Question

Find the fourier cosine transform of e^-ax * e^-ax.

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

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

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

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

I had been asked this question in a job interview.

My question is taken from Fourier Transform and Convolution topic in chapter Fourier Integral, Fourier Transforms and Integral Transforms of Engineering Mathematics

Answer 1

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

The explanation is: = fourier cosine transform of \(e^{-ax} = \frac{p}{a^2+p^2}\)

fourier cosine transform of \(e^{-ax}* e^{-ax} = \frac{p}{a^2+p^2} . \frac{p}{a^2+p^2} \)

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