lim_(s -> 0^+) int_0^(oo) a(t)e^(-st) dt. What is the meaning of the limit of this integral as s -> 0^+.

Kenna Stanton 2022-11-20 Answered
lim s 0 + 0 a ( t ) e s t d t
0 a ( t ) e s t d t = f ( s )
What is the meaning of the limit of this integral as s 0 + .
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Answers (1)

Samsonitew7b
Answered 2022-11-21 Author has 15 answers
I am guessing that you would like to know lim s 0 + f ( s ) = lim s 0 + 0 a ( t ) e s t d t. Let ( s n ) be any sequence of positive numbers converging to 0 and set f n ( t ) = a ( t ) e s n t a ( t ) as n . Moreover, | f n ( t ) | | a ( t ) | so if a ( t ) is Lebesgue integrable, then the limit equals 0 a ( t ) d t by Lebesgue's Dominated convergence theorem. On the other hand, if a(t) is not Lebesgue integrable, then the limit need not exist. Consider a ( t ) = | sin ( t ) | t , for example
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