Complex Analysis : Complex Analysis

Study concepts, example questions & explanations for Complex Analysis

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Example Questions

Example Question #11 : Complex Numbers

Evaluate:

\displaystyle \left| ( 8 + 5 i ) ( -10 + 3 i ) \right|

Possible Answers:

\displaystyle \sqrt{9701}

\displaystyle - \sqrt{9701}

\displaystyle 9701

\displaystyle \frac{\sqrt{9701}}{2}

Correct answer:

\displaystyle \sqrt{9701}

Explanation:

The general formula to figure out the modulus is

\displaystyle \left|a+bi\right|=\sqrt{a^2+b^2}

We apply this to get

\displaystyle \left| ( 8 + 5 i ) ( -10 + 3 i ) \right| = \sqrt{ 64 + 25 }\cdot \sqrt{ 100 + 9 }

\displaystyle = \sqrt{89} \cdot \sqrt{109}

\displaystyle = \sqrt{9701}

Example Question #11 : Complex Analysis

Evaluate: \displaystyle \left| 9 - 7 i \right|

Possible Answers:

\displaystyle 65

\displaystyle \sqrt{130}

\displaystyle -\sqrt{130}

\displaystyle 130

Correct answer:

\displaystyle \sqrt{130}

Explanation:

The general formula to figure out the modulus is

\displaystyle \left|a+bi\right|=\sqrt{a^2+b^2}.

We apply this notion to get.

\displaystyle \left| 9 - 7 i \right| = \sqrt{ 81 + 49 }

\displaystyle = \sqrt{130}

Example Question #12 : Complex Analysis

Evaluate \displaystyle (\sqrt{3} - i)^6

Possible Answers:

-64i

64

64i

-64

None of the other answers

Correct answer:

-64

Explanation:

Converting from rectangular to polar coordinates gives us

\displaystyle r = \sqrt{x^2+y^2} = \sqrt{3+1} = 2

\displaystyle \theta = tan^{-1}(y/x) = tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6}

So 

\displaystyle (\sqrt{3}-i)^6 = (2e^{-\frac{\pi}{6}i})^6 = 64e^{-\pi i} = -64

Example Question #14 : Complex Numbers

Compute \displaystyle (-16)^\frac{1}{4}

Possible Answers:

\displaystyle \pm\sqrt{2}(1-i)

\displaystyle \pm(\sqrt{3} - i) \text{ and } \pm(1+\sqrt{3}i)

\displaystyle \pm 2 \text{ and } \pm 2i

\displaystyle \pm\sqrt{2}(1+i)

\displaystyle \pm\sqrt{2}(1+i) \text{ and } \pm\sqrt{2}(1-i)

Correct answer:

\displaystyle \pm\sqrt{2}(1+i) \text{ and } \pm\sqrt{2}(1-i)

Explanation:

Converting from Rectangular to Polar Coordinates

 

\displaystyle (-16)^\frac{1}{4} = (16e^{i\pi + 2\pi k})^\frac{1}{4} = 2e^{i\frac{\pi}{4} + \frac{\pi}{2}k}

Evaluating for \displaystyle k = 0,1,2,3

we get that 

\displaystyle \pm\sqrt{2}(1+i) \text{ and } \pm\sqrt{2}(1-i)

Example Question #13 : Complex Analysis

The 5th roots of unity are the five unique solutions to which equation?

Possible Answers:

\displaystyle (z-1)^5=0

\displaystyle z^5+1=0

\displaystyle z^5-1=0

\displaystyle z^5+z^4+z^3+z^2+z+1=0

None of these

Correct answer:

\displaystyle z^5-1=0

Explanation:

A 5th root of unity is a complex number \displaystyle z such that \displaystyle z^5=1. Manipulating this equation yields \displaystyle z^5-1=0.

Example Question #16 : Complex Numbers

What is the magnitude of the following complex number?

\displaystyle 5+2i

Possible Answers:

\displaystyle \sqrt{14}

\displaystyle \sqrt{29}

\displaystyle 7

None of these

\displaystyle 1

Correct answer:

\displaystyle \sqrt{29}

Explanation:

The magnitude of a complex number \displaystyle a+bi is defined as

\displaystyle \vert a+bi\vert=\sqrt{a^2+b^2}.

So the modulus of \displaystyle 5+2i is

\displaystyle \vert5+2i\vert=\sqrt{5^2+2^2}=\sqrt{25+4}=\sqrt{29}.

Example Question #14 : Complex Numbers

What is the magnitude of the following complex number?

\displaystyle 3-i

Possible Answers:

\displaystyle \sqrt{10}

\displaystyle 2

None of these

\displaystyle \sqrt{8}

\displaystyle 4

Correct answer:

\displaystyle \sqrt{10}

Explanation:

The magnitude of a complex number \displaystyle a+bi is defined as

\displaystyle \vert a+bi\vert=\sqrt{a^2+b^2}.

So the modulus of \displaystyle 3-i is

\displaystyle \vert3-i\vert=\sqrt{3^2+(-1)^2}=\sqrt{9+1}=\sqrt{10}.

Example Question #17 : Complex Numbers

What is the magnitude of the following complex number?

\displaystyle 8

Possible Answers:

\displaystyle 2\sqrt2

\displaystyle 0

\displaystyle 8

\displaystyle 64

None of these

Correct answer:

\displaystyle 8

Explanation:

The magnitude of a complex number \displaystyle a+bi is defined as

\displaystyle \vert a+bi\vert=\sqrt{a^2+b^2}.

Because the complex number \displaystyle 8 has no imaginary part, we can write it in the form \displaystyle 8+0i. Then the modulus of \displaystyle 8+0i is

\displaystyle \vert8+0i\vert=\sqrt{8^2+0^2}=\sqrt{64+0}=\sqrt{64}=8.

Example Question #18 : Complex Numbers

What is the argument of the following complex number?

\displaystyle 1+i

Possible Answers:

\displaystyle 2

\displaystyle \frac{-3\pi}{4}

\displaystyle \frac{\pi}{4}

None of these

\displaystyle \frac{\pi}{2}

Correct answer:

\displaystyle \frac{\pi}{4}

Explanation:

Note that the complex number \displaystyle 1+i lies in the first quadrant of the complex plane.

 

For a complex number \displaystyle z=a+bi, the argument of \displaystyle z is defined as the real number \displaystyle \theta such that

\displaystyle \tan(\theta)=\frac{b}{a},

where \displaystyle -\pi< \theta\leq\pi is in radians.

 

Then the argument of \displaystyle 1+i is

\displaystyle \tan(\theta)=\frac{1}{1}=1

\displaystyle \theta=\frac{-3\pi}{4},\frac{\pi}{4}.

 

The angle \displaystyle \frac{-3\pi}{4} lies in the third quadrant of the complex plane, but the angle \displaystyle \frac{\pi}{4} lies in the first quadrant, as does \displaystyle 1+i. So \displaystyle \theta=\frac{\pi}{4}.

Example Question #19 : Complex Numbers

What is the argument of the following complex number in radians, rounded to the nearest hundredth?

\displaystyle 2-3i

Possible Answers:

\displaystyle 3.61

None of these

\displaystyle -0.98

\displaystyle -0.59

\displaystyle 2.16

Correct answer:

\displaystyle -0.98

Explanation:

Note that the complex number \displaystyle 2-3i lies in the fourth quadrant of the complex plane.

 

For a complex number \displaystyle z=a+bi, the argument of \displaystyle z is defined as the real number \displaystyle \theta such that

\displaystyle \tan(\theta)=\frac{b}{a},

where \displaystyle -\pi< \theta\leq\pi is in radians.

 

Then the argument of \displaystyle 2-3i is

\displaystyle \tan(\theta)=\frac{-3}{2}

\displaystyle \theta=-0.98,2.16.

 

The angle \displaystyle 2.16 lies in the second quadrant of the complex plane, but the angle \displaystyle -0.98 lies in the fourth quadrant, as does \displaystyle 2-3i. So \displaystyle \theta=-0.98.

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