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 Recognitions: Gold Member Science Advisor The general solution to the DE $r'' + \omega ^{2}r = 0$ would be $r(t) = Ae^{i\omega t} + Be^{-i\omega t}$. For simplicity of visualization, take A = 1, B = 0 so that $r(t) = e^{i\omega t}$. You will recognize this as being $exp:I\rightarrow S^{1}$ where $S^{1}$ is the unit circle (we are viewing it as being a subset of the complex plane) and $I$ is an appropriate interval in $\mathbb{R}$. Now, with regards to the physics of simple harmonic motion, we just take the real part of the above expression for $r(t)$ but you can visualize $\omega$, the angular frequency, as being the rate at which the unit circle is being swept out in the complex plane.