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In my course solid state physics, there is a part about the Lorentz oscillator. At a certain part, this is written:

"X(t) = X_0sin(-ωt+α)

This changes into:

X(t) = X_0 exp(-iωt)

by choosing X_0 as a complex number and putting the phase factor into the complex amplitude."

But I just don't see how you can do/prove this mathematically?

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# Putting phase factor in amplitude in Lorentz oscillator

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