PX155 - D6 - SHM and complex exponentials

using euler's formula

e^{i θ} = \cos θ + i \sin θ

so, $$x=A' \cos{(\omega t+\phi)}= \mathbb Re(A'e^{i(\omega t + \phi)})$$

x = R e (A^{'} e^{i θ} e^{i ω t})

let complex amplitude $a = A^{'} e^{i ϕ}$

x = R e (a e^{i ω t})

- $a$ encodes both amplitude and the phase $ϕ$ in one complex number

the complex number $z = a e^{i ω t}$ is a solution of $\ddot{z} + ω^{2} z = 0$
- ie: $\dot{z} = i ω a e^{i ω t} = i ω z$
  $\ddot{z} = - ω^{2} a e^{i ω t} = - ω^{2} z$
- so, $\ddot{z} + ω^{2} z = 0$
so, can solve $S H M$ equation for complex variables $z$ and take the real part of the solution