Problem 25 - Advanced Engineering Mathematics

Question

Expand \(f(z)=\frac{7 z-3}{z(z-1)}\) in a Laurent series valid for the indicated annular domain. \(0<|z|<1\)

Step-by-Step Solution

Answer

Laurent series is \(f(z) = \frac{-3}{z} - 10 - 10z - 10z^2 - 10z^3 - \ldots\) for \(0 < |z| < 1\).

Step 1: Analyze the Function

The function given is \( f(z) = \frac{7z - 3}{z(z-1)} \). This suggests a partial fraction decomposition to handle the singularities.

Step 2: Perform Partial Fraction Decomposition

Express \( f(z) = \frac{7z - 3}{z(z-1)} \) as partial fractions: \[ \frac{7z-3}{z(z-1)} = \frac{A}{z} + \frac{B}{z-1} \]. Solving for \(A\) and \(B\) involves matching coefficients after rewriting \( 7z - 3 = A(z-1) + Bz \).Solving the equations gives: \(A = -3\), \(B = 10\). So the decomposition is: \[ \frac{7z-3}{z(z-1)} = \frac{-3}{z} + \frac{10}{z-1} \].

Step 3: Expand Each Term in the Series

For the domain \(0 < |z| < 1\), series expansions must reflect the condition where \(|z| < 1\). - The term \(\frac{-3}{z}\) is already in Laurent series form.- The term \(\frac{10}{z-1}\) must be expanded in a geometric series as \[ \frac{10}{z-1} = -10\left( \frac{1}{1-z} \right) = -10 (1 + z + z^2 + z^3 + \ldots) \].

Step 4: Combine Series Terms

Combine the series expansions obtained in Step 3:\[ \frac{-3}{z} - (10 + 10z + 10z^2 + 10z^3 + \ldots) \]This creates the Laurent series valid in the domain \(0 < |z| < 1\).

Step 5: Write the Final Laurent Series

The final representation of \( f(z) \) in the Laurent series for \(0 < |z| < 1\) is:\[ f(z) = \frac{-3}{z} - 10 - 10z - 10z^2 - 10z^3 - \ldots \]

Key Concepts

Complex Analysis

Complex analysis is a branch of mathematics focused on the study of functions that operate on complex numbers. A complex number is typically denoted as \(z = x + iy\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\). Functions of complex variables exhibit fascinating behaviors not seen in purely real functions. These functions are infinitely differentiable, which lends them a smoothness that makes complex analysis both beautiful and powerful.

One of the critical tools in complex analysis is the concept of a power series, which provides a way to express complex functions as infinite sums of terms that are powers of the variable. When dealing with regions that contain singularities (points where the function is not defined), we use the Laurent series. The Laurent series, unlike the Taylor series, can include terms with negative powers of \(z\), allowing us to accurately represent functions with singularities.

In the case of the function \(f(z) = \frac{7z - 3}{z(z-1)}\), complex analysis helps us understand how the function behaves near its singular points, which in this exercising, are at \(z=0\) and \(z=1\). The goal is to find a representation of the function that is valid in the annular region where \(0 < |z| < 1\). This involves exploring how functions change and transforming them into forms that provide insights into their behavior in complex domains.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used in calculus and algebra to break down complex rational expressions into simpler components. This is particularly useful in integrating complex rational functions or finding series expansions similar to our exercise. For a function expressed as a fraction, where both the numerator and denominator are polynomials, partial fraction decomposition helps express the function as the sum of simpler fractions.

We begin by identifying the singularities of the rational function \( \frac{7z - 3}{z(z-1)} \), once these are determined, the next step involves expressing the function as a sum:

• \( \frac{A}{z} \) corresponds to the singularity at \(z = 0\).
• \( \frac{B}{z-1} \) corresponds to the singularity at \(z = 1\).

After determining the form of the partial fractions, we solve for the constants \(A\) and \(B\) by equating coefficients or by substituting convenient values of \(z\). This results in the equation:

• \(7z - 3 = A(z-1) + Bz\).

By comparing coefficients, we find \(A = -3\) and \(B = 10\). Thus, the partial fraction decomposition is \( \frac{-3}{z} + \frac{10}{z-1} \). This step is essential as it breakdowns the complex function into manageable parts that can be expanded separately in the next steps.

Series Expansion

Series expansion is a method to express functions as infinite sums of terms. In complex analysis, one common type is the Laurent series, which accommodates terms with negative powers. This is particularly useful for functions with singularities, as seen in our exercise.

The goal is to find a series that accurately represents the function \(f(z)\) within a specified domain. For the domain \(0 < |z| < 1\), we need to ensure that our expansion reflects the conditions for \(|z| < 1\). For the term \(\frac{-3}{z}\), it is already in a proper form for the Laurent series because it is a negative power. The interesting part comes with \(\frac{10}{z-1}\), which must be rewritten to fit within the series approach.

To expand \(\frac{10}{z-1}\) for \(|z|<1\), rewrite it using the geometric series formula:

• \( \frac{10}{z-1} = -10 \left(\frac{1}{1 - z}\right) \).
• This expression transforms into \(-10 (1 + z + z^2 + z^3 + \ldots)\), using the formula for a geometric series.

Finally, to find the Laurent series, combine \(\frac{-3}{z}\) and the expanded form of \(-10(1 + z + z^2 + z^3 + \ldots)\), creating the series:

• \( \frac{-3}{z} - 10 - 10z - 10z^2 - 10z^3 - \ldots \).

This representation is valid for \(0 < |z| < 1\) and gives us the desired Laurent series.