Year 2 Chapter 1: Complex Numbers

Question 1

What is cosθ as an infinite series of powers of θ?

Answer 1

cosθ = 1 - (θ^2)/(2!) + (θ^4)/(4!) - (θ^6)/(6!) + …

Question 2

What is sinθ as an infinite series of powers of θ?

Answer 2

sinθ = θ - (θ^3)/(3!) + (θ^5)/(5!) - (θ^7)/(7!) + …

Question 3

What is e^𝑥 as a series expansion in powers of 𝑥?

Answer 3

e^𝑥 = 1 + 𝑥 + (𝑥^2)/(2!) + (𝑥^3)/(3!) + (𝑥^4)/(4!) + (𝑥^5)/(5!) + …

Question 4

What is Euler’s Relation?

Answer 4

e^iθ = cosθ + isinθ

Question 5

How can you use Euler’s relation to write a complex number “z” in exponential form?

Answer 5

z = re^iθ

r = |z|
θ = arg z

Question 6

What is the modulus argument form when multiplying z1 by z2?

Answer 6

z1z2 = r1r2(cos(θ1 + θ2) + isin(θ1 + θ2)

Question 7

What is the modulus argument form when dividing z1 by z2?

Answer 7

z1/z2 = r1/r2(cos(θ1 - θ2) + isin(θ1 - θ2)

Question 8

When z1 = r1e^iθ1and z2 = r2e^iθ2, what is the modulus argument form when multiplying z1 by z2?

Answer 8

z1z2 = (r1r2)e^i(θ1 + θ2)

Question 9

When z1 = r1e^iθ1and z2 = r2e^iθ2, what is the modulus argument form when dividing z1 by z2?

Answer 9

z1/z2 = (r1/r2)e^i(θ1 - θ2)

Question 10

What is the formula for De Moivre’s Theorem

Answer 10

(r(cosθ + isinθ))^n = r^n(cosnθ + isinnθ)

Question 11

How can you use Euler’s Relation to prove De Moivre’s Theorem?

Answer 11

(r(cosθ + isinθ))^n = (re^iθ)^n

r^n(e^(inθ))

r^n(cosnθ + isinnθ)