Question

\(F(x) = x^{(\frac{-1}{2})} \)is self reciprocal under Fourier cosine transform.

(a) True

(b) False

This question was addressed to me in class test.

Question is taken from Fourier Transform and Convolution in portion Fourier Integral, Fourier Transforms and Integral Transforms of Engineering Mathematics

Answer 1

Correct answer is (a) True

Easy explanation: \(Fc \{x^{(\frac{-1}{2})}\}  = \int_0^∞ x^{(\frac{-1}{2})} cos(px)dx = constant * p^{(\frac{-1}{2})} \)

Inverse fourier transform of \( p^{(\frac{-1}{2})} = constant * x^{(\frac{-1}{2})} \)

Hence the  function \(x^{(\frac{-1}{2})} \)is self reciprocal.