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Chapter 7: Problem 3

Write the following in rectangular form, \(z=a+i b\). a. \(4 e^{i \pi / 6}\) b. \(\sqrt{2} e^{5 i \pi / 4}\) c. \((1-i)^{100} .\)

Short Answer

Expert verified
a. \(2\sqrt{3} + 2i\), b. \(-1 + i\), c. \(-1\)

Step by step solution

01

Problem a: Convert \(4 e^{i \pi / 6}\) to Rectangular Form

From Euler's formula, this can be written as \(4(\cos(\pi / 6) + i \sin(\pi / 6))\). Then, using the values cos(\(\pi / 6\) = \(\sqrt{3}/ 2\), and \sin(\pi / 6) = 0.5, simplification gives \(2\sqrt{3} + 2i\).
02

Problem b: Convert \(\sqrt{2} e^{5 i \pi / 4}\) to Rectangular Form

Using Euler's formula, this can be written as \(\sqrt{2}(\cos(5\pi / 4) + i \sin(5 \pi / 4))\). Knowing that \cos(\(5 \pi / 4\)) = -\(\frac1{\sqrt2}\), and \sin(\(5 \pi / 4\)) = \(\frac1{\sqrt2}\), simplification gives \(-1 + i\).
03

Problem c: Convert \((1-i)^{100}\) to Rectangular Form

The first step is to write \(1 - i\) in polar form. This gives \(e^{7i\pi / 4}\), due to \(\cos(7 \pi / 4) = \sqrt{2}/ 2\) and \(\sin(7 \pi / 4) = - \sqrt{2}/ 2\). Raising this to the power 100 gives \(e^{175i\pi} = (e^{i\pi})^{175}\).\nThe 175 is an odd number, and we know that \(e^{i\pi} = -1\), so it can be simplified as \(-1\).

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Most popular questions from this chapter

Consider the circle \(|z-1|=1\) a. Rewrite the equation in rectangular coordinates by setting \(z=\) \(x+i y\) b. Sketch the resulting circle using part a. c. Consider the image of the circle under the mapping \(f(z)=z^{2}\), given by \(\left|z^{2}-1\right|=1\) i. By inserting \(z=r e^{i \theta}=r(\cos \theta+i \sin \theta)\), find the equation of the image curve in polar coordinates. ii. Sketch the image curve. You may need to refer to your Calculus. II text for polar plots. [Maple might help.]

What parametric curve is described by the function $$ \gamma(t)=(t-3)+i(2 t+1) $$ \(0 \leq t \leq 2\) ? [Hint: What would you do if you were instead considering the parametric equations \(x=t-3\) and \(y=2 t+1]\)

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.].

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 \(e^{i \theta}\) ).

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