# The proof of Time Scaling, Laplace transform

 Math Emeritus Sci Advisor Thanks PF Gold P: 39,353 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 $$\frac{1}{a}\int_0^\infty e^{(s/a)t'} f(t)dt'$$