I don't understand why Mathematica has to be such a pain in the ***.

ComplexExpand[(I*ω)^(-3/4), TargetFunctions->{Re, Im}]

The result is

$$\frac{\cos \left(\frac{3}{4} \tan ^{-1}(0,\omega )\right)}{\left(\omega ^2\right)^{3/8}}-\frac{i \sin \left(\frac{3}{4} \tan ^{-1}(0,\omega )\right)}{\left(\omega ^2\right)^{3/8}}$$

Now, all good BUT the arctangent.

I have tried EVERYTHING I could to FORCE Mathematica to understand that $\omega$ is a NATURAL number, and nothing.

It doesn't get it. Assuming, Assumption, Conditions...

Is there a way to get rid of that useless arctangent terms, which is, by the way, nothing but $\frac{\pi}{2}$ considering that indeed $\omega >0$ ?

Thank you!