45 and 90-degree Phase shift

Question

Assume $i=\sqrt{-1}$, $\omega$ is the angular frequency. Then $i\omega$ corresponds to a 90-degree phase shift, whereas $\sqrt{i\omega}$ corresponds to a 45-degree phase shift. Can anybody explain why? Math derivation is also welcome.

Answer 1

You may be familiar with Euler's formula: $$e^{ix}=\cos(x)+i\sin(x)$$

When $x=\frac{\pi}{2}$ (90 degrees), this formula tells you that $e^{i\frac{\pi}{2}}=i$ (look at @user14717's comment). If you take square roots, you get $$\sqrt{i}=(i)^{\frac{1}{2}}=e^{i\frac{\pi}{4}}$$ and now you see that $\sqrt{i}$ corresponds to an angle of $\frac{\pi}{4}$ (45 degrees).