The discussion revolves around finding the Laplace transform of a step function defined piecewise, specifically for the function $\displaystyle f(t)=1$ for $\displaystyle 1\le t\le 4$ and $\displaystyle f(t)=0$ otherwise. The conversation includes considerations of how the transform might change if the endpoints of the interval are adjusted.
Participants generally agree that changing the definition of the function at specific points does not alter the integral, but the discussion does not reach a consensus on the implications of endpoint exclusions in all contexts.
The discussion assumes familiarity with the properties of integrals and the definition of the Laplace transform, but does not explore the implications of these properties in detail.
Alexmahone said:Find the Laplace transform of $\displaystyle f(t)=1$ if $\displaystyle 1\le t\le 4$; $\displaystyle f(t)=0$ if $\displaystyle t<1$ or if $\displaystyle t>4$.
CaptainBlack said:Straight forward application of the definition:
\[ F(s)=\int_0^{\infty}f(t)e^{-st}\; dt=\int_1^4e^{-st}\;dt \]
CB
Alexmahone said:Thanks. How would the answer differ if one of the endpoints 1 or 4 (or both) were excluded?
Ackbach said:Do you mean if your function were defined as, for example, $f(t)=1$ if $1<t\le 4$; $f(t)=0$ if $t\le 1$ or if $t>4$? It would make no difference. The reason is that the changing of one point in a function does not alter the integral of that function. In fact, changing the function at countably many points does not change the value of the integral.
Alexmahone said:That's exactly what I meant. Thanks.