Problem 16 - Advanced Engineering Mathematics

Question

(a) For \(f(z)=\left(z^{2}-1\right)^{1 / 2}\) \\[ f(t)=\left|t^{2}-1\right|^{1 / 2} \cos \left(\frac{1}{2} \operatorname{Arg}\left(t^{2}-1\right)\right)+i\left|t^{2}-1\right|^{1 / 2} \sin \left(\frac{1}{2} \operatorname{Arg}\left(t^{2}-1\right)\right) \\] $$\text { and so } f(t)=\left\\{\begin{array}{ll} \left|t^{2}-1\right|^{1 / 2}, & |t|>1 \\ i\left|t^{2}-1\right|^{1 / 2}, & |t|<1 \end{array} \text { Hence } \operatorname{Im}(G(z))=0 \text { on the boundary of } R\right.$$ $$\text { (b) } x=\left|(t+i c)^{2}-1\right|^{1 / 2} \cos \left(\frac{1}{2} \operatorname{Arg}\left((t+i c)^{2}-1\right)\right), \quad y=\left|(t+i c)^{2}-1\right|^{1 / 2} \sin \left(\frac{1}{2} \operatorname{Arg}\left((t+i c)^{2}-1\right)\right) \text { for } c>0$$ c.

Step-by-Step Solution

Answer

For (a), \(f(t)\) is real for \(|t| > 1\) and imaginary for \(|t| < 1\). For (b), \(x\) and \(y\) describe the real and imaginary parts for complex arguments.

Step 1: Understanding the Problem

The problem involves a complex function \( f(z) = (z^2 - 1)^{1/2} \) and requires determining how it behaves for different values of \( z \). We have a specific representation of the function in terms of its modulus and argument for a real variable \( t \), and conditions defined for when \(|t| > 1\) and \(|t| < 1\).

Step 2: Analyze Part (a) Behavior for |t| > 1

For \(|t| > 1\), \(f(t)\) simplifies to \( |t^2 - 1|^{1/2} \). This means that for values of \(t\) outside the interval \([-1, 1]\), the function is purely real and its imaginary part is zero. This is because for \(|t| > 1\), the argument \( \operatorname{Arg}(t^2-1) = 0 \) or \( \pi \), so the sine term vanishes.

Step 3: Analyze Part (a) Behavior for |t| < 1

For \(|t| < 1\), \(f(t)\) becomes \( i |t^2 - 1|^{1/2} \), which means the function is purely imaginary with its real part being zero. This is because when \(|t| < 1\), the term \( t^2 - 1 \) is negative and \( \operatorname{Arg}(t^2-1) = \pi \), making \( \cos \left(\frac{1}{2} \pi\right) = 0 \).

Step 4: Understanding Imaginary Part on Boundary for R

Given that \( \text{Im}(G(z)) = 0 \) on the boundary of some region \( R \), this implies that the function \( G(z) \) has no imaginary component on the boundary, suggesting \( R \) might involve real values of \( z \), possibly related to the unit circle or a similar geometric locus on the complex plane.

Step 5: Analyzing Part (b) for Complex Argument

For part (b), with \( x = |(t+ic)^2 - 1|^{1/2} \cos\left(\frac{1}{2} \operatorname{Arg}((t+ic)^2 - 1)\right) \), and \( y = |(t+ic)^2 - 1|^{1/2} \sin\left(\frac{1}{2} \operatorname{Arg}((t+ic)^2 - 1)\right) \) for \( c > 0 \), \( x \) and \( y \) represent the real and imaginary parts of \( f(t+ic) \) respectively, calculated from the modulus and half-angle argument form for complex square roots.

Step 6: Final Thoughts

Working with complex functions often involves understanding the transformation of real and imaginary parts across certain domains. The square root function introduces branch cuts which determine the piecewise behavior of \( f(z) \), highlighting where the function transitions between purely real and imaginary values.

Key Concepts

Complex Analysis

Complex analysis is the study of functions that operate using complex numbers. It extends many concepts from real analysis to the complex plane. In complex analysis, functions of a complex variable—like the one in our exercise, \( f(z)=(z^2-1)^{1/2} \)—are examined for their properties and behaviors. What makes these functions unique is how they are represented and manipulated using complex arithmetic.

Complex numbers are represented as \( z = x+iy \), where \( x \) is the real part and \( iy \) is the imaginary part. Functions like \( f(z) \) can present multidimensional insights due to the extra dimension (the imaginary axis) they possess. This feature allows analysis techniques, like contour integration and series expansion, to be developed, resulting in vast applications in physics and engineering.

In the context of our exercise, complex analysis helps determine how recurring principles—like limits, continuity, and differentiability—apply. Complex analysis introduces the concept of an analytic function, a function that is differentiable at every point in its domain. This ties into our problem where understanding the nature of \( f(z) \) is crucial to solving it. With a complex function like \( f(z) \), we can visualize its behavior over different regions to predict its value, making complex analysis an important tool for comprehending the problem's solution.

Modulus and Argument

The modulus and argument are fundamental properties when dealing with complex numbers. The modulus of a complex number \( z = x + iy \) is given by \( |z| = \sqrt{x^2 + y^2} \), representing the distance from the origin (0,0) to the point \( (x,y) \) in the complex plane.

The argument, typically denoted as \( \operatorname{Arg}(z) \), describes the angle formed with the positive x-axis, calculated by \( \operatorname{Arg}(z) = \tan^{-1}\left(\frac{y}{x}\right) \). However, care must be taken as the angle can wrap around the circle, leading to different values (differing by multiples of \( 2\pi \)). This concept is crucial to solving our exercise as it involves evaluating the function \( f(z) = (z^2-1)^{1/2} \) based on its modulus and argument.

In this problem, we express the complex square root of \( z^2 - 1 \) through its modulus \( |z^2-1|^{1/2} \) and half-angle argument \( \frac{1}{2}\operatorname{Arg}(z^2-1) \). By breaking down the function into its real and imaginary parts using these properties—\( f(t) = |t^2-1|^{1/2} \cos\left(\frac{1}{2}\operatorname{Arg}(t^2-1)\right) + i |t^2-1|^{1/2}\sin\left(\frac{1}{2}\operatorname{Arg}(t^2-1)\right) \)—we gain insights into why its behavior differs based on \( |t| \) values. This understanding is key to analyzing the nature of boundary regions where \( \text{Im}(G(z))=0 \), as it influences whether the function appears purely real or imaginary.

Branch Cuts

Branch cuts are critical when dealing with multi-valued complex functions, like the square root. A branch cut is a curve starting at a branch point along which a function is discontinuous. For square root functions, branch cuts serve to prevent \( f(z) \) from being ill-defined in its multi-valued nature.

In our exercise, the function \( f(z) = (z^2-1)^{1/2} \) provides a clear example of where branch cuts are necessary. The term \( z^2-1 \) indicates branch points at \( z = \pm1 \). To make the function single-valued and thus well-defined, branch cuts along the real axis between these points are typically established, influencing how the function behaves for \( |t| > 1 \) (purely real) and \( |t| < 1 \) (purely imaginary).

The correct placement of branch cuts ensures that each point in the plane maps uniquely to a value of \( f(z) \). Without branch cuts, values could continuously change, resulting in an infinitely looping route around branch points. In solving the exercise, recognizing branch cuts helps determine where the function transitions and resolves ambiguity when evaluating plots or specific values of \( G(z) \) and their imaginary components. Understanding branch cuts is crucial for demonstrating both theoretical prowess and practical application in complex analysis.