Laplace transforms of Heavyside functions

[iceman_ch]

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.

 

[matticus]

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.

 

[sutupidmath]

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.