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A Level Further Maths Pure 2 > Complex Numbers > Flashcards

Flashcards in Complex Numbers Deck (23)
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1
Q

Exponential form

A

re^iθ = r(cos θ + i sin θ)

2
Q

Complex to exponential

A

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

3
Q

Exponential to complex

A

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

4
Q

Multiply and divide in exponential form

A

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

5
Q

When to simplify mod-arg

A

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

6
Q

De Moivre’s theorem

r(cos(θ) + isin(θ))^n

A

r^n(cos(nθ) + isin(nθ))

7
Q

Proving de Moivre’s theorem by induction

A

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

8
Q

de Moive’s theorem

re^iθ

A

n inθ

r^e^

9
Q

(x + yi)^n

A

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

10
Q

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

A
  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
Q

1
z^n + —-
z^n

A

2cosnθ

12
Q

1
z^n - —-
z^n

A

2i sin(nθ)

13
Q

cos/sin^nθ in terms of cos/sin nθ

A
  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
Q

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

A

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

15
Q

n-1
∑ w z^r
r=0

A

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

16
Q


∑ w z^r if |z| < 1
r=0

A

w/(z-1)

17
Q

Simplifying sums to an-1

A
  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
Q

e^πi

A

-1

19
Q

z^n= x + iy method

A
  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
Q

Geometric problems

A

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
Q

Series of cos and sin from an infinite series

A

Find the real and imaginary parts of the series

22
Q

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

A

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

23
Q

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

A

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