Laplace Transforms of periodic/piecewise functions

[UglierBetty]

I understand Laplace transforms pretty well, and I've been doing well in the course.


The one thing I just can't wrap my head around is taking the Laplace transform of a piecewise or periodic function. I have notes on it from lectures, but I just can't make sense of them. I have tried the textbook, but I couldn't make sense of that either. It's all just too much notation for me:
L{u(t-a)f(t-a)}(s)=e^(-as)L{f(t)}(s) and all that nonsense, but nowhere is there an actual description "in English" of how to solve those types of problems.

I've also tried finding YouTube videos on the topic, but have been unable to locate any.



We didn't have a question on our midterm of this form, but it's appearing on a lot of the past finals I'm doing and I don't know how to do it. I worry a problem may end up on the final that I don't know how to do.



For example: the problem


f(t)= t if 0<=t<=1
2-t if 1<=t<=2
0 if t>2

I have to take the Laplace transform of the above piecewise function, but I don't know how to do it.

 

[HallsofIvy]

I am puzzled by this giving you any trouble at all! The definition of the "Laplace transform of F(x)" is \(\displaystyle \int_0^\infty F(t)e^{-st}dt\).

If F(x)= f(x) for \(\displaystyle x\le x_0\), g(x) for \(\displaystyle x\ge x_0\), the Laplace transform is exactly that:
\(\displaystyle \int_0^\infty F(t)e^{st}dt= \int_0^{x_0} f(t)e^{-st}dt+ \int_{x_0}^\infty g(t)e^{-st}dt\)