How to calculate the following Laplace transform: L[(1-J_0(t))/(t)]?

Paul Reilly 2022-09-12 Answered
How to calculate the following Laplace transform: L [ 1 J 0 ( t ) t ]? where J 0 ( t ) is the zeroth Bessel function.
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Answers (1)

Mateo Tate
Answered 2022-09-13 Author has 18 answers
If
L t [ F ( t ) ] ( s ) = f ( s )
then:
L t [ F ( t ) t ] ( s ) = s f ( u ) d u
provided: lim t 0 F ( t ) t exist.
Since:
L t [ 1 J 0 ( t ) ] ( s ) = 1 s 1 1 + s 2
and:
lim t 0 1 J 0 ( t ) t = 0
we have:
L t [ 1 J 0 ( t ) t ] ( s ) = s ( 1 u 1 1 + u 2 ) d u = ln ( 1 2 + 1 2 1 + 1 s 2 )
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