A-Level Complex Numbers

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exponential form

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19 Terms

1

exponential form

(r)e^iθ

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2

argument of a real number

-π, 0, π

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3

argument of imaginary number

π/2, -π/2

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4

expression for cosθ

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5

expression for sinθ

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6

derive the expression for cosθ from Euler’s relation

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7

derive the expression for sinθ from Euler’s relation

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8

how to derive a formula for sin(nθ) and cos(nθ)

  • cos(nθ)+isin(nθ) = (cosθ+isinθ)ⁿ (de Moivre’s theorem)

  • expand this bracket (let cosθ=c and sinθ=s for ease of layout)

  • the real parts are cos(nθ) and the imaginary parts are sin(nθ)

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9

how to change cosⁿx/sinⁿx into an integrable form and what is this form

  • form is sum of multiple angles

  • express trig function in exponential form

  • let e^iθ = z so that

  • cosⁿ(θ) = (½)ⁿ[z+(1/z)]ⁿ
    sinⁿ(θ)=(1/2i)ⁿ[z-(1/z)]ⁿ

  • expand this using binomial expansion

  • gather inverse terms together

  • zⁿ±(1/zⁿ)= 2cos(nθ)/2isin(nθ) - de Moivre

  • done!

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10

how to turn a combination of sin and cos into an integrable form

same procedure as above, but include difference of two squares to make calculations easier

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11

integral of cos(nθ)

(1/n)sin(nθ)

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12

integral of sin(nθ)

(-1/n)cosnθ

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13

when to employ “the trick”

when there is a 1±e^iϕ

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14

what is “the trick”

for 1±e^iϕ, multiply the whole expression by e^(-iϕ/2) over itself

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15

what to do when something isn’t “trickable”

for n±k(e^iϕ), multiply by n±k(e^-iϕ) [ie the complex conjugate] over itself

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16

how to find sums of trig series for cos and sin, given one of the series

  • find the corresponding cos/sin series

  • find cos+i sin

  • convert into exponential form to form geometric series

  • apply formula then manipulate it (using trick or non-trick)

  • sum of cos series is real part/ sum of sin series is imaginary part

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17

what to do for trig series with binomial coefficients

  • C + iS

  • convert to exponential form (still with binomial coefficients)

  • figure out what bracket is being expanded

  • apply “the trick” to the bracket and manipulate

  • apply the power to each term in the bracket

  • cos series is real part/ sin series is imaginary part

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18

the nth roots of unity are

1, e^i(2π/n), e^i(4π/n),……, e^i(2(n-1)π/n)

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19

how to find the nth roots of a number (two different ending methods) ie how to solve zⁿ=w

  • express equation as rⁿe^inθ=Re^iϕ

  • solve rⁿ=R (r must be positive as r=|z|)

  • solve nθ=ϕ

  • either solve up to nθ=ϕ+2(n-1)π

  • or let ω=e^i2π/n and x=ₙ√R e^iϕ/n, then the roots are x, xω, xω²….xωⁿ⁻¹

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