\magnification = \magstep1
\centerline{Mathematics 174 -- Applied Mathematics 2}
\centerline{A Short Table of Fourier Transforms}
\centerline{April 4, 2000}
\bigskip
\noindent
{\bf Notes}
\bigskip
This table is constructed using the following convention for 
the Fourier transform:
$${\cal F}[f](\omega) = {1\over 2\pi}\int_{-\infty}^\infty f(x) e^{i \omega x}\, dx $$
and the Fourier inversion formula in the form
$$f(x) = \int_{-\infty}^\infty  {\cal F}[f](\omega) e^{-i \omega x}\, d\omega.$$
You will find similar tables in many places, but {\it beware in comparing 
what's here with those other tables}.  There are MANY different 
ways people choose to incorporate factors of $2\pi$ in the 
definition of ${\cal F}$ and ${\cal F}^{-1}$, and most people
write the exponential in ${\cal F}$ with a 
negative sign rather than a positive.  Depending how all that 
is done, the results will change accordingly.  The first column of the table 
gives a time domain function or combination of functions.  The second column 
gives the corresponding Fourier transform.  
If you are clever, you can also see how to compute direct Fourier
transforms by reading this table from right to left(!)  (Hint:
what is ${\cal F}({\cal F}[f])$ if $f$ is suitably nice?)
\bigskip
\noindent
{\bf Table -- Special Functions}
\bigskip
$$\matrix{
&f(x) & {\buildrel {\cal F} \over \longrightarrow} & {\cal F}[f](\omega)\cr 
&&&\cr
&\Pi_a(x) = \cases{1& if $|x| \le a$\cr
                0& otherwise\cr} 
&\qquad\rightarrow\qquad 
&a {\rm sinc}(a \omega) = {\sin(a \omega )\over \pi \omega}\cr
&&&\cr
&\Lambda_a(x)=\cases{2a - |x| & if $|x| \le 2a$\cr
                  0& otherwise\cr} 
&\qquad\rightarrow\qquad 
&2\pi a^2{\rm sinc}^2(a\omega) = {2\sin^2(a\omega)\over \pi \omega^2}\cr
&&&\cr     
&e^{-x^2} 
&\qquad\rightarrow\qquad
&{1\over \sqrt{4\pi}} e^{-\omega^2/4}\cr
&&&\cr
&\delta(x)
&\qquad\rightarrow\qquad
&{1\over 2\pi}\cr
&&&\cr
&e^{-|x|}
&\qquad\rightarrow\qquad
&{1\over \pi}{1\over 1 + \omega^2}\cr
&&&\cr
&&&{\rm (continued\ on\ back)}\cr}$$
\vfill\eject
\noindent
{\bf Table -- General Properties}
\bigskip
$$\matrix{
(Linearity)&cf(x) + g(x)
&\qquad\rightarrow\qquad 
&c{\cal F}[f](\omega) + {\cal F}[g](\omega)\cr
&&&\cr
(Scaling)& f(ax)&\qquad\rightarrow\qquad 
&{1\over |a|} {\cal F}[f](\omega/a)\cr
&&&\cr
(Shifting)& f(x + a) 
&\qquad\rightarrow\qquad 
&{\cal F}[f](\omega) e^{- i a \omega}\cr
&&&\cr
(Modulation)& f(x)e^{i a x} 
&\qquad\rightarrow\qquad 
& {\cal F}[f](\omega + a)\cr
&&&\cr
(Convolution)& (f\ast g)(x) 
&\qquad\rightarrow\qquad 
&2\pi {\cal F}[f](\omega){\cal F}[g](\omega)\cr
&&&\cr
(Derivatives)&f^{(n)}(x) 
&\qquad \rightarrow\qquad 
& (-i\omega)^n {\cal F}[f](\omega)\cr}$$
\bye