Problem 4 - Advanced Engineering Mathematics

Question

Use a Laurent series to find the indicated residue. \(f(z)=(z+3)^{2} \sin \frac{2}{z+3} ;\) Res \((f(z),-3)\)

Step-by-Step Solution

Answer

The residue of \(f(z)\) at \(z = -3\) is \(-\frac{4}{3}\).

Step 1: Identify the singularity and substitute

The singularity occurs at \(z = -3\). Substitute \(z+3 = w\) to simplify the function, yielding \(f(z) = w^2 \sin \frac{2}{w}\).

Step 2: Expand the sine function as a power series

Recall the power series expansion for sine: \[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\] Substitute \(x = \frac{2}{w}\) into the series:\[\sin \frac{2}{w} = \frac{2}{w} - \frac{(\frac{2}{w})^3}{3!} + \frac{(\frac{2}{w})^5}{5!} - \cdots = \frac{2}{w} - \frac{8}{3!w^3} + \frac{32}{5!w^5} - \cdots\]

Step 3: Multiply the series by \(w^2\)

Multiply the series expansion from Step 2 by \(w^2\):\[w^2 \left( \frac{2}{w} - \frac{8}{3!w^3} + \frac{32}{5!w^5} - \cdots \right) = 2w - \frac{8}{3!w} + \frac{32}{5!w^3} - \cdots\]

Step 4: Identify the coefficient of \(\frac{1}{w}\)

In the expression from Step 3, look for the \(\frac{1}{w}\) term, which is crucial in finding the residue. The term is:\[- \frac{8}{3!w} = -\frac{4}{3w}\]Thus, the coefficient of \(\frac{1}{w}\) is \(-\frac{4}{3}\).

Step 5: Conclude with the residue at the singularity

The residue of a function at a simple pole is the coefficient of the \(\frac{1}{w}\) term. Therefore, the residue at \(-3\) is \(-\frac{4}{3}\).

Key Concepts

Residue

In complex analysis, the residue is an essential concept when dealing with complex functions. It helps in evaluating complex integrals, particularly within the context of contour integrals in the complex plane. The residue of a function at a given singularity is essentially the coefficient of the term \[ \frac{1}{z-a} \] in the function's Laurent series expansion. Here, \(a\) is the point of the singularity.
This coefficient plays a significant role in calculating integrals using the residue theorem, which allows us to evaluate complex integrals by summing up the residues within a contour.

• To find the residue, expand the function around the point of interest using a Laurent series.
• Identify the necessary term \(\frac{1}{z-a}\) and note its coefficient.
• This coefficient is the residue at the singularity.

Understanding residues helps simplify complex problems, reducing integrals to simple calculations involving residues.

Singularity

A singularity is a point at which a mathematical object is not defined or not well-behaved. In the context of complex functions, singularities are points where the function becomes either infinite or undefined. These points are crucial because they affect the behavior of the entire function.

• There are different types of singularities, including removable singularities, poles, and essential singularities.
• In this exercise, the function \(f(z)\) has a singularity at \(z = -3\). This is where the function's behavior changes significantly.
• To effectively work with singularities, we often transform the variable to simplify the function.

Handling singularities allows mathematicians to precisely explore and understand the behavior of complex functions around these problematic points.

Power Series

Power series are infinite series that are used to represent functions as an infinite sum of terms. Each term is a power of a variable that scales alongside a coefficient. They are fundamental in approximating functions and analyzing their behavior around specific points.

• Power series have the form \(a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots\).
• They allow us to express functions like sine, cosine, and exponential functions as an infinite series.
• In complex analysis, the power series expansion is often used to study the local behavior of functions around singularities.

By expressing parts of the original function in terms of a power series, we can simplify complex functions and focus on significant features like residues.

Complex Analysis

Complex analysis is the study of complex numbers and functions of complex variables. It combines techniques and theorems that allow for a deeper understanding of function behavior in the complex plane. This branch of mathematics is known for its elegance and applications across various scientific fields.

• Involves evaluating complex functions, which are functions that depend on complex numbers.
• Analyzes singularities and residues, using tools like the Laurent series and residue theorem.
• Applications include fluid dynamics, quantum physics, and electrical engineering.

By using concepts like residues and singularities, complex analysis allows mathematicians to tackle integral problems in novel ways, simplifying seemingly complicated computations.