Finding Inverse Laplace Transform using Taylor Series

Question

Find the inverse Laplace transform

$F(t)=\mathcal{L}^{-1}(s^{-\frac{1}{2}}e^{-\frac{1}{s}})$

using each of the following techniques:

1. Expand the exponential in a Taylor series about s=∞, and take inverse Laplace transforms term by term (this is allowable since the series is uniformly convergent.).

2. Sum the resultant series in terms of elementary functions.

Answer 1

you should use the gamma function to find the inverse of each of the 1/sqrt(s) instead of the basic integer factorial. then take out a common factor of sqrt(t) and find a general series sum from n=0 to infinity. you also seem to be missing a 1/n! in the series.

then use the given gamma formula from the standard formulae to sub for the Gamma(n+1/2) function. you can then use wolfram to get a standard function that the series will converge to.

Then suggest what I can do for part b :)

Answer 2

$\frac{\cos 2(\sqrt{t})}{\sqrt{\pi t}}$