Y2, C1 - Complex Numbers Flashcards
If z = x + iy, what is the argument
θ = arg(z) = tan^-1 (y / x)
If z = x + iy, what is the modulus
r = l z l = root(x^2 + y^2)
What is the range for the principal argument
-pi < theta < pi
What is the modulus argument form of a complex number
z = r(cos(θ) + isin(θ)
What is the cartesian form of a complex number
z = x + iy
What is the exponential form of a complex number
z = re^iθ
Where r is the modulus
θ is the argument
Using Euler’s formula how can e^ix be written
e^ix = cosx + i sinx
What is Euler’s identity
e^ipi + 1 = 0
What constant is cosθ + isinθ equal to (exponential form)
e^iθ
How do you multiply complex numbers
Multiply the moduli and add the arguments
How do you divide complex numbers
Divide the moduli and subtract the arguments
How can cosθ - isinθ be written
cos(-θ) + isin(-θ)
What is De Moivre’s theorem
z^n = r^n (cos(nθ) + isin(nθ))
z^n = r^n * e^inθ
Steps for expressing cos(3θ) in terms of powers of cos(θ)
1) Create a De Moivre statement that includes cos(3θ) on the RHS
2) Binomial expansion
3) Compare real / imaginary parts
What is (cosθ + isinθ)^6 equal to
cos6θ +isin6θ
What is the De Moivre statement when expressing cos(3θ) in terms of powers of cos(θ)
(cosθ + isinθ)^3 = cos3θ + isin3θ
If z = cosθ + isinθ, what is z + 1/z
2cosθ
If z = cosθ + isinθ, what is z - 1/z
2isinθ
If z = cosθ + isinθ, what is z^n + 1/z^n
2cos(nθ)
If z = cosθ + isinθ. what is z^n - 1/z^n
2isin(nθ)
How would you express cos^5(θ) in the form acos5θ + bcos3θ + ccosθ
1) Raise RHS to required power (2cosθ)^5)
2) Raise LHS to the same power
(z + 1/z)^5)
3) Binomial expansion
4) Use the identities once again
5) Remember to isolate by dividing by any coefficient on LHS
What is 2cosnθ equal to
(exponential form)
e^niθ + e^-niθ
What is 2sinnθ equal to
(exponential form)
e^niθ - e^-niθ
How would you express 3 / (e^2iθ - 1) in hyperbolic form
Multiply by e to the power of half the negated power (e^-iθ)
Then sub in 2isinθ as the denominator
How would you express 3 / (e^4iθ + 2) in hyperbolic form
Multiply by the same expression but with the power of e negated (e^-4iθ + 2)
When would you multiply by e to the power of half the negated power to get an imaginary number into hyperbolic form
When there is a 1+, 1-, or -1 with the coefficient of e^inθ as 1
How to solve z^3 = 1
z^3 = e^0i
z = (e^0i)^1/3, +2pi/3 = z = (e^i2pi)^1/3
z = 1, z = e^i2pi/3, z = e^i4pi/3
How would solutions to z^n = 1 look on an Argand diagram
n-sided polygon
centre at origin
1 always a root
What is the difference in argument for each root of a root of unity (z^n = 1)
2pi / n
What is omega (w) in roots of unity
The first root (z1 = 1)
e^i2pi/n = w
What do the roots of unity sum to
0
What does multiplying by omega (w) do
Rotates the complex number by 2pi / n radians
Moves it along to the next vertex of the root
How do you find the vertexes of a polygon which doesn’t have its centre at the origin
Move the centre to the origin
Move the known vertex to the corresponding point
Work out the other vertexes
Transform them back to the correct place
How many times do you have to multiply by omega (w) to reach z1 (1)
n times
w^n = 1
If z^6 = 1
w^6 = 1
