this final short video is just
a little bit of a warning about
the way the fourier transform is defined
and the kinds of things you're likely to
see
if you're looking at other materials
about the fourier transform
the problem is there's a factor of two
pi you've got to put somewhere when
you're doing a fourier transform
and depending on where it ends up you
can end up with quite a few different
versions of the fourier transform and
you'll see these when you look at other
books
and other materials online you'll see
the fourier transform defined in
different ways
it's all the same fourier transform it's
just a bit of a technicality you've got
to look out for
and if using mathematica to look at
fourier transforms it's also critical to
remember this because you've got if you
want to get the fourier transforms that
we've defined
in these videos you've got to tell
mathematica what you want explicitly
so the fourier transform using
mathematica
is defined using this command here
fourier parameters
1 -1 if you want to see what that means
then just look at this
link here and this gives us the fourier
transform defined in this way as we've
done in the videos
so for example if i take the fourier
transform of the cosine of omega naught
times t
i would do it like this the command is
fourier transform
this is the the real space variable this
is the frequency variable
and i give it this command fourier
parameters one minus one
and that gives me the correct fourier
transform and the answer
is pi times the direct delta function at
plus omega naught and minus omega naught
you could also define it as -1 1 like
this and that would be backwards
to our version that would give us the
forward transform with a factor of 2 pi
here
and a plus i here instead of a minus
and here the factor of 2 pi has gone
away and we've got a minus
the default mathematica fourier
transform
is given with fourier parameters 0 1. so
if you leave
this command out here if you leave this
option out altogether this is what it
defaults to
and this is symmetric so it's got a
factor of 1 and root 2 pi
in the forward and reverse transforms
and it's a
plus i k a negative ik so the opposite
of what we use
and finally one that's also very
commonly used is
the fourier parameters 0 minus 2 pi
so here we've got the negative sign in
the forward transform
and positive in the reverse and the
factor of 2 pi is taken up
into the exponent here and there's no
factor of 2 pi that appears in front of
the transform
so as i say the fourier transform comes
in really too many flavors
and this is just a little bit of an
instruction how to do the fourier
transform
using mathematica and how you get the
answer you expect
and finally it may come as something of
a relief to know that
very rarely will you ever have to
compute
a fourier transform by hand pretty much
the whole time
you can refer to a table like this and
that will be enough for the purpose of
this course and
probably for the rest of your physics
career unless you do something very
unusual
because actually a fairly limited number
of functions you ever really care about
as it turns out so here are some
functions in real space
and here is the corresponding fourier
transform i've written it down in terms
of
time and angular frequency omega but if
you replace
t with x and omega with k
you get the fourier transforms in
spatial coordinates