Question

Let f be one of the following functions on R. Compute f-hat. (a) f(x)=cosx, xER. (b) f(x)=sinx, xER. (c) f(x)=e^(iax), xER, where aER.

Answer 1

Fourier transform of function is:

$\hat{f}(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} dx\, e^{ikx}f(x)$

and integral representation of Dirac delta-function is

$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ikx}\, dk$

a)

$f(x)= \cos(x)\\ \hat{f}(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} dx\, e^{ikx}\cos(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dx\, e^{ikx}\frac{e^{ix}+e^{-ix}}{2}=\\ \sqrt{\frac{\pi}{2}}\delta(k-1)+\sqrt{\frac{\pi}{2}}\delta(k+1)$

b)

$f(x)= \sin(x)\\ \hat{f}(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} dx\, e^{ikx}\sin(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dx\, e^{ikx}\frac{e^{ix}-e^{-ix}}{2i}=\\ i\sqrt{\frac{\pi}{2}}\delta(k-1)-i\sqrt{\frac{\pi}{2}}\delta(k+1)$

c)

$f(x)= e^{iax}\\ \hat{f}(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} dx\, e^{ikx}e^{iax} =\\ =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} dx\, e^{i(k+a)x} = \sqrt{2\pi}\delta(k+a)$