I'm trying to calculate the following:
$$ |a^{(\frac{1}{2}+ib)}|, $$ where $a,b\in\mathbb{R}$, and $a<0$. So what I am doing is:
$$ |a^{(\frac{1}{2}+ib)}|=\left|(-|a|)^{(\frac{1}{2}+ib)}\right|=|a|^{\frac{1}{2}}(-1)^{ib} $$
My problem now is that I think I can set $-1=e^{i\pi}$ or $-1=e^{-i\pi}$, and the results for each choice seem to be different:
- If I set $-1=e^{i\pi}$, then the result is $|a|^{\frac{1}{2}}e^{-b}$.
- If I set $-1=e^{-i\pi}$, then the result is $|a|^{\frac{1}{2}}e^{b}$.
Am I doing something wrong or is this natural?