Obligatory Equations Flashcards
(x-a)^2 + (y - b)^2 = radius^2
equation of a circle,
where a = x coordinate of the center of a circle
b = y coordinate of the center of a circle
x^2 + y^2 has its center …
in the origin
wn = …
e^(i*2pi/n)
e^(i*theta) = …
cos(theta) + i*sin(theta)
** in radians
When finding complex roots,
r =
theta =
r = nth root(r0) theta = theta0/n + 2pik/n
where k = 0, +/-1, +/- 2, …
z = ….
r(cos theta + i*sin theta)
The complex term “a” = …
r*cos(theta)
The complex term “b” = …
r*sin(theta)
When talking about complex numbers, theta means the same as …
arg(z)
Two values of the argument of a complex number will differ from each other by an integer multiple of
2*pi
The principal value of a negative number is ….
pi
The principal value of the argument (sometimes called the principal argument) is …
the unique value of the argument that is in the range -pi < arg z <= pi
The principal value is denoted by …
Arg(z)
arg(z) =
Arg(z) + 2pin
if θ1 and θ2 are two values of arg z, then for some integer k, we will have …
θ1 − θ2 = 2πk
If you’re using a calculator to calculate the principal value, be careful because …
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.
The principal value for a purely imaginary number is …
π/2
Euler’s Formula
e^(iθ) = cosθ + i*sinθ
z = …
z = r*e^(iθ)
Find the modulus and an argument of 5, 8i, −3, and −7i.
|5| = 5, θ = 0 |8i| = 8, θ= π/2 |−3| = 3, θ = π |−7i| = 7, θ = 3π/2
If using arctan to calculate the principal value, then ... First quadrant: θ = Second quadrant: θ = Third quadrant: θ = Fourth quadrant: θ =
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).
Be careful when a=0, i.e. the complex number has no real part because …
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.
