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Chapter 7: Problem 1

Write the following in standard form. a. \((4+5 i)(2-3 i)\) b. \((1+i)^{3}\) c. \(\frac{5+3 i}{1-i}\).

Short Answer

Expert verified
The answers are: \n(a) -7 + 22i \n(b) -2 + 2i \n(c) 4 + 2i.

Step by step solution

01

Multiply

For (a), multiply the given complex numbers as per the rule of multiplication of complex numbers. (a+bj)(c+dj) = ac-bd + j(ad+bc). Thus, \((4+5i)(2-3i)\) = 4*2 - 5*3 + i(4*3 + 5*2) = -7 + 22i.
02

Exponentiate

For (b), calculate the cube of the complex number as (a+bj)^3 = a^3 + 3a^2bj + 3ab^2j^2 + b^3j^3. Given number is \(1+i\), so (1+i)^3 = 1 + 3*1*1i + 3*1*1*(-1) + 1*(-1) = -2 + 2i.
03

Divide

For (c), divide the complex numbers as per the rule of division of complex numbers. \(\frac{a+bj}{c+dj}\) = \(\frac{ac+bd}{c^{2}+d^{2}}\) + j\(\frac{bc-ad}{c^{2}+d^{2}}\). Thus, \(\frac{5+3i}{1-i}\) = \(\frac{5*1 + 3*1}{1^2 + 1^2}\) + j\(\frac{3*1 - 5(-1)}{1^2 + 1^2}\) = 4 + 2i.

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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.]

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

Find series representations for all indicated regions. a. \(f(z)=\frac{z}{z-1},|z|<1,|z|>1\). b. \(f(z)=\frac{1}{(z-1)(2+2)},|z|<1,1<|z|<2,|z|>2\). [Hint Use partial fractions to write this as a sum of two functions first.]

. 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. \(\cos \frac{\pi}{\pi-\pi}, \quad z=\pi\)

Evaluate the following integrals: a. \(\int_{C} \bar{z} d z\), where \(C\) is the parabola \(y=x^{2}\) from \(z=0\) to \(z=1+i\). b. \(\int_{C} f(z) d z\), where \(f(z)=2 z-\bar{z}\) and \(C\) is the path from \(z=0\) to \(z=2+i\) consisting of two line segments from \(z=0\) to \(z=2\) and then \(z=2\) to \(z=2+i\) c. \(\int_{C} \frac{1}{x^{2}+4} d z\) for \(C\) the positively oriented circle, \(|z|=2\). [Hint: Parametrize the circle as \(z=2 e^{i \theta}\), multiply numerator and denominator by \(e^{-i \theta}\), and put in trigonometric form.]
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