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A Level Further Maths Pure 2 > Complex Numbers > Flashcards

Flashcards in Complex Numbers Deck (23)
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1

Exponential form

re^iθ = r(cos θ + i sin θ)

2

Complex to exponential

Find the mod (r) and the argument (θ) and substitute into the exponential format

3

Exponential to complex

Put into modulus-argument form and then into the complex form

4

Multiply and divide in exponential form

Multiply/divide the mods and add/subtract the args before writing in exponential form

5

When to simplify mod-arg

When there are simple values for θ that give exact values e.g. π/2

6

De Moivre's theorem
r(cos(θ) + isin(θ))^n

r^n(cos(nθ) + isin(nθ))

7

Proving de Moivre's theorem by induction

do z^k x z and simplify to r^k+1(cos(k+1)θ + isin(k+1)θ) using the addition formulae

8

de Moive's theorem
re^iθ

n inθ
r^e^

9

(x + yi)^n

Write x + yi in mod-arg and apply de Moivre's theorem

10

Express cos/sin nθ in terms of powers of cos/sin nθ

1. Use de Moivre's theorem with n as the power
2. Expand (cosθ + isinθ)^n using binomial expansion
3. Set the real/imaginary part of each side equal
4. Simplify to be in terms of cos/sin

11

1
z^n + ----
z^n

2cosnθ

12

1
z^n - ----
z^n

2i sin(nθ)

13

cos/sin^nθ in terms of cos/sin nθ

1. Use (2cosθ)^n = (z + 1/z)^n or (2sinθ) = (z - 1/z)^n
2. Expand both sides remembering the 2/2i
3. Group the RHS with z^n +/- 1/z^n
4. Use the identities for (z +/- 1/z) and substitute
5. Divide both sides by the coefficient on the LHS

If it has both cos and sin expand both terms and multiply

14

Showing z^n +/- z^-n is 2cosθ or 2isinθ

Use de Moivre’s theorem with n and -n, simplifying the negative signs

15

n-1
∑ w z^r
r=0

w(z^n-1)/(z - 1)

16


∑ w z^r if |z| < 1
r=0

w/(z-1)

17

Simplifying sums to an-1

1. Write out the sum of series rule
2. Replace w with the first term and z with its exponential form, using e^πi = -1
3. Multiply top and bottom by e^(-1/2 x power in denominator)
4. Use the 2cosθ and 2isinθ rules in the denominator
5. Multiply top and bottom by i, use I^2 = -1 and simplify
6. Put numerator in mod-arg and simplify to the needed form

18

e^πi

-1

19

z^n= x + iy method

1. Put the RHS in mod-arg form
2. Write each θ as (θ + 2kπ)
3. Use de Moivre's theorem to raise each side by 1/n
4. Substitute k=0, k=1... for n values of k and put in principal argument form

20

Geometric problems

No matter what the roots are the ratio between them is the same as the roots of unity for that power
ω = cos(2π/n) + isin(2π/n)
Multiply by ω to get as many points as necessary, use exponential form

21

Series of cos and sin from an infinite series

Find the real and imaginary parts of the series

22

Infinite series from (cos θ + cos 2θ + cos 3θ) + i(sin θ + sin 2θ + sin 3θ)

Write as z + z^2 + z^3 + …
Where z = e^iθ
A 1 first can be z^0

23

Infinite series from (cos θ + kcos 2θ + k^2cos 3θ) + i(sin θ + ksin 2θ + k^2sin 3θ)

Write as regular sum of series with e^iθ as the numerator
Multiply by the denominator with the power of e inversed
Write in mod arg and simplify with real and imaginary