Need to evaluate the Laplace integral transform (definition below) of sqrt(t) using complex analysis. How is that possible? hat(f)(s)=int_0^(oo) sqrt(t) e^(-st) dt

profesorluissp 2022-09-12 Answered
Need to evaluate the Laplace integral transform (definition below) of t using complex analysis. How is that possible?
f ^ ( s ) = 0 t e s t d t
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Answers (1)

Peyton Atkins
Answered 2022-09-13 Author has 13 answers
For s > 0
0 t 1 / 2 e s t d t = 0 ( u / s ) 1 / 2 e s ( u / s ) d ( u / s ) = s 3 / 2 0 u 1 / 2 e u d u = s 3 / 2 Γ ( 3 / 2 ) = s 3 / 2 Γ ( 1 / 2 ) 1 / 2 = s 3 / 2 π 1 / 2 / 2
(the last step is with the reflection formula)
Since
0 t 1 / 2 e s t d t s 3 / 2 π 1 / 2 / 2
is complex analytic for ( s ) > 0 and it vanishes for s > 0, it vanishes for ( s ) > 0 and you got your Laplace transform.

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