Toronto Math Forum

APM346-2012 => APM346 Math => Term Test 2 => Topic started by: Victor Ivrii on November 15, 2012, 08:23:51 PM

Title: TT2--Problem 4
Post by: Victor Ivrii on November 15, 2012, 08:23:51 PM

Find Fourier transform of the  function
\begin{equation*}
f(x)= \left\{\begin{aligned}
&1-|x| &&|x|<1\\
&0 &&|x|>1.
\end{aligned}\right.
\end{equation*}
and write this function $f(x)$ as a Fourier integral.

Post after 22:30

Title: Re: TT2--Problem 4
Post by: Ian Kivlichan on November 15, 2012, 11:46:03 PM

Hopeful solution attached! :)

Title: Re: TT2--Problem 4
Post by: Victor Ivrii on November 16, 2012, 07:02:54 AM

Actually since $f$ is an even function so is $\hat{f}$ and $f(x)$ could be written as $\cos$-Fourier integral.

BTW plugging $x=0$ we can calculate $\int_0^\infty \frac{1-\cos(\omega)}{\omega^2}\,d\omega$.