Question

What is the fourier transform of e^-a|x| * e^-b|x|?

(a) \(\frac{4ab}{(a^2+p^2)(b^2+p^2)} \)

(b) \(\frac{2ab}{(a^2+p^2)(b^2+p^2)} \)

(c) \(\frac{4}{(a^2+p^2)(b^2+p^2)} \)

(d) \(\frac{a^2 b^2}{(a^2+p^2)(b^2+p^2)} \)

I had been asked this question in final exam.

I'm obligated to ask this question of Fourier Transform and Convolution in division Fourier Integral, Fourier Transforms and Integral Transforms of Engineering Mathematics

Answer 1

Correct option is (a) \(\frac{4ab}{(a^2+p^2)(b^2+p^2)} \)

The explanation is: Fourier transform of \( e^{-a|x|} =  \frac{2a}{a^2+p^2} \)

Fourier transform of \(e^{-b|x|} = \frac{2b}{b^2+p^2} \)

fourier transform of \(e^{-a|x|}* e^{-b|x|}= \frac{2a}{a^2+p^2}.\frac{2b}{b^2+p^2} \)

\( = \frac{4ab}{(a^2+p^2)(b^2+p^2)} \).