Inverse Laplace transform of $f(s)={\frac{1}{s^{3/2}}}$ using complex integration

Question

I want to find the inverse Laplace transform of $$f(s)={\frac{1}{s^{3/2}}}$$ Refer to the Laplace transform table, and I found that the result is $$F(t)=2\sqrt{\frac{t}{\pi}}$$ But I do not know how to get this result. I tried to use the Bromwich integral $$F(t)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{1}{s^{3/2}}e^{st}\,ds$$ My progress so far has been stunted by the fact that we have a branch point at s=0. The contour should be like this, but I don't know how to perform the integration.

Any help is appreciated.

Answer 1

$s=0$ is not an essential singularity. It is a branch point. Choose a branch to calculate your integral, for example, choose branch $-\pi<\arg z<\pi$ and integrate along the contour that made up of:

1. straight line from $c-iR$ to $c+iR$
2. along large quarter circle to approximately $(c-R)+\varepsilon i$
3. straight line to $\varepsilon i$
4. right half of circle $\lvert z\rvert=\varepsilon$, to $-\varepsilon i$
5. straight line to approximately $(c-R)-\varepsilon i$
6. another large quarter circle to $c-iR$

Can you finish from here? Be careful when you take square-root.

Answer 2

Haven't studied about the Bromwich integral yet but wouldn't it be sufficient to use the identity:

$L(t^n) = \dfrac{\Gamma(n+1)}{s^{n+1}}$

with $n = \dfrac{1}{2}$ and deduce hence?