# Mathematica and its use of Conditions, Assumptions and so on

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!

• I think you're looking for Simplify with the Assumptions option: Simplify[ComplexExpand[(I*\[Omega])^(-3/4)], Assumptions -> {\[Omega] > 0}] May 15, 2017 at 15:35
• @jjc385 That is very beautiful but then it gives me an expression in which real part and imaginary part are not separated anymore! May 15, 2017 at 15:36
• Ahh done! I found it out! May 15, 2017 at 15:37

ExpToTrig[Simplify[ComplexExpand[(I*ω)^(-3/4)], Assumptions -> {ω > 0}]]

And the output is: $$\frac{\sin \left(\frac{\pi }{8}\right)}{\omega ^{3/4}}-\frac{i \cos \left(\frac{\pi }{8}\right)}{\omega ^{3/4}}$$
• I'm glad you figured it out! Have you also seen the Assuming function? Note that it only has an effect on functions like Simplify which take an Assumptions option. May 15, 2017 at 15:41