Could someone help me find the Laplace transform of the time funtion f(t)=e^(-Kt^n) where K is real and positive and 0<n<1 ?

wurpenxd 2022-09-11 Answered
Could someone help me find the Laplace transform of the time funtion f ( t ) = e K t n where K is real and positive and 0<n<1 ?
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Answers (1)

Theodore Dyer
Answered 2022-09-12 Author has 10 answers
We have that :
Γ ( ω ) m ω = 0 x ω 1 e m n d x
Now applying the Laplace Transform of e K t n , we get :
(1) L ( e K t n ) = 0 e K t n e s t d t
Since :
(2) e K t n = r = 0 ( K ) r r ! t n r
Then by substituting (2) in (1) we get :
0 r = 0 ( K ) r r ! t n r e s t d t
Hence, we get that :
F ( s ) = r = 0 ( K ) r r ! Γ ( 1 + n r ) s 1 + n r
[Note] : 1 + n r > 0 to avoid Γ diverging to ±

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