I am currently exploring the use of inverse laplace transform and inverse fourier transform in statistics. From what I have read, for a random variable $ X $ with $ f(x) $ and $ F(x) $ as its PDF and CDF, we can use inverse fourier transform to recover its PDF, which is
$$ f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-itx} \phi(t) dt $$
If I want to calculate the CDF of $ X $ at point a, I can integrate above formula
$$ F(a) = \frac{1}{2\pi} \int_{-\infty}^{a} \int_{-\infty}^{\infty} e^{-itx} \phi(t) dt dx $$
with $ \phi(t) = E[e^{itX}] $ is a characteristic function of $ X $. From what I have read from Wikipedia, we can use inverse laplace transform to calculate a CDF of a random variable directly by using this formula
$$ F(a) = \mathcal{L}^{-1} \bigg{\{} \frac{1}{s} E \Big{[}e^{-sX} \Big{]} \bigg{\}} $$ $$ F(a) = \frac{1}{2 \pi i} \int_{\gamma - i \infty}^{\gamma + i \infty} \frac{e^{sa}}{s} E \Big{[}e^{-sX} \Big{]} ds $$
I know inverse fourier transform is equal to inverse laplace transform if we take $ s = \gamma + it $. I am trying to prove the formula for $ F(a) $ is equal if we use inverse fourier transform and inverse laplace transform, but somehow, it is obvious that they are different, one with one integral and the other with two integrals. Or is there something wrong in the formula that I have stated so both of the are not the same? If it is already correct, kindly need your help to prove it. Thank you.