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Find the Laplace transform

Alexmahone

Active member
Jan 26, 2012
268
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

Well-known member
Jan 26, 2012
890
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 $\displaystyle t>4$.
Straight forward application of the definition:

\[ F(s)=\int_0^{\infty}f(t)e^{-st}\; dt=\int_1^4e^{-st}\;dt \]

CB
 

Alexmahone

Active member
Jan 26, 2012
268
Straight forward application of the definition:

\[ F(s)=\int_0^{\infty}f(t)e^{-st}\; dt=\int_1^4e^{-st}\;dt \]

CB
Thanks. How would the answer differ if one of the endpoints 1 or 4 (or both) were excluded?
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,193
Thanks. How would the answer differ if one of the endpoints 1 or 4 (or both) were excluded?
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

Active member
Jan 26, 2012
268
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.
That's exactly what I meant. Thanks.
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,193