The proof of Time Scaling, Laplace transform

[killahammad]

Hi

I understand most of the steps in the determination of the time scale. But i don't really understand the step in equation 6.96.

The first attachment is the full details of the time scale, and the second attachment is the part which I am stuck on.

I just want to know, how they get from the LHS to the RHS of the equation

Thanks for any help :)

 

[HallsofIvy]

to go from
$$\int_0^\infty e^{-st}f(at)dt$$
You do 2 things: Multiply -st in the exponential by a/a= 1 and rewrite (st)(a/a)= (s/a)(at); also write (a/a)dt= (1/a)(adt). Since a is a constant, we can take the 1/a outside the integral and write adt= d(at). That gives you
$$\frac{1}{a}\int_0^\infty e^{-\frac{s}{a}}(at)f(at)d(at)$$
Now, put in a new variable: let t'= at. Then dt'= d(at). When t= 0, u= t' and, as t goes to infinity, t' goes to infinity. In the variable t', f(at)= f(t') so we have
$$\frac{1}{a}\int_0^\infty e^{(s/a)t'} f(t)dt'$$

 

[killahammad]

thank you very much, that helped me alot