Calculate the laplace transform of following function, f(t)=(1)/(1+p(t/tau)^(beta)) where beta<1.

clovnerie0q 2022-10-01 Answered
Calculate the laplace transform of following function,
f ( t ) = 1 1 + p ( t / τ ) β
where β < 1. Any ideas of how to go about the calculation?
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Answers (1)

Matteo Estes
Answered 2022-10-02 Author has 9 answers
The Laplace transform can be evaluated as an integral of a product of two Meijer G-functions. For β > 0, we obtain
0 e s t 1 + t β d t = 0 G 1 , 1 1 , 1 ( t β | 0 0 ) G 0 , 1 1 , 0 ( s t | 0 ) d t = 1 s H 2 , 1 1 , 2 ( s β | ( 0 , 1 ) , ( 0 , β ) ( 0 , 1 ) ) .
For β < 0
0 e s t 1 + t β d t = 0 G 1 , 1 1 , 1 ( t β | 1 1 ) G 0 , 1 1 , 0 ( s t | 0 ) d t = 1 s H 2 , 1 1 , 2 ( s β | ( 1 , 1 ) , ( 0 , β ) ( 1 , 1 ) ) ,
which, incidentally, is the same as formally extending the first result to negative β.
The resulting Fox H-function can be converted to a G-function when β is rational.
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