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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:

$\displaystyle \Re(x(t))$ $\displaystyle =$ $\displaystyle \Re(x_{0+} e^{+i\omega t} + x_{0-} e^{-i\omega t})$ (84)
  $\displaystyle =$ $\displaystyle X_0 \cos(\omega t + \phi)$ (85)

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!

Robert G. Brown 2011-04-19