Question

Find the fourier sine transform of \( \frac{x}{(a^2+x^2)}. \)

(a) \(2πe^{-ap} \)

(b) \(\frac{π}{2} e^{-ap} \)

(c) \(\frac{2}{π} e^{-ap} \)

(d) \(πe^{-ap} \)

I got this question during a job interview.

Question is from Fourier Transform and Convolution topic in division Fourier Integral, Fourier Transforms and Integral Transforms of Engineering Mathematics

Answer 1

The correct answer is (b) \(\frac{π}{2} e^{-ap} \)

For explanation: Fourier transform of \( e^{-ax} \, is \, \frac{p}{(a^2+p^2)} \)

Substitute x=m and p=x.

\(\frac{π}{2} e^{-am}= \int_0^∞ \frac{x}{x^2+a^2} sin(mx)dx \)

Therefore, fourier sine transform of \(\frac{x}{(a^2+x^2)} \, is \, \frac{π}{2} e^{-ap}.\)