Problem 18 - Advanced Engineering Mathematics

Question

Since \(w=(1+i) z=\sqrt{2} e^{i \pi / 4} z,\) the mapping is the composite of a rotation through \(45^{\circ}\) and a magnification by \(\alpha=\sqrt{2} .\) The image of the first quadrant is therefore the angular wedge \(\pi / 4 \leq \operatorname{Arg} w \leq 3 \pi / 4.\)

Step-by-Step Solution

Answer

The image is the wedge \(\pi/4 \leq \operatorname{Arg}(w) \leq 3\pi/4\).

Step 1: Understand the Mapping Expression

The given expression is \(w = (1+i)z = \sqrt{2} e^{i \pi / 4} z\). This indicates that the mapping is a combination of a complex multiplication that rotates the angle by \(\pi/4\) radians (which is equivalent to a \(45^\circ\) rotation) and scales the magnitude by \(\sqrt{2}\).

Step 2: Identify the Effects of Rotation and Scaling

The term \(e^{i \pi / 4}\) represents a rotation of \(45^\circ\), which affects the direction of the complex number. The term \(\sqrt{2}\) before this exponential term indicates the magnitude of \(z\) is scaled by \(\sqrt{2}\). This means every point on the complex plane is moved \(\sqrt{2}\) times further from the origin.

Step 3: Analyze the Image of the First Quadrant

The first quadrant in the complex plane consists of points \(z\) such that \(0 \leq \operatorname{Arg}(z) \leq \pi/2\). Under this transformation, the argument of \(z\), when multiplied by \(e^{i \pi / 4}\), would shift each angle \(z\) by \(45^\circ\) (or \(\pi/4\) radians).

Step 4: Calculate the Resulting Arguments

For the points in the first quadrant, their arguments, initially between \(0\) and \(\pi/2\), are transformed as follows: \(\operatorname{Arg}(w) = \operatorname{Arg}(z) + \pi/4\). Therefore, \(\operatorname{Arg}(w)\) will range from \(\pi/4\) to \(\pi/4 + \pi/2 = 3\pi/4\).

Step 5: Determine the Image of the Angular Wedge

By observing the transformed arguments from \(\pi/4\) to \(3\pi/4\), we conclude that the image of the first quadrant under the given transformation is an angular wedge in the range \(\pi/4 \leq \operatorname{Arg}(w) \leq 3\pi/4\).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers and are used to solve problems that real numbers cannot. They are in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\). The component \(a\) is the real part, and \(b\) is the imaginary part.

• They can be visualized as points or vectors in a two-dimensional space called the complex plane, where the horizontal axis is the real part, and the vertical axis is the imaginary part.
• A complex number \(z\) can be represented in polar form as \(r\,e^{i\theta}\), where \(r\) is the magnitude (or modulus) \(\sqrt{a^2 + b^2}\) and \(\theta\) is the angle (or argument) made with the positive real axis.

Understanding how complex numbers behave is fundamental to many areas of mathematics, including engineering and physics, as they allow us to describe phenomena like electrical currents and waves.

Angle Rotation

In complex analysis, angle rotation refers to changing the direction of a vector in the complex plane by a certain angle. This transformation is achieved through multiplication by an exponential term \(e^{i\theta}\), where \(\theta\) is the rotation angle.

• When multiplying a complex number \(z\) by \(e^{i\theta}\), it rotates its direction by \(\theta\) radians counterclockwise around the origin.
• The magnitude (length) of the vector remains unchanged during the rotation.
• In the given expression \(w = (1+i)z = \sqrt{2} e^{i \pi / 4} z\), the term \(e^{i \pi / 4}\) denotes a \(45^\circ\) or \(\pi/4\) radian rotation, which affects how the complex number is positioned in the plane.

This rotation is crucial in understanding movements within the complex plane, as it shows how a simple multiplication can effortlessly adjust an angle.

Scaling Transformation

Scaling transformation in the context of complex numbers refers to changing the size, or magnitude, of the vector representing the number. This change is achieved through multiplication by a real constant.

• The scaling factor determines how much larger or smaller the new vector will be compared to the original.
• In the equation \(w = (1+i)z = \sqrt{2} e^{i \pi / 4} z\), the term \(\sqrt{2}\) is the scaling factor that stretches all vectors by \(\sqrt{2}\) times their original length.
• The angle or direction of the vector stays constant during scaling; only its length changes.

Understanding scaling is important because it explains how complex numbers can be expanded or shrunk while maintaining their original direction, forming the basis for numerous practical applications such as signal processing and control systems.

Complex Plane Mapping

Complex plane mapping involves transforming points in the complex plane from one position to another using complex multiplication. It combines rotation and scaling transformations and can map a region to another.

• Mappings can be visualized as smooth warps or shifts on the plane, altering both the location and the space's angle.
• The given transformation \(w = (1+i)z = \sqrt{2} e^{i \pi / 4} z\) first rotates and then scales every point in a specified region.
• Such mappings are useful in identifying how functions can modify shapes and spaces, seen in graphics, animations, and various engineering challenges.

The concept of mapping within the complex plane helps us understand not just individual transformations but the interplay of different transformations to achieve desired results, providing insights into both mathematical theory and practical utility.