用Laplace Transform 求Fourier Transform

 $$求\int_0^\infty {\sin(at)\over t}\,dt\quad及\int_{-\infty}^\infty {\sin(at)\over t} e^{-j\omega t}\,dt\\令f(t)={\sin(at)\over t} \Rightarrow I(s)= L\{f(t)\} =\int_0^\infty {\sin(at) \over t}e^{-st}\,dt =\int_s^\infty {a\over x^2+a^2}\,dx\\ =\left.\left[\tan^{-1}{x\over a} \right] \right|_s^\infty ={\pi\over 2}-\tan^{-1}{s\over a} \Rightarrow \int_0^\infty {\sin(at)\over t}\,dt=I(0)=\bbox[red,2pt]{\pi\over 2}\\又 f(-t)={-\sin(at)\over -t} ={\sin(at) \over t}=f(t) \Rightarrow f \text{ is even function} \Rightarrow \int_{-\infty}^\infty f(t)\sin(\omega t)\,dt =0\\ \Rightarrow  \text{Fourier transform }F(\omega)=\int_{-\infty}^\infty f(t) e^{-j\omega t}\,dt =\int_{-\infty}^\infty f(t)(\cos(\omega t) -i\sin(\omega t))\,dt\\ =\int_{-\infty}^\infty f(t)\cos(\omega t)\,dt =2\int_0^\infty f(t)\cos(\omega t)\,dt =2\int_0^\infty {\sin(at) \over t}\cos(\omega t)\,dt \\=\int_0^\infty {\sin(a+\omega) \over t} +{\sin(a-\omega)\over t} \,dt ={\pi\over 2}+{\pi\over 2}= \bbox[red, 2pt]\pi$$