Problem 7 - Advanced Engineering Mathematics

Question

The potential \(\phi\) on the semicircle \(|z| \leq 1, y \geq 0\), satisfies the boundary conditions \(\phi(x, 0)=0,-1

Step-by-Step Solution

Answer

The potential is \( \phi(x, y) = \frac{1}{\pi} \operatorname{Arg}\left(\frac{z-1}{z+1}\right)^{2} \), and equipotential lines are arcs of circles due to mapping properties of \( \frac{z-1}{z+1} \).

Step 1: Analyze Boundary Conditions

The given boundary conditions are \( \phi(x,0)=0 \) for \(-1 < x < 1\), which means the potential along the x-axis within this interval is zero. The condition \( \phi(e^{i\theta})=1 \) for \( 0<\theta<\pi \) implies the potential on the upper half of the unit circle is one.

Step 2: Propose a Suitable Form for \( \phi(z) \)

We propose that the function \( \phi(z) \) be expressed in terms of the argument function, given by: \[ \phi(x, y) = \frac{1}{\pi} \operatorname{Arg}\left(\frac{z-1}{z+1}\right)^2 \]. The reason for this choice is to satisfy the nature of the argument function which changes smoothly around a circle.

Step 3: Verify Boundary Conditions for \( \phi(z) \)

Calculate the argument of \( \frac{z-1}{z+1} \) at the boundaries. When \( z=x \) ( x-axis, \(-1 < x < 1\) ), the expression becomes pure real, leading to \( \phi(x,0)=0 \), which matches the first boundary condition. For \( z = e^{i\theta} \) (upper semicircle), \( \phi(e^{i\theta}) \) results in 1, matching the second boundary condition \( 0 < \theta < \pi \).

Step 4: Mapping Properties of Linear Fractional Transformations

Consider the linear fractional transformation \( w = \frac{z-1}{z+1} \). Such transformations map circles (or lines) onto circles or lines. The expression \( \operatorname{Arg}(w^2) \), thus, represents angles subtended by circular arcs on the complex plane. Hence, by the mapping properties, the equipotential lines \( \phi(x,y) = constant \) are represented as arcs of circles.

Key Concepts

Boundary Conditions

Boundary conditions in complex analysis are crucial for defining and solving problems involving complex potentials and fields. They specify the behavior of a function along the boundary of a region.

In this exercise, two boundary conditions are given:

• The potential \( \phi(x,0)=0 \) for \( -1 < x < 1 \), indicating the potential is zero along the segment of the real axis that lies between -1 and 1.
• The potential \( \phi(e^{i\theta})=1 \) for \( 0<\theta<\pi \), suggesting that the potential is one along the upper half of the unit circle.

Boundary conditions are used to ensure that any proposed solution fits perfectly with the physical or mathematical constraints of the problem. By satisfying these conditions, the potential function can be regarded as accurately representing our system.

Linear Fractional Transformations

Linear fractional transformations, or Möbius transformations, are essential functions in complex analysis. They are of the form \( w = \frac{az + b}{cz + d} \) and map the complex plane in significant ways.

These transformations have distinct characteristics:

• They are conformal, meaning they preserve angles—but not necessarily distances.
• They map circles and lines in the complex plane to other circles or lines.

For our exercise, we consider the transformation \( w = \frac{z-1}{z+1} \). This transformation helps simplify the domain geometry, adjusting the complex plane so the mappings of points become more intuitive. Specifically, this transformation maps the unit semicircle in the complex plane to alternative forms, making it easier to analyze potential functions and boundary behavior.

Argument Function

The argument function, denoted as \( \operatorname{Arg}(z) \), is used to determine the angle between a point and the origin in the complex plane. It’s a vital concept in connecting complex numbers and trigonometry.

The key aspects of the argument function include:

• It provides a measure for the rotational component of a complex number.
• The angle represents the direction from the origin towards the point \( z \) over the complex plane.
• The argument is principally taken in the interval \((-\pi, \pi]\).

In this exercise, the function \( \phi(x, y) = \frac{1}{\pi} \operatorname{Arg}\left(\frac{z-1}{z+1}\right)^2 \) utilizes the argument property to smoothly adjust potential values across the semicircle, while maintaining the correct boundaries.

Equipotential Lines

Equipotential lines are curves or lines along which a potential function is constant. In complex analysis, these are often represented as arcs or closed shapes that illustrate the field's potential across a surface.

Considerations for equipotential lines:

• They reveal the symmetry of the potential field.
• For a given complex map, certain circles or arcs on the plane will maintain constant potential.

In this specific problem, the equipotential lines formed by the function \( \phi(x, y) \) are indeed arcs of circles. This is a consequence of the properties of linear fractional transformations, which dictate that such mappings convert lines and circles onto other lines or circles, preserving the nature of equipotential lines as circular arcs on the complex plane's mapped view.