#### 6.3.3 Complex-valued analysis

\begin{align} \text e^{j \omega t} &= \cos \omega t + j \sin \omega t \\ \frac {\text d}{\text d t} \text e^{j \omega t} &= j \omega \text e^{j \omega t} \\ \frac {\text d^2}{{\text d t}^2} \text e^{j\omega t} &= - \omega^2 \text e^{j\omega t} \end{align}
$\mathbf K \mathbf u + \mathbf C \dot {\mathbf u} + \mathbf M \ddot{\mathbf u} = \mathbf f$

For harmonic vibrations, the equation becomes $\mathbf K \mathbf u + j \omega \mathbf C \mathbf u - \omega^2 \mathbf M \mathbf u = \mathbf f$ so for a given frequency it has the same form as the static case but with a complex-valued system matrix: $( \mathbf K + j \omega \mathbf C - \omega^2 \mathbf M ) \mathbf u = \mathbf f .$

R. Funnell