Looking for Fourier transform of x^-(n/m), where n,m are integers but n/m is not
Jay R. Yablon
the need to evaluate the Fourier transform of the general form:
F(x^-(n/m)) = $ x^-(n/m) exp^-x.w dx (1)
For example, what might be F(x^-(10/3)), which contains a cubed root,
and is (x^10)^-(1/3)?
At the link
http://74.125.93.132/search?q=cache:fmiTYnSjDbgJ:en.wikipedia.org/wiki/Fourier_transform+fourier+transform&cd=1&hl=en&ct=clnk&gl=us,
there is a very complete table including (#308):
F(x^n) = i^n sqrt(2pi) delta^(n) (w) (2)
where delta^(n) is the n-th distribution derivative of the Dirac delta
function. And (#310):
F(1/x^n) = F(x^-n) = -i sqrt(pi/2) [(-iw)^n-1/(n-1)!] sgn(w) (3)
And even the square root (#311):
F(1/sqrt(x)) = F(x^-(1/2)) = 1/sqrt(w) (4)
But I see nothing with the form (1), which involves a root other than
the square root, nor is it immediately clear to me how to easily
calculate (1) using the various ingredients in the table shown in this
link. And, I'd prefer to avoid having to convolve anything, because
that just replaces one integral with another.
Any help is appreciated -- preferably a link or reference to a table
which does have this function, or a good boost on how to combine the
ingredients at the above link to get to (1).
Thanks,
Jay
____________________________
Jay R. Yablon
Email: jya...@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm
Jay R. Yablon
line 310 of:
http://74.125.95.132/search?q=cache:fmiTYnSjDbgJ:en.wikipedia.org/wiki/Fourier_transform+fourier+transform&cd=1&hl=en&ct=clnk&gl=us,
for the Fourier transform F(1/x^n), where n-->m/n where each of m and n
are integers.
The "fractional factorial" threw me at first, but it seems as if one can
sensibly define this Fourier transform for non-integer factorials using
the gamma function, as described in both
http://74.125.95.132/search?q=cache:UFun5xqdxlgJ:en.wikipedia.org/wiki/Factorial+factorial&cd=1&hl=en&ct=clnk&gl=us
and http://en.wikipedia.org/wiki/Gamma_function. (There are also a
number of gamma function calculators for this infinitely-recursive
function when you get to other than half-integer factorials, see, e.g.,
http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Gamma)
Am I correct that one can use line 310 of that first reference, with m/n
type factorials evaluated via the gamma function, to evaluate the
Fourier transform F(1/x^n), ?
Thanks,
Jay
"Jay R. Yablon" <jya...@nycap.rr.com> wrote in message
news:7mdi55F...@mid.individual.net...
Jay R. Yablon
> Per the query below, it seems to me that the answer may in fact be on
> line 310 of:
> http://74.125.95.132/search?q=cache:fmiTYnSjDbgJ:en.wikipedia.org/wiki/Fourier_transform+fourier+transform&cd=1&hl=en&ct=clnk&gl=us,
> for the Fourier transform F(1/x^n), where n-->m/n where each of m and
> n
> are integers.
>
> The "fractional factorial" threw me at first, but it seems as if one
> can
> sensibly define this Fourier transform for non-integer factorials
> using
> the gamma function, as described in both
> http://74.125.95.132/search?q=cache:UFun5xqdxlgJ:en.wikipedia.org/wiki/Factorial+factorial&cd=1&hl=en&ct=clnk&gl=us
> and http://en.wikipedia.org/wiki/Gamma_function. (There are also a
> number of gamma function calculators for this infinitely-recursive
> function when you get to other than half-integer factorials, see,
> e.g.,
> http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Gamma)
>
> Am I correct that one can use line 310 of that first reference, with
> m/n
> type factorials evaluated via the gamma function, to evaluate the
> Fourier transform F(1/x^n), ?
>
> Thanks,
>
> Jay
That is, referring to
http://74.125.95.132/search?q=cache:fmiTYnSjDbgJ:en.wikipedia.org/wiki/Fourier_transform+fourier+transform&cd=1&hl=en&ct=clnk&gl=us,
#311 is indeed a special case of #310 for n=1/2.
To see this, substitute n=1/2 into #310. Use -.5!=sqrt(pi) (You can
deduce that from what is on
http://74.125.95.132/search?q=cache:UFun5xqdxlgJ:en.wikipedia.org/wiki/Factorial+factorial&cd=1&hl=en&ct=clnk&gl=us).
Use sqrt(i)=(1/sqrt(2))(1+/-i).
The result you get from #310 is (with no |abs| sign)"
Fourier(1/x^.5) = .5sgn(w)[+/-1+i] (1/sqrt(w)) (1)
Now, this is for the integral from x=-oo to +oo. The first term is for
the +x half of the integral, the second for the -x half (which is why
the i shows up, because that is what happens for square roots of
negative numbers).
If you now take |x| and |w|, then sgn(w)=1, the real part of (1) gets
duplicated twice over and so the factor of .5 is doubled, and up to the
+/- sign, one has, precisely, #311.
So, #310 DOES work for non-integer n, using non-integer factorials, and
#311 is indeed a special case of this. Do the full calculation using
n=1/2 in #310 and you will
see that this is correct.
So, based on this known special case, it appears that #310 applies for
fractional n as well as integer n, using the fractional factorials and
the associated Gamma functions.
Jay.
Igor Khavkine
> In the course of working through a physics exercise, I have come across
> the need to evaluate the Fourier transform of the general form:
>
> F(x^-(n/m)) = $ x^-(n/m) exp^-x.w dx (1)
I am supposing that your limits of integration take x from -oo to +oo.
Have you considered the possibility that this integral is not
absolutely convergent and hence not necessarily well defined?
Igor
Jay R. Yablon
news:a08c503f-67bc-459d...@j24g2000yqa.googlegroups.com...
Yes, -oo to +oo. I use the convention that this is understood, and only
show explicit limits if they are *other than* -oo to +oo.
And, absolutely, yes, I considered this. And for about a day or so I
thought that this was non-convergent. But I have found that #310 at
http://74.125.93.132/search?q=cache:fmiTYnSjDbgJ:en.wikipedia.org/wiki/Fourier_transform+fourier+transform&cd=1&hl=en&ct=clnk&gl=us
does apply even to a fractional n, and that #311 is a special case of
#310 for the case of n=1/2, see the details provided in my most recent
post to this thread. It is a straightforward calculation; you need to
use (-1/2)! and sqrt(i)=(1 +/- i) in #310, and then realize that the
result is in two parts, one from x=-oo to 0 where the sqrt is imaginary
and the other from x=0 to +oo where the sqrt is real. Then you take
absolute values, so sgn(x)=1, and the imaginary part is then discarded
and the real part doubled, yielding #311, with the only difference being
that there is a +/- sign which of course is an endemic feature of taking
a square root of a real number.
Jay.
Jay R. Yablon
As a follow up, I am very pleased to note that this discussion got
somebody's attention, because somebody changed entry #310 at
http://en.wikipedia.org/wiki/Fourier_transform to expressly deal with
F(1/|x|^a), *with a being a non-integer,* by writing this expressly in
terms of the Gamma function which is designed for non-integer
factorials.
This entry, which makes sense to me, if correct, tells us that aside
from a constant coefficient C which is a function of the Gamma function,
the "answer" I was looking for is:
Fourier(1/|x|^a) = C |w|^(a-1) (1)
So, whatever |x|^a is, it gets inverted, turned into w (omega), and
drops down one power.
Substantively, this is the same as the result I have been discussing,
but for the exact magnitude of the constant coefficient.
Importantly, it is finite, convergent.
Jay
Peter
> In the course of working through a physics exercise, I have come across
> the need to evaluate the Fourier transform of the general form:
>
> F(x^-(n/m)) = $ x^-(n/m) exp^-x.w dx � (1)
>
>
> Thanks,
>
> Jay
Hello Jay,
the fractional Fourier transform is used in Optics (introduced, iirc,
by Lohmeyer), for refs. see my Huygens' principle paper (revised and
extended in Latin Am. J. Phys. Teach. 2009),
Good luck!
Peter
Igor Khavkine
> "Igor Khavkine" <igor...@gmail.com> wrote in message
> news:a08c503f-67bc-459d...@j24g2000yqa.googlegroups.com...
>
> > On Nov 16, 10:42 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
> >> In the course of working through a physics exercise, I have come
> >> across
> >> the need to evaluate the Fourier transform of the general form:
>
> >> F(x^-(n/m)) = $ x^-(n/m) exp^-x.w dx (1)
>
> > I am supposing that your limits of integration take x from -oo to +oo.
> > Have you considered the possibility that this integral is not
> > absolutely convergent and hence not necessarily well defined?
> And, absolutely, yes, I considered this. And for about a day or so I
> thought that this was non-convergent. But I have found that #310
> [at http://en.wikipedia.org/wiki/Fourier_transform ...].
[...]
> Importantly, it is finite, convergent.
What you've found is a formula, with no indication of why it should be
trusted. The integral is very obviously not absolutely convergent.
Therefore, it is not defined according to normal rules of integration.
It so happens that sense can be made out of this integral under very
special circumstances. Whether these circumstances are applicable to
your needs is not clear, since you've not stated what they are.
Igor
Jay R. Yablon
"Peter" <end...@dekasges.de> wrote in message
news:1774b88b-3d14-4c88...@p35g2000yqh.googlegroups.com...
Thank you for the reference, I will certainly take a look.
I should note, however, that what I thought last week was a need for me
to use the fractional Fourier transform, turned out to not be so.
Nonetheless, I had the chance to learn something that I would not have
looked at otherwise, and maybe I will find some other need for the
fractional FT and fractional derivatives (also a very interesting
subject) in the future.
Jay