An Important Relation

A relation I will state without proof that is very important to this course is that the real part of the $x(t)$ derived above:

$$\Re(x(t)) = \Re(x_{0+} e^{+i\omega t} + x_{0-} e^{-i\omega t})$$ (55)

$$= X_0 \cos(\omega t + \phi)$$ (56)

where $\phi$ is an arbitrary phase. You can prove this in a few minutes or relaxing, enjoyable algebra from the relations outlined above - remember that $x_{0+}$ and $x_{0-}$ are arbitrary complex numbers and so can be written in complex exponential form!