Problem 41 - A First Course in Complex Analysis with Applications

Question

Consider (6) with the symbol \(z\) replaced by \(e^{i z}\) : $$ \frac{1}{1-e^{i z}}=e^{i z}+e^{2 i z}+e^{3 i z}+\cdots $$ Give the region in the complex plane for which the foregoing series converges.

Step-by-Step Solution

Answer

The series converges for \(\text{Im}(z) > 0\), in the upper half of the complex plane.

Step 1: Identify the Series

The given series is a geometric series. The series is expressed as \(\sum_{n=1}^{\infty} e^{i n z}\). This is a geometric series with the common ratio, \(r = e^{i z}\).

Step 2: Compute Convergence Condition

Recall that a geometric series \( \sum_{n=0}^{\infty} r^n \) converges if \(|r| < 1\). In our case, the common ratio \( r = e^{i z} \) must satisfy this condition for convergence.

Step 3: Analyze Magnitude of Common Ratio

We need \(|e^{i z}| < 1\). Consider the property of complex exponentials: \(|e^{i z}| = e^{i x}e^{-y}\), where \(z = x + iy\). In this form, \(|e^{i z}| = e^{-y}\).

Step 4: Determine Convergence Region

For the series to converge, we require \(e^{-y} < 1\). This inequality holds when \(-y < 0\), or equivalently, \(y > 0\). Therefore, the series converges for all \(z = x + iy\) where \(y > 0\).

Step 5: Conclude with the Convergence Region

By finalizing our analysis, we determined that the series converges in the upper half of the complex plane. Specifically, the region for convergence is \(\{z \in \mathbb{C} : \text{Im}(z) > 0\}\).

Key Concepts

Geometric Series

A geometric series is a sum of terms in which each term is a constant multiple of the previous one. In simple terms, each term after the first is obtained by multiplying the previous term by a constant called the common ratio. Geometric series are fascinating due to their capability to model exponential growth or decay.

• The expression for a geometric series is typically written as \( \sum_{n=0}^{\infty} r^n \), where \( r \) is the common ratio.
• Convergence of a geometric series depends on the absolute value of the common ratio. Specifically, the series converges if \( |r| < 1 \).
• In our context, the series is \( \sum_{n=1}^{\infty} e^{i n z} \) with \( r = e^{i z} \).

Understanding geometric series is crucial because it appears frequently in analysis and has practical applications in areas like interest calculations and signal processing.

Convergence Region

The convergence region of a series is the set of points in the complex plane where the series converges to a finite sum. This concept is crucial when dealing with functions of a complex variable, especially to understand where they are well-defined.

• For a geometric series to converge, the condition \( |r| < 1 \) must be met.
• In our specific exercise, this translates to needing \( |e^{i z}| < 1 \).

The analysis showed that \( |e^{i z}| = e^{-y} \), leading to the requirement that \( y > 0 \) for convergence.
This condition tells us that the series will converge for all complex numbers \( z = x + iy \) where the imaginary part \( y \) is positive. Thus, the convergence region is the upper half of the complex plane, emphasizing the importance of the imaginary component in determining convergence.

Complex Plane

The complex plane is a two-dimensional plane where each point represents a complex number. It is a vital tool in complex analysis, as it allows us to visualize and understand complex numbers and their behaviors.
The plane is structured with:

• The real part of the complex number \( z = x + iy \) represented on the horizontal axis.
• The imaginary part \( y \) represented on the vertical axis.

This system is akin to a Cartesian plane but configured for complex numbers, aiding in analyzing their properties and behaviors, such as convergence regions. Understanding this view lets us see why the positive imaginary axis indicates convergence in our exercise: it shows the "upper half" of the complex plane where \( \, y > 0 \), thereby guiding us toward the solution.

Exponential Function

The exponential function, especially when involving complex numbers, plays a significant role in complex analysis. Defined generally as \( e^z \), this function extends the familiar exponential functions from real to complex numbers, and it has unique properties.
Some key characteristics include:

• For a complex number \( z = x + iy \), the exponential function is expressed as \( e^z = e^x e^{iy} \).
• Due to Euler's formula, \( e^{iy} = \cos(y) + i\sin(y) \), the exponential function links complex numbers with trigonometric functions.

In our exercise, the term \( e^{i z} \) represents an oscillating component when embedded in a series. The magnitude of this component, \( |e^{i z}| = e^{-y} \), directly impacts the convergence. Higher mathematics leverages these properties to study various phenomena, from quantum mechanics to electrical engineering, all grounded in understanding how complex exponential functions behave.