-- Sinc relatives, with applications
David W. Cantrell
of the sine cardinal function and shows two nice applications of
those functions.
------------------------------------------------
The sine cardinal function, abbrev. sinc, may be defined by
| { 1 if x = 0,
| sinc(x) = {
| { sin(x)/x otherwise.
[Readers might be interested in
<http://mathworld.wolfram.com/SincFunction.html>. Among the references
listed there, I particularly recommend the paper by Gearhart and Shultz.]
This function removes the singularity of sin(x)/x, a commonly
occurring function. But there are other commonly occurring functions
having removable singularities for which the corresponding functions,
with those singularities removed, do not have convenient notations in
common use. When needed, those corresponding functions would typically
be defined piecewise. In this first section, we show how several of
those functions can be written nicely in terms of the sine cardinal
function, thereby obviating the need to define them piecewise ad hoc.
In the table below, the left column gives a function having a removable
singularity at x=0 and the right column gives the corresponding function,
written using sinc, with that singularity removed.
sin(x)/x sinc(x)
tan(x)/x sinc(x)/cos(x)
(1 - cos(x))/x^2 1/2 (sinc(x/2))^2
asin(x)/x 1/sinc(asin(x))
atan(x)/x cos(atan(x))/sinc(atan(x))
-----------------------------------
sinh(x)/x sinc(i x)
tanh(x)/x sinc(i x)/cos(i x)
(cosh(x) - 1)/x^2 1/2 (sinc(i x/2))^2
asinh(x)/x 1/sinc(i asinh(x))
atanh(x)/x cos(i atanh(x))/sinc(i atanh(x))
-----------------------------------
(exp(x) - 1)/x sinc(i x) + x/2 (sinc(i x/2))^2
log(x + 1)/x 1/(sinc(i log(x+1)) + log(x+1)/2 (sinc(i log(x+1)/2))^2)
-----------------------------------
Notes:
1. Many readers will already have noticed a relationship between
certain pairs of lines (e.g., the pair above). Namely, if removing the
discontinuity of f(x)/x at x=0 yields g(x), then removing the
discontinuity of invf(x)/x at x=0 yields 1/g(invf(x)).
2. The table is, of course, not "complete". Suggestions for functions
to be added to the table are welcome.
3. Although a few of the functions do have special notations, they
are not in common use. See, for example, sinhc
<http://mathworld.wolfram.com/SinhcFunction.html> and tanc
<http://mathworld.wolfram.com/TancFunction.html> at MathWorld.
------------------------------------------------
Application: The surface area of a spheroid
For the surface area of a spheroid, separate formulae are often given,
depending on whether the spheroid is prolate or oblate. Here, we let r
and p denote the equatorial and polar radii, resp. One sometimes sees
S = 2 pi r (r + p asin(ec)/ec) where ec = sqrt(1 - (r/p)^2)
given for the prolate case (i.e., p > r) and
S = 2 pi r (r + p asinh(ec)/ec) where ec = sqrt((r/p)^2 - 1)
given for the oblate case (i.e., p < r). However, either formula works in
either case, as long as dealing with complex values during the intermediate
calculations is allowed. So why not give just a single formula which works
for _all_ spheroids? Because neither formula above works in the simplest
case: when the spheroid happens to be a sphere (i.e., p = r).
But using sinc, we can write a single formula which is correct for all
nondegenerate spheroids -- prolate, spherical or oblate. Such a formula
could be presented in different ways, depending on which inverse (circular
or hyperbolic) function is chosen. Two of my favorites are
S = 2 pi r (r + p/sinc(atan(sqrt((p/r)^2 - 1)))) [1]
S = 2 pi r (r + p/sinc(acos(r/p))) [2]
which are both valid for all positive p and r.
Perhaps something should also be said about the two degenerate cases: when
p or r, but not both, is zero. Unfortunately, a detailed discussion of
those cases would disturb the flow of this article. But very briefly,
both [1] and [2] are also valid in both of those degenerate cases if sinc
and the inverse function are defined "correctly", in particular, at certain
infinities. Obviously, if p = 0, we want S = 2 pi r^2 and, if r = 0, we
want S = 0.
------------------------------------------------
Application: Antiderivatives
Several common indefinite integrals which tables typically present either
with restrictions or in separate cases can instead be given, using sinc, in
a single case without restriction. As simple examples, consider integrating
cos(r x), sin(r x) and exp(r x) with respect to x. A table would typically
give the antiderivatives as sin(r x)/r, -cos(r x)/r and exp(r x)/r, resp.,
together with the restriction that r be nonzero. But using sinc, we have
the following, without restriction on r:
int(cos(r x), x) = x sinc(r x)
int(sin(r x), x) = r/2 (x sinc(r x/2))^2
int(exp(r x), x) = x (sinc(i r x) + r x/2 (sinc(i r x/2))^2)
As another example, consider integrating sin(m x) sin(n x) with
respect to x. A table would typically give an antiderivative as
sin((m-n)x)/(2(m-n)) - sin((m+n)x)/(2(m+n)) if m^2 <> n^2, and refer
the reader, in the event that m^2 = n^2, to a separate entry for the
antiderivative of (sin(n x))^2. But using sinc, we have the following,
without restriction on m and n:
int(sin(m x) sin(n x), x) = x/2 (sinc((m-n) x) - sinc((m+n) x))
Similarly, we also have
int(cos(m x) cos(n x), x) = x/2 (sinc((m-n) x) + sinc((m+n) x))
Giving such results in terms of sinc could be useful for computer
algebra systems. Most systems currently give results as they are given
in tables _but without stating the necessary restrictions_. We have
shown a way to avoid that problem in the examples above: using sinc so
that there are no restrictions. (By the way, Sinc is newly implemented
in version 6 of Mathematica. However, as an example, for
Integrate[Cos[r x], x], it still gives just Sin[r x]/r, without any
restriction on r being stated. Perhaps Sinc has not been fully
"integrated" into the system yet, but will be in some later version.)
As our last example, we consider integrating x^p with respect to x.
The result is typically stated in two cases: an antiderivative is
log(x) if p = -1, x^(p + 1)/(p + 1) otherwise. Note that, for a given
positive x, that result is discontinuous at p = -1. That discontinuity,
however, can be removed by altering a constant of integration, giving now
log(x) if p = -1, (x^(p + 1) - 1)/(p + 1) otherwise
as an antiderivative. Finally, if we wish to avoid having two cases, we can
use sinc to state a single result, valid for all p: int(x^p, x) =
log(x) (sinc(i(p+1)log(x)) + (p+1)log(x)/2 (sinc(i(p+1)log(x)/2))^2)
Needless to say, the examples above do not constitute a complete list
of common antiderivatives for which the use of sinc is helpful.
-------------------------------------------------
Further thoughts
In the example above, we would have liked to have had a simpler looking
result. Of course, that could have been achieved if we had expressed the
result in terms of, say, the function f(x) obtained by removing the
singularity of (exp(x) - 1)/x. [Recall that
f(x) = sinc(i x) + x/2 (sinc(i x/2))^2.] We would then have had, more
neatly, int(x^p, x) = log(x) f((p+1)log(x)). Should we perhaps devise a
standard notation for functions like those in our table? Of course, the
table showed how some of those functions can be written using sinc, but a
more compact notation would sometimes be nice. [As noted earlier, MathWorld
has already taken a step more-or-less in that direction (with sinhc and
tanc, for example). But I'm not certain that represents a good way to go.]
Furthermore, there are, of course, many functions of interest which
have removable singularities but which presumably cannot be written in
terms of sinc. And some of those functions are simple ones involving
elementary functions. Consider 1/x - cot(x), for example, which has a
removable singularity at x = 0. As far as I know, the function g(x)
obtained by removing that singularity cannot be expressed using sinc
itself. However, g(x) can be expressed nicely if we are also allowed
to use the derivative of sinc, namely, g(x) = -sinc'(x)/sinc(x).
------------------------------------------------
Thoughtful comments on this article are welcome!
David W. Cantrell
I.N. Galidakis
Very nice.
> David W. Cantrell
--
I.N. Galidakis
David W. Cantrell
Thank you, Ioannis, for your comment.
See below for an important function added to the table.
> This article discusses useful functions which can be written in
> terms of the sine cardinal function and shows two nice
> applications of those functions.
>
> ------------------------------------------------
>
> The sine cardinal function, abbrev.sinc, may be defined by
>
> | { 1 if x = 0,
> | sinc(x) = {
> | { sin(x)/x otherwise.
>
> [Readers might be interested in
> <http://mathworld.wolfram.com/SincFunction.html>. Among the
> references listed there, I particularly recommend the paper by
> Gearhart and Shultz.]
>
> This function removes the singularity of sin(x)/x, a commonly
> occurring function. But there are other commonly occurring
> functions having removable singularities for which the
> corresponding functions, with those singularities removed, do
> not have convenient notations in common use. When needed, those
> corresponding functions would typically be defined piecewise.
> In this first section, we show how several of those functions
> can be written nicely in terms of the sine cardinal function,
> thereby obviating the need to define them piecewise ad hoc.
> In the table below, the left column gives a function having a
> removable singularity at x=0 and the right column gives the
> corresponding function, written usingsinc, with that
> singularity removed.
>
> sin(x)/x sinc(x)
>
> tan(x)/x sinc(x)/cos(x)
>
> (1 - cos(x))/x^2 1/2 (sinc(x/2))^2
>
> asin(x)/x 1/sinc(asin(x))
>
> atan(x)/x cos(atan(x))/sinc(atan(x))
>
> -----------------------------------
>
> sinh(x)/x sinc(i x)
>
> tanh(x)/x sinc(i x)/cos(i x)
>
> (cosh(x) - 1)/x^2 1/2 (sinc(i x/2))^2
>
> asinh(x)/x 1/sinc(i asinh(x))
>
> atanh(x)/x cos(i atanh(x))/sinc(i atanh(x))
>
> -----------------------------------
>
> (exp(x) - 1)/x sinc(i x) + x/2 (sinc(i x/2))^2
>
> log(x + 1)/x 1/(sinc(i log(x+1)) + log(x+1)/2 (sinc(i log(x+1)/2))^2)
>
> -----------------------------------
[snip]
> The table is, of course, not "complete". Suggestions for
> functions to be added to the table are welcome.
Here's one to be added. It has a different "character" from the functions previously in the table. Removing the singularity of
x log|x|
at x = 0, we get the function
sign(x)(x^2 - 1)/(2 sinc(i atanh((x^2 - 1)/(x^2 + 1))))
[snip]
At the end of the section
Application: The surface area of a spheroid,
I had said that two formulae for the surface area
> are also valid in both of those degenerate cases if sinc
> and the inverse function are defined "correctly", in
> particular, at certain infinities.
Perhaps I should say a little about that now, because the function newly added to the table depends on that behavior. Of course, to remove the singularity of x log|x|, we need the new function to have the value 0 when x = 0. Here, step by step, I show how that is achieved when we substitute 0 for x in the function above:
sign(0)(0^2 - 1)/(2 sinc(i atanh((0^2 - 1)/(0^2 + 1)))) =
0 (-1)/(2 sinc(i atanh(-1))) =
0/(2 sinc(i (-oo))) =
0/(2 sinc(-i oo)) =
0/(2 oo) =
0/oo =
0, as desired.
Of course, I still welcome suggestions, including those for more functions to be added to the table.
David W. Cantrell