Obligatory Equations

Question 1

(x-a)^2 + (y - b)^2 = radius^2

Answer 1

equation of a circle,

where a = x coordinate of the center of a circle
b = y coordinate of the center of a circle

Question 2

x^2 + y^2 has its center …

Answer 2

in the origin

Question 3

wn = …

Answer 3

e^(i*2pi/n)

Question 4

e^(i*theta) = …

Answer 4

cos(theta) + i*sin(theta)

** in radians

Question 5

When finding complex roots,

r =
theta =

Answer 5

r = nth root(r0)
theta = theta0/n + 2pik/n 

where k = 0, +/-1, +/- 2, …

Question 6

z = ….

Answer 6

r(cos theta + i*sin theta)

Question 7

The complex term “a” = …

Answer 7

r*cos(theta)

Question 8

The complex term “b” = …

Answer 8

r*sin(theta)

Question 9

When talking about complex numbers, theta means the same as …

Answer 9

arg(z)

Question 10

Two values of the argument of a complex number will differ from each other by an integer multiple of

Answer 10

2*pi

Question 11

The principal value of a negative number is ….

Answer 11

pi

Question 12

The principal value of the argument (sometimes called the principal argument) is …

Answer 12

the unique value of the argument that is in the range -pi < arg z <= pi

Question 13

The principal value is denoted by …

Answer 13

Arg(z)

Question 14

arg(z) =

Answer 14

Arg(z) + 2pin

Question 15

if θ1 and θ2 are two values of arg z, then for some integer k, we will have …

Answer 15

θ1 − θ2 = 2πk

Question 16

If you’re using a calculator to calculate the principal value, be careful because …

Answer 16

You will need to compute both θ1 and θ2 then determine which falls into the correct quadrant to match the complex number we have because only one of them will be in the correct quadrant.

Question 17

The principal value for a purely imaginary number is …

Answer 17

π/2

Question 18

Euler’s Formula

Answer 18

e^(iθ) = cosθ + i*sinθ

Question 19

z = …

Answer 19

z = r*e^(iθ)

Question 20

Find the modulus and an argument of 5, 8i, −3, and −7i.

Answer 20

|5| = 5, θ = 0
|8i| = 8, θ= π/2
|−3| = 3, θ = π
|−7i| = 7, θ = 3π/2

Question 21

If using arctan to calculate the principal value, then ...
First quadrant: θ =
Second quadrant: θ = 
Third quadrant: θ =
Fourth quadrant: θ =

Answer 21

First quadrant: θ = arctan(b/a).
Second quadrant: θ = arctan(b/a) + π.
Third quadrant: θ = arctan(b/a) − π.
Fourth quadrant: θ = arctan(b/a).

Question 22

Be careful when a=0, i.e. the complex number has no real part because …

Answer 22

In this case, the arctan method doesn’t work, but the argument is either π/2 or −π/2 for numbers with positive and negative imaginary parts, respectively.