Inverse Laplace transform?

Question

What is the inverse Laplace transform of the following:
e^(-a * sqrt(s))

I've tried doing this using computer software, and it's not giving me an answer. An explanation would be helpful.

a is constant

Answer 1

I used Mathematica, with the input

InverseLaplaceTransform[Exp[-a * Sqrt[s]],s,t]

and it gave me

a * Exp[-(a^2)/(4t)]/(2 * Sqrt[Pi] * t^(3/2))

or

a * e^(-a^2/(4t)) all over 2 * sqrt(pi) * t^(3/2).

I thought about it and tried looking up tables online, but I couldn't find anything about this type of function, so I have no explanation as of yet. If you know how, maybe a Bromwich Integral would be useful, but I doubt it.

Answer 2

e^(-asqr[s]) can be written as an infinite series:
e^(-sqr[x]) = 1 - x^(1/2) + x/2! - x^(3/2)/3! +-...
Let x = sqr[a^2 s]
Now take the inverse of each term in the series, collect terms, then you will get
L^(-1)|e^(-as^[1/2)) = e^(-1/4at)/{2sqr[pi] (at)^(3/2)}
I might be off by a sign or have missed an a, but you should get the idea