Draw argand and show z* and iz
z* - 90˚ right
iz - 90˚ left
Show different forms of z: de moivre, with e
(cosθ + i sin θ)
re^iθ
Show roots of r^n = 1 and formula for the angles
distributed around a circle at angles 2π/n
(z)(z*)
|z|^2
z1^z2
e^(z2 ln z1)
z + z*
2 re(z)
z - z*
2 im(z)
|1/z|
1/r
arg(1/z)
-arg(z)
State cos n𝜃 and sin n𝜃
Re(cos + isin)^n
Im(cos + isin)^n
State cos 𝜃 and sin 𝜃 in terms of e and i
1/2 (e^i𝜃 + e^-i𝜃)
1/2i (e^i𝜃 - e^-i𝜃)
cos^n 𝜃 and sin^n 𝜃
[1/2 (e^i𝜃 + e^-i𝜃)]^n
[1/2i (e^i𝜃 - e^-i𝜃)]^n
SN = ∑k=0N-1 cos k 𝜃
Re[a (1-e^iN𝜃 / 1 - e^i𝜃)]
SN = 0∑N-1 sin k 𝜃
Im[a (1-e^iN𝜃 / 1 - e^i𝜃)]
SN = 1∑N-1 cos k 𝜃
Re[a e^ik𝜃 (1-e^iN𝜃 / 1 - e^i𝜃)]
SN = 1∑N-1 sin k 𝜃
Im[a e^ik𝜃 (1-e^iN𝜃 / 1 - e^i𝜃)]
Evaluate ln z
z = 2^i = cos ln 2 + i sin ln 2
Find roots of z^n = a
e^inθ = e^ix
θ = (x+2πk)/n, 0≤ k ≤ n-1
Give oscillation equation and define each part
x(t) = a cos wt + b sin wt = Re[A e^iwt]
A = a - ib
Give velocity, amplitude, phase
v(t) = dx/dt = Re[iwA e^iwt]
Amplitude = |A|
phase = θ
State fundamental theory of algebra
P(z) fo degree n has n complex roots
