Problem 1 - Complex Analysis

Question

Expand the function given by the formula \(f(z)=z /\left(z^{2}+1\right)\) in\\} $$ \mathcal{A}=\\{z \in \mathbb{C} ; \quad 0<|z-\mathrm{i}|<2\\} $$ into a LAURENT series. What kind of singularity has \(f\) at \(a=\mathrm{i}\) ?

Step-by-Step Solution

Answer

The Laurent series is centered at \( z = i \) with a pole at \( z = i \).

Step 1: Identify the Function Components

We start by looking at the given function, which is \( f(z) = \frac{z}{z^2 + 1} \). Notice that \( z^2 + 1 \) can be factored as \((z - i)(z + i)\). This will help in identifying singularities.

Step 2: Find Singularity Locations

The singularities occur where the denominator equals zero. So, set \( z^2 + 1 = 0 \), giving solutions \( z = i \) and \( z = -i \). The singularities are at \( i \) and \( -i \).

Step 3: Determine the Region of Interest

We are given the region \( \mathcal{A} = \{ z \in \mathbb{C} : 0 < |z-i| < 2 \} \). This is an annular region centered at \( i \) with an inner radius of 0 and an outer radius of 2.

Step 4: Identify the Singularities in the Region

Within the region \( \mathcal{A} \, (|z-i|<2) \), the point \( z=-i \) lies outside while \( z=i \) is excluded (it's a hole), hence focus on expanding around \( z=i \).

Step 5: Rewrite Function for Laurent Series

For a Laurent series around \( z = i \), re-express \( f(z) = \frac{z}{z^2 + 1} \) in terms of \( w = z - i \): \( f(z) = \frac{z}{(z-i)(z+i)} = \frac{z}{(w)(z+i)} \).

Step 6: Expand the Denominator in Series

Express \( \frac{1}{z+i} \) in a geometric series suitable for \( |z+i| > |w| \). It becomes \( \sum_{n=0}^{\infty} \left(-\frac{w}{z+i}\right)^n \).

Step 7: Complete the Laurent Series Expansion

Substitute \( z = w+i \) into the expanded \( \frac{1}{z+i} \) and multiply by \( \frac{z}{w} \) to get the full Laurent series, i.e., \( \frac{(w+i)}{iw} \sum_{n=0}^{\infty} \left(-\frac{w}{z+i}\right)^n \). Simpllify the expression to write out the series fully.

Step 8: Classify the Singularity at z=i

Upon expanding the series, check for terms with negative powers of \( w \). If such terms exist, they correspond to a principal part indicating a pole. If this part is non-zero, \( i \) is a pole.

Key Concepts

Complex Analysis

Complex analysis is a branch of mathematics that focuses on complex numbers and functions derived from them. It extends concepts of calculus and algebra to complex plane (also known as the Argand plane), which is built from the field of real and imaginary numbers.
A simple yet significant use of complex analysis is evaluating integrals around complex contours. There are various techniques and theorems involved, like Cauchy's theorem and residue theorem, which allow solving complex integrals more effectively.

• Complex analysis elegantly links real and imaginary parts and provides insights through visual diagrams in the complex plane.
• It helps in dealing with functions that are not easily defined by real numbers alone. This makes it indispensable in many areas of physics and engineering.

Understanding how functions behave in the complex plane gives us new tools like the Laurent series, which we can use to analyze function behavior, particularly around singularities.

Singularity

In complex analysis, a singularity is a point at which a given function is not defined or does not behave normally. These are critical for comprehending the function's overall behavior. There are different types of singularities that we might encounter, such as poles, removable singularities, and essential singularities.
Specifically, for the function given in the exercise, a singularity occurs at a point where the denominator becomes zero, indicating undefined behavior.

• A removable singularity is a point where a function might not be defined but could become continuous or differentiable if value adjusted properly.
• A pole is a more aggressive type of singularity where the function tends to infinity as it approaches the singularity.
• In this exercise, the singularity at \( z = i \) needs to be understood and handled correctly using methods like the Laurent series.

Identifying and classifying these singularities allow us to understand the "holes" or "peaks" in the behavior of complex functions.

Complex Function Expansion

Complex function expansion refers to expressing a complex function in a series format, like a Taylor or Laurent series. These expansions help us to study functions around specific points, especially around singularities where functions misbehave under normal definitions.
When expanding functions as Taylor or Laurent series:

• Taylor series allow expansion at points where the function is analytic without singularities.
• Laurent series, which includes negative powers, become crucial when addressing singularities.

In the given exercise, we focused on the Laurent series to explore the function around the singularity at \( z = i \). This expansion helps separate regular parts from singular parts, giving clear sight of how the function behaves precisely around such points.
The process involves:

• Rewriting the function as a series.
• Identifying principal and regular parts.
• Classifying singularities based on the series' outcome.

The outcome of the Laurent expansion directly tells us about singularity types, their classifications, and overall function behavior in complex planes.