What is the Fourier transform G(ω) of the signal of \(\rm g(t)=\frac{1}{1+jt}\)?

Concept:

As per the property of duality

For, x(t) \(\overset{FT}{\rightleftharpoons}\) x(ω)

x(t) \(\overset{FT}{\rightleftharpoons}\) 2πx (-ω)

Calculation: 

For the given question g(t) = \(\frac{1}{1 + jt}\)

we know e^-at u(t) \(\rightleftharpoons \) \(\frac{1}{a + jω}\)

then by property of duality \(\frac{1}{a + jt} \rightleftharpoons \) 2πe^aω u(-ω)

∴ for a = 1 = \(\frac{1}{1 + jt} \overset{FT}{\rightleftharpoons}\) 2πe^ωu(-ω)