Problem 31 - A First Course in Complex Analysis with Applications

Question

In Problems 31 and 32 , expand the given function in Taylor series centered at each of the indicated points. Give the radius of convergence \(R\) of each series. Sketch the region within which both series converge. $$ f(z)=\frac{1}{2+z}, z_{0}=-1, z_{0}=i $$

Step-by-Step Solution

Answer

Series at \(z_0 = -1\) converges for \(|z+1| < 3\), radius 3. Series at \(z_0 = i\) converges for \(|z-i| < \sqrt{5}\), radius \(\sqrt{5}\).

Step 1: Recall the Taylor Series Formula

To expand a function into a Taylor series centered at a point, use the formula:\[ f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n \]where \(f^{(n)}(z_0)\) is the nth derivative of \(f(z)\) evaluated at the center \(z_0\).

Step 2: Evaluate the Function at the Center

First, find the Taylor series centered at \(z_0 = -1\):Given \(f(z)=\frac{1}{2+z}\), compute \(f(z+y)=\frac{1}{3+y}\) where \(y=z+1\), and rewrite it as a geometric series:\[ f(y) = \frac{1}{3} \cdot \frac{1}{1 + \frac{y}{3}} = \frac{1}{3} \sum_{n=0}^{\infty} \left(-\frac{y}{3}\right)^n \]Thus, the series becomes:\[ \frac{1}{3} \sum_{n=0}^{\infty} (-1)^n \left( \frac{z+1}{3} \right)^n \]

Step 3: Evaluate Radius of Convergence for \(z_0 = -1\)

For a geometric series \(\sum_{n=0}^{\infty} a r^n\) to converge, \(|r| < 1\).Here, \(\left| \frac{z+1}{3} \right| < 1\), giving \(|z+1| < 3\), hence the radius of convergence \(R = 3\).

Step 4: Evaluate the Function at the Next Center

Now find the Taylor series centered at \(z_0 = i\):Set \(f(z+i)=\frac{1}{2+(z+i)} = \frac{1}{2+i+z}\) and express as a geometric series:\[ \frac{1}{2+i} \cdot \frac{1}{1 + \frac{z}{2+i}} = \frac{1}{2+i} \sum_{n=0}^{\infty} \left(-\frac{z}{2+i}\right)^n \]

Step 5: Evaluate Radius of Convergence for \(z_0 = i\)

For convergence, \(|- \frac{z}{2+i}| < 1\implies |z| < |2+i|\).Calculate \(|2+i| = \sqrt{4+1} = \sqrt{5}\), resulting in a radius of convergence \(R = \sqrt{5}\).

Step 6: Sketch the Regions of Convergence

The first series centered at \(z_0 = -1\) converges within a circle of radius 3 centered at \(-1\).The second series centered at \(z_0 = i\) converges within a circle of radius \(\sqrt{5}\) centered at \(i\).To sketch, draw two overlapping circles with given radii and centers on the complex plane.

Key Concepts

Complex Analysis

Complex analysis is a branch of mathematics focused on functions that operate on complex numbers. Complex numbers are expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\). In this domain, we analyze functions of a complex variable, such as \(f(z)\), where \(z = x + yi\).

One of the beautiful aspects of complex analysis is the ability to extend many concepts from real calculus. For instance, we use derivatives and integrals similarly, but with complex variables. The function provided, \(f(z) = \frac{1}{2+z}\), is an example of a rational function. This involves identifying key points like singularities where the function cannot be defined. When expanding functions into series like the Taylor or Laurent series, complex analysis provides powerful tools to approximate these functions in certain regions of the complex plane, known as domains of convergence.

• Complex Plane: A two-dimensional plane used to visualize complex numbers with the real part on the x-axis and the imaginary part on the y-axis.
• Holomorphic Function: A complex function that is differentiable at every point in its domain, reminiscent of how differentiability works in calculus with real numbers.

Understanding complex analysis is crucial because it allows us to simplify difficult problems in physics and engineering by leveraging the properties of complex functions.

Radius of Convergence

The radius of convergence determines the region in which a power series representation of a function converges in the complex plane. For functions like a Taylor series, it is crucial to identify this radius to understand where the series effectively approximates the function.

In our exercise, when calculating the radius of convergence for the Taylor series at different centers, we used the formula for geometric series: \(|r| < 1\). When - Centered at \(z_0 = -1\), the expression \(\left| \frac{z+1}{3} \right| < 1\) implies that the radius \(R\) is 3.- Centered at \(z_0 = i\), the radius becomes \(R = \sqrt{5}\), determined from the simplification \(|z| < |2 + i|\).

The radius of convergence not only informs us about where the series is valid, but in complex analysis, it often marks the boundary beyond which the function may become multi-valued or cease to be holomorphic. Determining this radius is a fundamental step in extending and validating series expansions. It can be influenced by singularities, beyond which the function or its expansion cannot extend.

Geometric Series

A geometric series is an infinite sum of terms where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. It forms the backbone in many mathematical arguments due to its convergence properties. The series takes the form \( \sum_{n=0}^\infty ar^n = a + ar + ar^2 + \cdots \) where \(a\) is the first term and \(r\) is the common ratio.

In the provided exercise, we expressed the function \(f(z)\) in terms of a geometric series for both series centers:

• Centered at \(z_0 = -1\), the expression \((z+1)/3\) acts as the common ratio for the geometric series.
• Centered at \(z_0 = i\), \(-z/(2+i)\) defines the common ratio.

The key condition for the series' convergence is \(|r| < 1\), ensuring the terms tend towards zero as \(n\) goes to infinity, making the infinite sum converge to a finite value.

• Sum Formula: For \(|r| < 1\), the sum \(S\) of an infinite geometric series is \(S = \frac{a}{1-r}\).
• Practical Applications: Geometric series are widely used to model phenomena with exponential growth or decay, such as in financial calculations and physics.

Using geometric series simplifies problems by converting complex expressions into sums, making them easier to manipulate and evaluate.