Re: Probably haven't seen this one, but...

junoexpress <MTBrenneman@xxxxxxxxx> writes:

Hi,

I'm doing some work with a function that is the ratios of (normalized)
sinc functions, having the form:
f:x-> Sinc(Mx)/Sinc(x) for M a natural number and |x| < 1

I'm having to work out some properties of this function, which are not
that bad, but in the process, I keep wondering if this ratio has been
analyzed before (in other words, I hate to present a lot of detailed
derivations only to have someone else say, "Oh yeah, that's just the
Gluckenheimer function" and if it has been looked at before, maybe I
could get some deeper insight into the solution also.)

So I come to the gurus. Is this a function which anyone has seen
analyzed before? To my knowledge, it is not in Abrahamowitz and
Stegan, and the closest I can come to pinning it on anything known is
to say that it's the ratio of 2 spherical Bessel functions (which
doesn't seem like a function that's probably been analyzed).

Since sinc(x) = sin(x)/x (for x <> 0), your function is just
f(x) = sin(Mx)/(M sin(x)). This can also be written as
U_{M-1}(cos(x))/M where U_k is the k'th Chebyshev polynomial of
the second kind.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

Relevant Pages

  • Probably havent seen this one, but...
    ... sinc functions, having the form: ... that bad, but in the process, I keep wondering if this ratio has been ... could get some deeper insight into the solution also.) ... So I come to the gurus. ...
    (sci.math)
  • Re: Probably havent seen this one, but...
    ... Robert Israel wrote: ... sinc functions, having the form: ... that bad, but in the process, I keep wondering if this ratio has been ... could get some deeper insight into the solution also.) ...
    (sci.math)
  • Re: Probably havent seen this one, but...
    ... junoexpress wrote: ... I'm doing some work with a function that is the ratios of ... sinc functions, having the form: ... So I come to the gurus. ...
    (sci.math)