Problem 6 - Advanced Engineering Mathematics

Question

A curve in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image curve in the \(w\) -plane. Ray \(\theta=\pi / 4\) under \(w=\operatorname{Ln} z\)

Step-by-Step Solution

Answer

The image is a horizontal line at \( \operatorname{Im}(w) = \frac{\pi}{4} \).

Step 1: Understanding the Mapping

The given complex mapping is \( w = \operatorname{Ln} z \). This is the principal branch of the complex logarithm function. In terms of polar coordinates, if \( z = re^{i\theta} \), then the logarithm is given by \( w = \ln|r| + i\arg(z) \), where \( \arg(z) \) is the principal argument.

Step 2: Identifying the Input Curve

The input curve is the ray \( \theta = \pi / 4 \). In polar form, any point \( z \) on this ray can be expressed as \( z = re^{i\pi/4} \), where \( r \geq 0 \). This means that the ray emanates from the origin at an angle of \( \pi/4 \) radians.

Step 3: Mapping the Ray to the w-plane

Substitute \( z = re^{i\pi/4} \) into the mapping function: \( w = \operatorname{Ln}(z) = \ln|r| + i\frac{\pi}{4} \). The real part of \( w \) is \( \ln|r| \), which can take any real value, and the imaginary part is always \( \frac{\pi}{4} \).

Step 4: Describe the Image Curve

The image curve in the \( w \)-plane is described by the equation \( \operatorname{Im}(w) = \frac{\pi}{4} \), and \( \operatorname{Re}(w) = \ln|r| \). When \( r \) changes from 0 to infinity, \( \ln|r| \) covers the whole real line. Thus, the image is a horizontal line at \( \operatorname{Im}(w) = \frac{\pi}{4} \).

Key Concepts

Complex Mapping

In complex analysis, complex mapping is a function that takes a complex number from one plane (often called the \( z \)-plane) and transforms it to another complex number in another plane (the \( w \)-plane). A common form of complex mapping is expressed as \( w = f(z) \), where \( w \) is the image of \( z \) under the mapping function \( f \).
In our exercise, the mapping function is \( w = \operatorname{Ln} z \), which involves taking the logarithm of a complex number. Complex mappings like this can significantly alter the shape and position of curves or areas from the \( z \)-plane to the \( w \)-plane.
Understanding these mappings in terms of their effects on geometric shapes or curves can reveal important insights about both the structure and behavior of other complex functions.

Complex Logarithm

The complex logarithm is an extension of the logarithm function to complex numbers. Given a complex number \( z = re^{i\theta} \) in polar form, the complex logarithm is defined as \( \operatorname{Ln} z = \ln|r| + i\arg(z) \).
Here, \( \ln|r| \) is the natural logarithm of the modulus of \( z \), while \( \arg(z) \) represents the argument (or angle) of \( z \). In the context of the principal branch of the complex logarithm, the argument \( \arg(z) \) is restricted to the interval \( (-\pi, \pi] \).
This function can map rays emanating from the origin of the \( z \)-plane to horizontal lines in the \( w \)-plane by fixing the imaginary part, which corresponds to the angle \( \theta \), and letting the real part \( \ln|r| \) vary freely.

Principal Argument

The principal argument of a complex number is the angle of the complex number from the positive real axis within a specified range, typically \( (-\pi, \pi] \). For any complex number \( z = re^{i\theta} \), \( \theta \) is its argument. However, there are infinitely many arguments for a given complex number since adding any integer multiple of \( 2\pi \) results in the same value.
The principal argument, noted as \( \arg(z) \), is unique within its range and is often used in calculations to ensure consistency. In our example with \( \theta = \pi/4 \), this becomes the fixed imaginary component in the \( w \)-plane mapping, resulting in a horizontal line since only the real part varies.

Polar Coordinates

Polar coordinates offer an intuitive way to represent complex numbers and can significantly simplify complex analysis problems. Any complex number \( z \) can be expressed in the form \( z = re^{i\theta} \), where \( r \) is the magnitude (or modulus) of \( z \), and \( \theta \) is the argument or angle.
Using polar coordinates, operations like multiplication and logarithms are often simpler. For example, the product of two complex numbers simply requires multiplying their magnitudes and adding their angles.
In the given exercise, the ray represented in polar form (\( \theta = \pi/4 \)) maps directly to a line with an imaginary component in the \( w \)-plane, thanks to the logarithmic mapping converting radial growth \( r \) to an unrestricted real line \( \ln|r| \).