Laplace transforms of Heavyside functions

  • Thread starter iceman_ch
  • Start date
Ok, I have two question that have me stuck. I understand heavyside functions and how to do a laplace transform on them but, they've thrown me a curve ball. I'm sure I'm just making it more complicated then it needs to be.

Here is problem number one.

g(t) = 2t for 0 <= t < 1;
2 for 1 <= t;

This is the same as:

g(t) = 2tH(t) - 2tH(t-1) + 2H(t-1)

The next step is were I'm confused. The book finished setting up this problem for a laplace transform by changing the equation to this:

g(t) = 2tH(t) - 2(t-1)H(t-1)

Why did they change it like this and how did they do this. I'm know I'm over looking something but, I'm not sure what. Any help would be great.
 
to take the laplace transform of the heaviside function, you want it to be of the form f(t)H(t). Since 2tH(t-1) isn't of this form, the author uses some simple algebra to get something that is.
 
H(t)=1 if t>0 and 0 if t<0

Now H(t-a)=1 if t-a>0 and 0 if t-a<0=> H(t-a)=1 if t>a and 0 if t<a.

this means that the laplace transform from 0 to a is going to be zero, so there is nothing to worry about it. Just take the laplace transfrom from a to infty.
 

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