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Chapter 10: Problem 20

Find the curve defined by each of the following equations. (a) \(z=1-i t, \quad 0 \leq t \leq 2\), (b) \(z=t+i t^{2}, \quad-\infty

Short Answer

Expert verified
(a) results in a line segment from z=1 to z=1-2i, (b) forms a parabolic curve that covers the entire complex plane, (c) gives a semicircular arc from z=-ai to z=ai, and (d) yields a hyperbolic curve in the third and fourth quadrants.

Step by step solution

01

Solve (a)

For the equation \(z=1-it, \quad 0 \leq t \leq 2\), substitute the value of t from the given range into the equation to obtain the mapping of the curve. This results in the mapping of a line segment from the point z=1 in the complex plane to the point z=1-2i.
02

Solve (b)

For the equation \(z=t+it^{2}, \quad-\infty<t<\infty\), substitute varying values of t from the given range into the equation to plot the curve. This forms a parabolic curve due to the square of t in the imaginary part of the function. The curve encompasses the entire complex plane since t ranges from negative to positive infinity.
03

Solve (c)

For the equation \(z=a(\cos t+i \sin t), \quad \frac{\pi}{2} \leq t \leq \frac{3 \pi}{2}\), substitute varying values of t from the given range into the equation to plot the points of the curve. This forms a semicircular arc in the complex plane due to the cosine and sine function in the equation. The arc starts from the point z=-ai and ends at the point z=ai, centered at the origin.
04

Solve (d)

For the equation \(z=t+\frac{i}{t}, \quad-\infty<t<0\), substitute varying negative values of t into the equation to plot the points of the curve. This forms a hyperbolic curve in the complex plane. The curve is located in the third and fourth quadrants due to the negative values of t.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are numbers that consist of a real part and an imaginary part. They are typically written in the form of \( a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \).

Complex numbers can be plotted on a plane known as the complex plane, where the x-axis represents the real part of the number and the y-axis represents the imaginary part. This graphical representation makes it possible to visualize complex number operations and provides a deeper understanding of complex functions and their transformations.
Parametric Equations
Parametric equations are a set of equations that express the coordinates of a point as functions of a variable, referred to as the parameter. For complex numbers, parametric equations allow us to define a complex number \( z \) in terms of a real parameter \( t \), often as \( z(t) = x(t) + i y(t) \), where \( x(t) \) and \( y(t) \) are the real functions of \( t \) representing the real and imaginary parts of \( z \), respectively.

This approach is essential in plotting complex curves since it provides a way to express both the real and imaginary parts of a complex function in relation to a single variable, enabling the creation of a path in the complex plane as the parameter varies.
Complex Plane Mappings
Complex plane mappings are the transformations that occur when a complex function is applied to a number in the complex plane. These mappings can represent various shapes and paths, such as lines, circles, and more intricate curves.

For example, in exercise (a), the line segment map is created by transforming points through the complex function \( z(t) = 1 - it \). As the parameter \( t \) increases from 0 to 2, the imaginary part changes linearly, mapping out a straight line on the complex plane. In complex analysis, these mappings are fundamental concepts, as they reveal the structure of complex functions and how they alter the plane.
Graphing in Complex Analysis
Graphing in complex analysis involves plotting complex functions or sets of points on the complex plane. It's a visual process that aids in the understanding of the behavior of complex functions. By graphing, we can identify important features of functions, such as zeros, poles, and essential singularities, as well as observe the geometric transformations induced by the functions.

When graphing, it's important to consider the range of the parameter and the nature of the function being plotted. For instance, exercise (c) requires us to plot a semicircular arc, which is derived from the parametric equations involving cosine and sine functions. This arc represents how a circle, centered at the origin and with a radius \( a \), is mapped onto the complex plane within the specified interval for \( t \). The effective graphing of such curves is imperative for students to grasp the multifaceted nature of complex functions and their applications.

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Most popular questions from this chapter

Find the value of the integral \(\int_{C}[(z+2) / z] d z\), where \(C\) is (a) the semicircle \(z=2 e^{i \theta}\), for \(0 \leq \theta \leq \pi\), (b) the semicircle \(z=2 e^{i \theta}\), for \(\pi \leq \theta \leq 2 \pi\), and (c) the circle \(z=2 e^{i \theta}\), for \(-\pi \leq \theta \leq \pi\).

Use the definition of an integral as the limit of a sum and the fact that absolute value of a sum is less than or equal to the sum of absolute values to prove the Darboux inequality.

Let \(f(t)=u(t)+i v(t)\) be a (piecewise) continuous complex-valued function of a real variable \(t\) defined in the interval \(a \leq t \leq b\). Show that if \(F(t)=U(t)+i V(t)\) is a function such that \(d F / d t=f(t)\), then $$\int_{a}^{b} f(t) d t=F(b)-F(a) .$$ This is the fundamental theorem of calculus for complex variables.

What is the largest circle within which the Maclaurin series for tanh \(z\) converges to \(\tanh z\) ?

In this problem, you will find the capacitance per unit length of two cylindrical conductors of radii \(R_{1}\) and \(R_{2}\) the distance between whose centers is \(D\) by looking for two line charge densities \(+\lambda\) and \(-\lambda\) such that the two cylinders are two of the equipotential surfaces. From Problem \(10.10\), we have $$R_{i}=\frac{a}{\sinh \left(u_{i} / 2 \lambda\right)}, \quad y_{i}=a \operatorname{coth}\left(u_{i} / 2 \lambda\right), \quad i=1,2,$$ where \(y_{1}\) and \(y_{2}\) are the locations of the centers of the two conductors on the \(y\) -axis (which we assume to connect the two centers). (a) Show that \(D=\left|y_{1}-y_{2}\right|=\left|R_{1} \cosh \frac{u_{1}}{2 \lambda}-R_{2} \cosh \frac{u_{2}}{2 \lambda}\right|\). (b) Square both sides and use \(\cosh (a-b)=\cosh a \cosh b-\sinh a \sinh b\) and the expressions for the \(R\) 's and the \(y\) 's given above to obtain $$\cosh \left(\frac{u_{1}-u_{2}}{2 \lambda}\right)=\left|\frac{R_{1}^{2}+R_{2}^{2}-D^{2}}{2 R_{1} R_{2}}\right|$$ (c) Now find the capacitance per unit length. Consider the special case of two concentric cylinders. (d) Find the capacitance per unit length of a cylinder and a plane, by letting one of the radii, say \(R_{1}\), go to infinity while \(h \equiv R_{1}-D\) remains fixed.
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