How to use Complex Inversion Theorem to find the Inverse Laplace Transform?

Question

How to use Complex Inversion Theorem to find the Inverse Laplace Transform for the given $F(t)=L^{-1} \{s^{-1/2} e^{-1/s}\}$ ? Hint: make the radius $\epsilon$ of the inner circle $t-1/2$ rather than letting it vanish. Explain why you cannot let $\epsilon\to 0$.

Answer 1

Remember the formula for the complex inversion theorem/bromwich

Then draw the contours around where the branch points are

Then solve for the paths around the contour you have drawn