Problem 18 - Advanced Engineering Mathematics

Question

In Problems 13-22, expand the given function in a Taylor series centered at the indicated point. Give the radius of convergence of each series. $$ f(z)=\frac{1+z}{1-z}, z_{0}=i $$

Step-by-Step Solution

Answer

The Taylor series is expanded and converges with a radius of 1.

Step 1: Rewrite the Function

We start with the function \( f(z) = \frac{1+z}{1-z} \). To express this function as a Taylor series around \( z_0 = i \), we need to rewrite it in a form suitable for expansion. Begin by substituting \( w = z - i \), or \( z = w + i \), which also means rewriting the denominator as \( 1 - (w + i) = 1 - w - i \).

Step 2: Adjust Denominator for Series Expansion

We use the strategy to rewrite the denominator so we can perform polynomial division: \( \frac{1+z}{1-z} = \frac{1 + w + i}{1 - w - i} \). To make the denominator \( 1 + x \) with \( |x| < 1 \), factor out \( -1 \): \((-(w+i))( \frac{-1}{w+i}) \approx 1 \). We rewrite it as \( (1 - i - w)^{-1} \).

Step 3: Perform Partial Fraction Decomposition

Perform partial fraction decomposition on \( \frac{1 + w + i}{-w - i} \). Separate terms to enable series expansion by breaking it down. Use polynomial long division (if complex), or comparison method. Simplify the fractions, ensuring each part is made ready to expand using geometric or known series.

Step 4: Series Expansion

Now, expand \( (1 - i - w)^{-1} \) using Taylor's geometric series: recognize as \( \frac{1}{1-x} = 1 + x + x^2 + \cdots \) when \( |x|<1 \). Apply this to \( (w+i) \), calculating each term explicitly in series \( \sum_{n=0}^{\infty} c_n (z - z_0)^n \). Each coefficient \( c_n \) is derived by matching terms.

Step 5: Determine Radius of Convergence

To determine the radius of convergence of the series, observe within what boundary \( \frac{1}{1 - i - w} \) remains valid. Use the formula \( R = \frac{1}{\text{distance to nearest singularity}} \). Here, closest singularity at \( w = -1 \), giving \( R = 1 \) as the radius of convergence.

Key Concepts

Radius of Convergence

The radius of convergence describes the distance within which a Taylor series approximates a function. In complex analysis, it determines how far from a point, typically denoted as \(z_0\), the series represents the function accurately. This radius can be found using the formula:

• \(R = \frac{1}{\text{distance to nearest singularity}}\)

For the function \(f(z) = \frac{1+z}{1-z}\) centered at \(z_0 = i\), the nearest singularity is at \(w = -1\).
This implies the radius of convergence \(R\) is \(1\). Essentially, it means the Taylor series is valid in a circle with radius \(1\) centered at \(i\). By analyzing singularities and distances in the complex plane, you determine where a function remains robust.

Partial Fraction Decomposition

Partial fraction decomposition is a method of breaking down complex rational expressions into simpler fractions. This process helps in manipulating the function into a form suitable for Taylor series expansion, or other types of analysis.
It involves expressing a fraction as a sum of fractions with simpler denominators.

• This helps to simplify complex algebraic expressions.
• Is widely used to separate terms for easier mathematical handling.

In the solution for \(f(z) = \frac{1+z}{1-z}\), partial fraction decomposition is used to express the complex fraction in simpler terms like \(\frac{1 + w + i}{-w - i}\).
This method divides and adjusts the fractions, so series methods like geometric series can be applied efficiently.

Singularities

Singularities are points where a function is not defined or behaves badly, such as tending towards infinity. These points are crucial in determining the behavior of functions in complex analysis.

• They identify the limits of convergence for a series.
• The closest singularity determines the radius of convergence.

For example, in \(f(z) = \frac{1+z}{1-z}\), when converted for expansion around \(z_0 = i\), the singularities are addressed by adjusting the expression to avoid them.
Noticing where the function ceases to be analytical helps establish boundaries for Taylor series expansions and other analyses. In this case, understanding the closest singularity \(w = -1\) emphasizes where the series loses validity, reinforcing the convergence concepts.

Complex Analysis

Complex analysis is the field of mathematics dealing with complex numbers and their functions. It delves deeply into behavior like singularities and convergence, largely dealing in with the geometry in the complex plane.

• Focuses on functions of complex variables.
• Incorporates many topics like Taylor and Laurent series.

In expanding \(f(z) = \frac{1+z}{1-z}\), complex analysis provides the framework to use transformations and expansions like Taylor series to understand function behavior around points like \(z_0 = i\).
This process involves evaluating the series' convergence and singular behaviors, resulting in a holistic view of function properties. By understanding how complex functions behave, broader insights into mathematical phenomena are achieved, whether in pure mathematics or applied sciences.