s(x) = 4 pi x(1-x) / (x^2 - x + 2)^2
which satisfies
|s(x) - sin pi x| < 0.0259, 0 <= x <= 1, its maximum being at x = 1/2.
(The function is even simpler when expressed in a version symmetric about x
= 1/2: if y = x(1 - x), then it becomes (pi y) / (1 - y/2)^2. )
I enquired around to see if anyone had seen it before, without any success,
though I did have one interesting reaction, given below.
This approximation is in an article called 'A simple approach to the
factorial function: the next step' that I think will appear in the next
issue of the Mathematical Gazette. It will be the second in what seems to
be developing into a series of notes; the first was
A simple approach to the factorial function, Mathematical Gazette, 80
(1996), 378-381,
there is a third currently with the journal, and a fourth in my head.
Anyway I recently corrected the proofs of this second article and asked the
editor if he could squeeze a note added in proof which, in these
circumstances, must be in telegramese:
"David Handscomb of the Oxford University Computing Laboratory Numerical
Analysis Group has shown me how to use standard though more advanced
techniques to get a similar but more elaborate and rather better version of
the approximation s(x) to sin(pi x).
Express sin(pi x) as a function of X = x(1 - x) (which has the same
symmetry as sin(pi x) about x = 1/2), expand this as a MacLaurin series,
and take the Pade approximation of order 1/2 of this series. Unravelling
the details, this gives the approximation
sigma(x) = [4 pi x (1 - x)] / [(x^2 - x + 2)^2 + (2 pi^2/3 - 5) x^2 (1 - x)^2]
for which
|sigma(x) - sin(pi x)| < 0.00621, 0 <= x <= 1, its maximum error being at 1/2.
However, like everyone else I have asked, he has no recollection of seeing
the approximation s(x) of this article."
Can anyone add anything more? In particular the function looks as if it may
be a distant cousin of the class of functions considered by Nigel
Backhouse, described in his posting of Jan 19. As a reflexion on the
explosion of interesting things in print, I didn't know of his article even
though it appeared in the same journal, which I usually try to skim, and
even though it was voted 'Best Short Article of the Year'. Also equally
embarrassingly. my own was then voted 'Best Note of the Year' - perhaps
there is a message here. I tried to get the editor to change my certificate
to 'Best Footnote of the Year', but he was far too busy with much more
pressing and worthwhile things!
David Fowler
PS As an excuse for adding this bit of mathematics to the history list, I
should add that third notes contains a fair amount of historical comment,
and the next following one should also.