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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

. For the following, determine if the given point is a removable singularity, an essential singularity, or a pole (indicate its order). a. \(\frac{1-\cos z}{z^{2}}, \quad z=0\) b. \(\frac{\sin 2}{z^{2}}, \quad z=0\) c. \(\frac{z^{2}-1}{(z-1)^{2}}, \quad z=1\). d. \(z e^{1 / z}, \quad z=0\). e. \(\cos \frac{\pi}{\pi-\pi}, \quad z=\pi\)

Find the Laurent series expansion for \(f(z)=\frac{\sinh z}{2^{2}}\) about \(z=0\). [Hint You need to first do a MacLaurin series expansion for the hyperbolic sine.]

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Show that for \(g\) and \(h\) analytic functions at \(z_{0}\), with \(g\left(z_{0}\right) \neq 0, h\left(z_{0}\right)=0\), and \(h^{\prime}\left(z_{0}\right) \neq 0\) $$ \operatorname{Res}\left[\frac{g(z)}{h(z)} ; z_{0}\right]=\frac{g\left(z_{0}\right)}{h^{\prime}\left(z_{0}\right)} $$
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