Question

Find the fourier transform of \(\frac{∂^2 u}{∂x^2}\) . (u’(p,t) denotes the fourier transform of u(x,t)).

(a) (ip)^2 u’(p,t)

(b) (-ip)^2 u’(p,t)

(c) (-ip)^2 u(p,t)

(d) (ip)^2 u(p,t)

The question was asked in an online interview.

The query is from Fourier Transform and Convolution topic in portion Fourier Integral, Fourier Transforms and Integral Transforms of Engineering Mathematics

Answer 1

Correct option is (a) (ip)^2 u’(p,t)

To elaborate: \( F\{\frac{∂^2 u}{∂x^2}\} = \int_{-∞}^∞ \frac{∂^2 u}{∂x^2} e^{ipx} dx \)

\( = e^{ipx}  \frac{∂u}{∂x} \) from (-infinity to infinity) \(– \int_{-∞}^∞ ip \, e^{ipx} u \)

\( = (ip)^2u’(p,t) \)