What is a complex phase shift? In a complex methods course I am taking, we were given an equation for a particular driven harmonic oscillator where the driving force is trigonometric. I have worked out the math and obtained an equation that tells me that the driving frequency at resonance is the natural frequency multiplied by i. My tutor tells me that this is a 90 degree phase shift, but I don't really understand why. Isn't a phase shift obtained by adding or subtracting 90 degrees? And how can a frequency, which is a measurable physical value, take on imaginary values? I would understand if we were talking about velocity. Because velocity has a direction, addition or scalar multiplication by a real value would not describe a 90 degree rotation of the vector. But frequency is a scalar quantity. What does it mean to have an imaginary frequency?
 A: If your oscillating function is of the form $e^{i\omega t}$, a phase shift looks like $e^{i(\omega t+\phi)}$, which can be rewritten as $e^{i\omega t}e^{i\phi}$.
Now, recall that $e^{i\phi}=\cos\phi + i\sin\phi$. A 90 degree phase shift corresponds to $\phi=\frac{\pi}{2}$.
Thus,
$$e^{i\frac{\pi}{2}}=\cos\frac{\pi}{2} + i\sin\frac{\pi}{2} = 0 + i = i.$$
So finally we have,
$$e^{i(\omega t+\frac{\pi}{2})}=e^{i\omega t}e^{i\frac{\pi}{2}}=ie^{i\omega t}.$$
So we see that a phase shift of 90 degrees corresponds to multiplication by $i$.
A: There is an article here: (the optimal driving force is shown to be $90^{\circ}$ out of phase of the motion)
Phase difference of driving frequency and oscillating frequency
Also any vector like $4j + 3i$ can be expressed in phasor form as $5 \angle 41^{\circ}$ or in complex form $4+3i$.  Adding $90^{\circ}$ is just a vector of same amplitude at $90^{\circ}$ to the original.
A: It is a phase shift by 90 degrees if multiplied by $i$ indeed. 
Note that $i=e^{i\pi/2}$. Writing whatever driving signal in complex form, since it is sinusoidally driven, it will have an $e^{i\omega t}$ in it, multiplying by $i$ multiplies by $e^{i\pi/2}$, and when you multiply the exponentials you add the exponents to get $e^{i(\omega t+\pi/2)}$. 
Taking the real part to get an answer that actually makes sense physically, you would have a $\cos(\omega t+\pi/2)$ dependency in your driving. 
I think this is what you are asking, hope this helps.
