# Inverse Laplace Transform

1. Mar 21, 2010

### NT123

1. The problem statement, all variables and given/known data Calculate the inverse Laplace transform of exp(-as)/s (a is a constant).

2. Relevant equations

3. 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.

2. Mar 21, 2010

### LCKurtz

What about using the time shifting formulas?

3. Mar 22, 2010

### vela

Staff Emeritus
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.
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.