Help find the laplace transformation

Question

Given that

$L\{J_0(t)\}=1/(s^2+1)$

where $J_0(t)=\sum\limits^{∞}_{n=0}(−1)n(n!)2(t2)2n$,

find the Laplace transform of $tJ_0(t)$.

$L\{tJ_0(t)\}=$___________---___?

Answer 1

From the tables:

$$\int_0^{\infty} dt\: J_0(t) e^{-s t} = (1+s^2)^{-1/2}$$

$$\int_0^{\infty} dt\:t J_0(t) e^{-s t} = -\frac{d}{ds}\int_0^{\infty} dt\: J_0(t) e^{-s t} = s (1+s^2)^{-3/2} $$

Answer 2

Hint: Use the property

$$ L(t^n f(x)) = (-1)^{n} F^{(n)}(s), $$

where $F(s)$ is the Laplace transform of $f(x)$.