Which is the inverse Fourier transform of $w^a\cdot e^{-\sqrt{|w|}}$ for $a\geq 0$?

Question

Which is the inverse Fourier transform of $w^a\cdot e^{-\sqrt{|w|}}$ for $a\geq 0$?

Please work with the following definitions of the Fourier transform: $$F(iw) = \int\limits_{-\infty}^{\infty} f(t)\,e^{-iwt}\,dt $$ and $$f(t) = \frac{1}{2\pi} \int\limits_{-\infty}^{\infty} F(iw)\,e^{iwt}\,dw $$

Using $F(iw)$ instead of $F(w)$ is just notation, I am following the definitions used in the book "Signals and Systems, 2nd Edition" (Alan V. Oppenheim, Alan S. Willsky, with S. Hamid) [1].

I want to know the inverse Fourier Transform for these specific cases:

1. $a=0$ so what is the result of $\frac{1}{2\pi} \int\limits_{-\infty}^{\infty} e^{-\sqrt{|w|}}\,e^{iwt}\,dw\,$????
2. $a=1$ so what is the result of $\frac{1}{2\pi} \int\limits_{-\infty}^{\infty} w\,e^{-\sqrt{|w|}}\,e^{iwt}\,dw\,$????
3. $a=2$ so what is the result of $\frac{1}{2\pi} \int\limits_{-\infty}^{\infty} w^2\,e^{-\sqrt{|w|}}\,e^{iwt}\,dw\,$????
4. There is a general case for $a\geq0$???

Actually, I am trying to find a bound here for which any of the three first transforms will be useful, so partial answers could be been chosen as the final answer.

Beforehand, thanks you very much.