeper insights into not only mathematics but also their applications across physics and engineering.