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Chapter 7: Problem 2

Write the following in polar form, \(z=r e^{i \theta}\). a. \(i-1\). b. \(-2 i\). c. \(\sqrt{3}+3 i\).

Short Answer

Expert verified
\(i-1 = \sqrt{2} e^{i \frac{3\pi}{4}}\), \(-2i = 2 e^{i \frac{3\pi}{2}}\), and \(\sqrt{3} + 3i = 2\sqrt{3} e^{i \frac{\pi}{3}}\).

Step by step solution

01

Compute the modulus

Find the modulus (or magnitude) of the complex number using the formula \(r=\sqrt{a^2 + b^2}\). For example, for the complex number \(i-1\), \(a=-1\) and \(b=1\). Hence, \(r=\sqrt{(-1)^2 + (1)^2} = \sqrt{2}\).
02

Calculate the Argument

Calculate the argument (or angle) of the complex number with the formula \(\theta = \arctan(\frac{b}{a})\) for \(a>0\), and \(\theta = \arctan(\frac{b}{a})+\pi\) for \(a<0\). For the complex number \(i-1\), \(a=-1\) and \(b=1\). Since \(a\) is negative, we get \(\theta = \arctan(\frac{1}{-1})+\pi = \frac{3\pi}{4}\).
03

Write the polar form

The polar form of a complex number is \(z=r e^{i \theta}\). For the complex number \(i-1\), \(r=\sqrt{2}\) and \(\theta = \frac{3\pi}{4}\), so the polar form is \(z=\sqrt{2} e^{i \frac{3\pi}{4}}\).
04

Repeat steps for all complex numbers

Repeat Steps 1, 2 and 3 for the remaining complex numbers. For \(-2i\), \(r=2\) and \(\theta = \frac{3\pi}{2}\), so the polar form is \(z=2 e^{i \frac{3\pi}{2}}\). For \(\sqrt{3} + 3i\), \(r=2\sqrt{3}\) and \(\theta = \frac{\pi}{3}\), so the polar form is \(z=2\sqrt{3} e^{i \frac{\pi}{3}}\).

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Most popular questions from this chapter

Let \(f(z)=u+i v\) be differentiable. Consider the vector field given by \(\mathbf{F}=v \mathbf{i}+u \mathbf{j}\). Show that the equations \(\nabla \cdot \mathbf{F}=\mathbf{0}\) and \(\nabla \times \mathbf{F}=\mathbf{0}\) are equivalent to the Cauchy-Riemann Equations. [You will need to recall from multivariable calculus the del operator, \(\left.\nabla=\mathbf{i} \frac{\partial}{\partial x}+\mathrm{j} \frac{\partial}{\partial y}+\mathbf{k} \frac{\partial}{\partial z} \cdot\right]\)

Show that $$ \int_{C} \frac{d z}{(z-1-i)^{n+1}}=\left\\{\begin{array}{cl} 0, & n \neq 0 \\ 2 \pi i, & n=0 \end{array}\right. $$ for \(C\) the boundary of the square \(0 \leq x \leq 2,0 \leq y \leq 2\) taken counterclockwise. [Hint: Use the fact that contours can be deformed into simpler shapes (like a circle) as long as the integrand is analytic in the region between them. After picking a simpler contour, integrate using parametrization.].

Show that \(\sin (x+i y)=\sin x \cosh y+i \cos x \sinh y\) using trigonometric identities and the exponential forms of these functions.

Let \(C\) be the positively oriented ellipse \(3 x^{2}+y^{2}=9\). Define $$ F\left(z_{0}\right)=\int_{C} \frac{z^{2}+2 z}{z-z_{0}} d z $$

Write the equation that describes the circle of radius 3 that is centered at \(z=2-i\) in (a) Cartesian form (in terms of \(x\) and \(y\) ); (b) polar form (in terms of \(\theta\) and \(r\) ); (c) complex form (in terms of \(z, r\), and \(e^{i \theta}\) ).
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