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
