Write each of the following complex numbers in the exponential form [math]|z|e^{it}[/math].

• [math]1+i[/math]
• [math]1-i[/math]
• [math]1+i\sqrt3[/math]
• [math]-2\sqrt3+2i[/math]
• [math]-5[/math]
• [math]e^x[/math], where [math]z=2+i\frac{pi}4[/math]
• [math]5[/math]
• [math]i[/math].

Let [math]z_1=\sqrt3+i[/math] and [math]z_2=1-i[/math]. Write each of the following complex numbers in the exponential form [math]|z|e^{it}[/math] and plot it in the complex plane.

• [math]z_1[/math]
• [math]z_2[/math]
• [math]z_1z_2[/math]
• [math]\frac{z_1}{z_2}[/math]
• [math]2z_1[/math]
• [math](z_1)^6[/math].

Find the real and imaginary parts of each of the following complex numbers.

• [math]e^{i\pi}[/math]
• [math]2e^{i(\frac{\pi}4)}[/math]
• [math]e^{i\pi}2e^{i(\frac{\pi}4)}[/math]
• [math]e^{i\pi} + 2e^{i(\frac{\pi}4)}[/math]
• [math]\sqrt{34}e^{it}[/math], where [math]t=\arcsin \frac{5}{\sqrt{34}}[/math]
• [math]\sqrt{13}e^{it}[/math], where [math]\sin t= \frac{-2}{\sqrt{13}}[/math].

If [math]z_1=|z_1|e^{it}[/math], what is the exponential form of its complex conjugate [math]\conj{z_1}[/math]?

Derive \ref{thm 6.7.3'} and \ref{thm 6.7.4'} using \ref{thm 6.7.1'} and \ref{thm 6.7.2'}.

Let [math]n[/math] be a positive integer, and let [math]z^n[/math] be defined as in the text. If [math]z \ne 0[/math], define [math]z^{-n} = \frac1{z^n}[/math], and then show that [math](e^z)^{-n} = e^{-nz}[/math]. As a result, we know that Theorem \ref{thm 6.7.7} holds for all integers.

Find and plot the [math]n[/math]th roots of [math]z[/math] in each of the following cases.

• [math]n=3[/math] and [math]z=8i[/math]
• [math]n=2[/math] and [math]z=i[/math]
• [math]n=3[/math] and [math]z=2[/math]
• [math]n=4[/math] and [math]z=1[/math]
• [math]n=5[/math] and [math]z=2i[/math]
• [math]n=3[/math] and [math]z=1+i\sqrt3[/math].

How would you define the function [math]5^z[/math]?

• Using the equation [math]e^{ix}=\cos x + i \sin x[/math] and the fact that [math](e^{ix})^n = e^{inx}[/math], prove that
  [[math]] \cos nx + i \sin nx = (\cos x + i \sin x)^n . [[/math]]
  This is known as de Moivre's Formula.
• Using de Moivre's Formula and the Binomial Theorem, find trigonometric identities for [math]\cos 3x[/math] and [math]\sin 3x[/math] in terms of [math]\cos x[/math] and [math]\sin x[/math].

Every complex-valued function [math]f[/math] of a real variable determines two real-valued functions [math]f_1[/math] and [math]f_2[/math] of a real variable defined by

Thus [math]f(x) = f_1(x) + if_2(x)[/math] for every [math]x[/math] in the domain of [math]f[/math]. We define the derivative [math]f^\prime[/math] by the formula

Applying this definition to the function [math]f(x) = e^{ix}[/math], show that