1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
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: Alternative form of the 2nd translation theorem proof

  1. Mar 3, 2013 #1
    hi guys, this is literally my first post here on physicsforums so i apologize in advance that my latex formatting sucks.

    1. The problem statement, all variables and given/known data
    prove the alternative form of the 2nd translation theorem of the laplace transform:
    [itex]L[f(t)u(t-a)]=e^{-sa}L[f(t+a)] [/itex]

    where [itex]u(t-a)[/itex] is the unit step function and [tex]L[f(t)]=\int^\infty_0e^{-st}f(t)dt[/tex] is the definition of the laplace transform of [tex]f(x)[/tex]

    2. Relevant equations

    3. The attempt at a solution
    [itex]L[f(t)u(t-a)]=\int^\infty_0 e^{-st}f(t)u(t-a)\,dt [/itex]
    [tex]=\int^a_0 e^{-st}f(t)u(t-a)\,dt + \int^\infty_a e^{-st}f(t)u(t-a)\,dt [/tex]
    the first integral is 0 since the unit step function 0 for any t < a, and [tex]u(t-a)=1[/tex] for t >= a so the 2nd integral becomes, [tex]\int^\infty_a e^{-st}f(t)\,dt[/tex]
    then since the laplace transform is an integral from 0 to [tex]\infty[/tex] we need to make the substitutions [tex]u = t-a, \ t = u+a,\ du = dt[/tex] to change the lower limit of integration from a to 0. then the integral becomes, [tex]\int^\infty_0 e^{-s(u+a)}f(u+a)\,du[/tex]
    then you pull the [tex]e^{-sa}[/tex] outside the integral to give [tex]e^{-sa}\int^\infty_0 e^{-su}f(u+a)\,du[/tex] and this is where i'm stuck. I don't know what to do here since the definition of the laplace transform is [tex]L[f(u)]=\int^\infty_0e^{-su}f(u)du[/tex] and i can't make any more substitutions without changing the limits of integration.
  2. jcsd
  3. Mar 4, 2013 #2
    What's [itex] L[f(t+a)] [/itex] then?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted