Is this integral correct? (expanding the Bessel function into power series )

[Rfael]

Homework Statement

integral equality: is this integral correct ?

Relevant Equations

$$ exp(-x)= \int_{0}^{\infty}J_0 (\sqrt{xt})exp(-t) $$

i have expanded the bessel function into power series and integration term by term

 

[renormalize]

Homework Statement: integral equality: is this integral correct ?

Almost! According to Mathematica your result is missing a factor of 4:

 

[Rfael]

thanks how about if i put $J_0 ( 2\sqrt{xt})$ instead my original bessel function ?

 

[renormalize]

thanks how about if i put $J_0 ( 2\sqrt{xt})$ instead my original bessel function ?

Yes, that removes the factor of 4 from the answer, as is evident by making a change-of-variables in the first integral above: