I want to take the inverse Fourier transform of the following function:

$$ \hat{f}(\omega) = \begin{cases}e^{-r  \sqrt{\omega}} &amp; \text{for } \omega &gt; 0 \\ 0 &amp; \text{otherwise}\end{cases},$$ 

such that

$$ f(t) = \int_{-\infty}^\infty \hat{f}(w)e^{i\omega t} d\omega= \int_0^\infty e^{-r \sqrt{\omega}} e^{i\omega t} d\omega,$$ 

where $r$ is a constant.

I am having trouble in integrating this. That $ \sqrt{\omega}$ is ruining my day. Any suggestions?

Kind regards


Question: Using sketch to find exact value of trigonometric expression
Answer: 

Print it and give it to your teacher. Or send him this link. Answer is $\pm\frac{12}{5}=\pm 2.4$