I have some troubles finding the Laplace transform of : $~~~f(t)=~e^{a|t|} ~~~(a\le0)~~$ from it's
Fourier transform : $~~\hat{f}(s)=\int_{-\infty }^{+\infty}f(t)e^{-i2\pi st}~dt= \frac{-1}{a+2\pi is}+\frac{1}{-a+2\pi is} = \frac{-1}{a+i\omega}+\frac{1}{-a+i\omega} $ .
I do know that the Laplace transform of : $ \mathcal{L}(f(t))(p)=H(i2\pi s)=H(i\omega)$
so $H(p)$ should be equal to : $ \frac{-1}{a+p}+\frac{1}{-a+p} $
which is quite different from the result I got from wolphram :
http://www.wolframalpha.com/input/?i=laplace+transform+of+exp(a*abs(t))
Any pointers to my mistake ?
Thank you .
Please take this answer for trial and check with care. Generally the two-sided Laplace transform is more general than Fourier transform and has larger freedom to choose the convergence domain.
We have the function $f(t)=~e^{a|t|}$ with $a\le0$ and we need to find its Laplace transform $F(s)$ in frequency domain $s=\sigma+i\omega$, then \begin{align} F(s)&=\mathcal L_{t\to s}[f(t)]=\mathcal F_{t\to\omega}[f(t)e^{-\sigma t}]\\ &=\int_{-\infty}^\infty e^{-\sigma t+a|t|}e^{-i\omega t}\mathsf dt\\ &=\int_{-\infty}^\infty e^{a|t|}e^{-st}\mathsf dt\\ &=\int_{-\infty}^0e^{-at}e^{-st}\mathsf dt+\int_{0}^\infty e^{at}e^{-st}\mathsf dt\\ &\overset{*}{=}\int_0^{\infty}e^{at}e^{-(-s)t}\mathsf dt+\int_{0}^\infty e^{at}e^{-st}\mathsf dt\\ &=\frac{1}{-s-a}+\frac{1}{s-a}\\ &=\frac{2a}{s^2+a^2} \end{align}