Fourier transform of $|t| \exp{(−|t|)}$

Question

How can I calculate the Fourier transform of $|t| \exp{(−|t|)}$. Can somebody show me the way?

Answer 1

By applying the definition \begin{align} g(\omega) &=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}f(t)e^{-i\omega t}dt=\\ &=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}|t|e^{-|t|}e^{-i\omega t}dt=\\ &=\frac{1}{\sqrt{2\pi}}\left(\int_{-\infty}^0(-t)e^{t}e^{-i\omega t}dt+\int_0^{+\infty}te^{-t}e^{-i\omega t}dt\right)=\\ &=\frac{1}{\sqrt{2\pi}}\left(\int_{+\infty}^0(t')e^{-t'}e^{+i\omega t'}(-dt')+\int_0^{+\infty}te^{-t}e^{-i\omega t}dt\right)=\\ &=\frac{1}{\sqrt{2\pi}}\left(\int_0^{+\infty}te^{-t}e^{+i\omega t}dt+\int_0^{+\infty}te^{-t}e^{-i\omega t}dt\right)=\\ &=\frac{2}{\sqrt{2\pi}}\int_0^{+\infty}te^{-t}\cos{\omega t}\,dt\\ \end{align}

can you proceed from here?