Calculating the Inverse Laplace Transform of exp(-as)/s

[NT123]

Homework Statement

Calculate the inverse Laplace transform of exp(-as)/s (a is a constant).

Homework Equations

The Attempt at a Solution

I need to calculate the integral (1/2*pi*i)int(c-i(inf), c+i(inf))(exp(s(t-a))/s).

I'm guessing I need to integrate around a circular contour centred at c, with a sufficiently large radius to contain zero.

The pole is at zero so I guess the residue is just exp(s(t-a))(0) = 1. I'm not sure how to show the rest of the integral ---> 0. Any help would be appreciated.

 

[LCKurtz]

What about using the time shifting formulas?

 

[vela]

Homework Statement

Calculate the inverse Laplace transform of exp(-as)/s (a is a constant).

The Attempt at a Solution

I need to calculate the integral (1/2*pi*i)int(c-i(inf), c+i(inf))(exp(s(t-a))/s).

I'm guessing I need to integrate around a circular contour centred at c, with a sufficiently large radius to contain zero.

Not quite. The integral is along the line Re(s)=c where c must be positive so that the line will be in the region of convergence. To use the residue theorem, you have to close the contour, and the choice of how to close it will depend on whether t>a or t<a.

I'm not sure how to show the rest of the integral ---> 0. Any help would be appreciated.

The complete contour consists of two parts: the line from the original integral and the piece you need to form a closed contour. Show that the integrand vanishes on that second piece.