Identify the function f(t) that has the following Laplace transform, bar(f)(s)=int_0^(oo)f(t)e^(-st)dt=((1+alpha s)/(1+alpha(s−s_0)))^p where alpha,s_0,p are positive parameters.

pobi1k 2022-09-13 Answered
Identify the function f(t) that has the following Laplace transform,
f ~ ( s ) = 0 f ( t ) e s t d t = ( 1 + α s 1 + α ( s s 0 ) ) p
where α , s 0 , p are positive parameters. Any suggestions about how to approach this problem?
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Answers (1)

Anabelle Hicks
Answered 2022-09-14 Author has 13 answers
Using Leucippus's hint, we have
L 1 [ 1 s ( 1 + 1 s ) p ] = L p ( t ) , L 1 [ ( 1 + 1 s ) p ] = d d t L p ( t ) = L p 1 ( 1 ) ( t ) + δ ( t ) , L 1 [ ( 1 + α s 1 + α ( s s 0 ) ) p ] = L 1 [ ( 1 + 1 s / s 0 + 1 / ( α s 0 ) 1 ) p ] = s 0 e ( s 0 1 / α ) t L p 1 ( 1 ) ( s 0 t ) + δ ( t ) ,
where L p 1 ( 1 ) is the generalized Laguerre function.
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