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12.4: Complex Exponential Solutions
https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_(Raymond)/12%3A_Harmonic_Oscillator/12.04%3A_Complex_Exponential_Solutions
For an equation like (12.2.1) which has real coefficients, if $\exp (i \omega t)$ is a solution, then so is $\exp (-\mathrm{i} \omega \mathrm{t})$ , so the superposition of these two solutions is a...For an equation like (12.2.1) which has real coefficients, if $\exp (i \omega t)$ is a solution, then so is $\exp (-\mathrm{i} \omega \mathrm{t})$ , so the superposition of these two solutions is also a solution. The quantity $\omega \equiv\left(\omega_{0}^{2}-\beta^{2}\right)^{1 / 2}$ is the actual frequency of oscillation of the damped oscillator, which one can see is less than the oscillation frequency ω 0 that occurs with the damping turned off.

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