Problem 31 - Advanced Engineering Mathematics

Question

(a) Letting \(z=x+i y\) and noting that in this case \(x^{2}+y^{2}=R^{2},\) the transformation \(w=z+k^{2} / z\) becomes \\[ \begin{aligned} w &=x+i y+\frac{k^{2}}{x+i y}=x+i y+\frac{k^{2}}{x^{2}+y^{2}}(x-i y) \\ &=x+i y+\frac{k^{2}}{R^{2}}(x-i y)=\left(1+\frac{k^{2}}{R^{2}}\right) x+i\left(1-\frac{k^{2}}{R^{2}}\right) y \end{aligned} \\] We identify \(u=\left(1+k^{2} / R^{2}\right) x\) and \(v=\left(1-k^{2} / R^{2}\right) y .\) Then \\[ \frac{u^{2}}{\left(1+\frac{k^{2}}{R^{2}}\right)^{2}}=x^{2} \quad \text { and } \quad \frac{v^{2}}{\left(1-\frac{k^{2}}{R^{2}}\right)^{2}}=y^{2}, \quad k \neq R ,\\] so that \\[ \frac{u^{2}}{\left(1+\frac{k^{2}}{R^{2}}\right)^{2}}+\frac{v^{2}}{\left(1-\frac{k^{2}}{R^{2}}\right)^{2}}=x^{2}+y^{2}=R^{2}, \quad k \neq R \\] (b) When \(R=k\) the circle \(z=k e^{i t}\) is transformed into \(w=z+k^{2} / z=k e^{i t}+k e^{-i t}=2 k \cos t+0 i .\) Thus, the circle \(z=k e^{i t}\) is transformed into the closed interval \([-2 k, 2 k]\) on the \(u\) -axis. (c) Letting \(w=z+k^{2} / z\) we have \\[ \frac{w-2 k}{w+2 k}=\frac{z+\frac{k^{2}}{z}-2 k}{z+\frac{k^{2}}{z}+2 k}=\frac{z^{2}+k^{2}-2 k z}{z^{2}+k^{2}+2 k z}=\frac{(z-k)^{2}}{(z+k)^{2}}=\left(\frac{z-k}{z+k}\right)^{2} .\\]

Step-by-Step Solution

Answer

The transformation maps the circle to an ellipse when \( R \neq k \) and to an interval \([-2k, 2k]\) on the u-axis when \( R = k \).

Step 1: Expressing w in terms of u and v

We start by expressing the transformation \( w = z + \frac{k^2}{z} \) for \( z = x + i y \), where \( x^2 + y^2 = R^2 \). The transformation leads to \( w = \left(1 + \frac{k^2}{R^2}\right) x + i\left(1 - \frac{k^2}{R^2}\right) y \), allowing us to identify \( u = \left(1 + \frac{k^2}{R^2}\right) x \) and \( v = \left(1 - \frac{k^2}{R^2}\right) y \).

Step 2: Relating u^2 and v^2 to x^2 and y^2

We relate \( u \) and \( v \) to \( x \) and \( y \) by setting \( \frac{u^2}{\left(1 + \frac{k^2}{R^2}\right)^2} = x^2 \) and \( \frac{v^2}{\left(1 - \frac{k^2}{R^2}\right)^2} = y^2 \). This enables us to write the ellipse equation as \( \frac{u^2}{\left(1 + \frac{k^2}{R^2}\right)^2} + \frac{v^2}{\left(1 - \frac{k^2}{R^2}\right)^2} = R^2 \).

Step 3: Special Case: R = k

When \( R = k \), the transformation simplifies because \( z = k e^{i t} \). The transformation \( w = k e^{i t} + k e^{-i t} = 2k \cos t \) maps the circle to the closed interval \([-2k, 2k]\) on the real axis.

Step 4: Finding Complex Ratio for w/z

For the transformation \( w=z+\frac{k^{2}}{z} \), consider the expression \( \frac{w-2k}{w+2k} \). Substituting for \( w \) gives the fraction \( \left(\frac{z-k}{z+k}\right)^2 \), showing the transformation is a M"obius transformation squared.

Key Concepts

Elliptic Transformation

Elliptic transformations are an interesting type of map used in various mathematical fields, especially in complex analysis. In the given exercise, the transformation is expressed as \( w = z + \frac{k^2}{z} \). This maps points \( z \) on the complex plane to a new point \( w \). For the elliptic transformation, the circle condition \( x^2 + y^2 = R^2 \) is crucial, as it stands for a locus where points are transformed into an ellipse. These transformations often result in geometric figures such as ellipses or intervals, depending on specific parameters and conditions.
This transformation is exhausting the elliptical form from the circles given in the complex plane. When considering specific values like \( R = k \), the resulting shape becomes a straight interval rather than an ellipse. This is an interesting case where concave transformations produce linear outcomes.

Mobius Transformation

Mobius transformations are a subclass within complex transformations that are characterized by their rational form. The main equation \( w = z + \frac{k^2}{z} \) can be linked to a Mobius transformation because of its structure. They represent maps from the extended complex plane to itself, facilitating transformations like circles to circles, or lines.
In the given exercise, this mapping shows a significant geometry change. Applying the complex ratio \( \frac{w-2k}{w+2k} \) yields further exploration into the Mobius transformation. It reveals the transformation as \( \left(\frac{z-k}{z+k}\right)^2 \), showing its relation to traditional Mobius transformations, except squared. This exercise highlights the flexibility of Mobius transformations to encompass various types of geometric conversions. Understanding this helps students appreciate how versatile and robust Mobius transformations are within complex analysis.

Complex Analysis

Complex analysis is the mathematical study of complex numbers and functions based on them. It's immensely useful for transformations of complex variables, offering vast applications across different fields of mathematics and engineering. When handling transformations like \( w = z + \frac{k^2}{z} \), complex analysis ensures we properly map and manipulate complex numbers, considering both real and imaginary components.
In the context of this exercise, complex analysis tools allow us to package the complex equation into more manageable forms enabling transformations that may represent geometric objects such as ellipses or straight intervals. Mastery of complex analysis allows for seamless handling of transformations, understanding their properties, and analyzing their geometric implications.

Geometric Interpretation

Geometric interpretation plays a vital role in understanding complex transformations. By analyzing the transformation \( w = z + \frac{k^2}{z} \), it shifts its depiction from algebraic notation to visual geometric interpretation. This exercise demonstrates how a circle on the complex plane morphs into an ellipse or a real interval during transformation.
For instance, when \( R = k \), this transformation results in a simple line segment between \(-2k\) and \(2k\) on the real axis, rather than a more complex shape, providing a straightforward visual explanation of the transformation outcome. As such, understanding these geometric interpretations offers clarity on the transformations' impact, enriching one's comprehension of complex analysis and the real-world phenomena it models.