Finding f(0) given F(s)

Baqling

Baqling

Answered question

2022-09-10

We've used this property a couple times:
f ( 0 ) = lim s s F ( s )
where F ( s ) is the Laplace transform of f ( t )
But I can't find an explanation or proof of why or when it works.

Answer & Explanation

Teagan Sutton

Teagan Sutton

Beginner2022-09-11Added 12 answers

We know that
L { f ( t ) } = 0 f ( t ) e s t d t = s F ( s ) f ( 0 )
Then if one takes limits on both sides we get that
0 = lim s s F ( s ) f ( 0 )
assuming that f ( t ) was bounded so we could apply the dominated convergence theorem and move the limit inside of the integral.

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