Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Inverse Laplace Transform

  1. Mar 21, 2010 #1
    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. jcsd
  3. Mar 21, 2010 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    What about using the time shifting formulas?
  4. Mar 22, 2010 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook