Higher order "trigonometric" functions
Mike
I have been studying some very interesting functions that arise
as higher order analogues of the trig functions. The idea is quite
simple. Consider the Fermat curve x^n + y^n = 1. For any
parametrization we must have x^{n-1}x' + y^{n-1}y' = 0. Surely the
simplest and most natural possibility is x' = -y^{n-1} and y' = x^
{n-1} together with initial conditions x(0) = 1 and y(0) = 0. For n=2
this is how we get the ordinary cosine and sine.
I find it strange that these functions are not better known. Any
bright calculus student could think of the idea. There is almost
NOTHING in the literature on these functions, which is quite
surprising given the obvious connection with the Fermat-Wiles
theorem. I know of only one thorough study of these functions. In
1890 Alfred Cardrew Dixon wrote a long rather thorough analysis of a
slight generalization of the case n=3. Dixon wrote from an advanced
point of view for a reader who is thoroughly knowledgeable about
elliptic function theory as it was understood and articulated in the
late 19th century. In his paper Dixon stated that Cauchy knew a bit
about these functions and used them as a counterexample to a plausible
conjecture. But if Cauchy ever did a thorough study he kept it to
himself. I have worked out that most of Dixon's results can be
developed using only basic calculus rather than the high-powered
methods he used. I also know a bit about the case n=4 and damn little
about n > 4.
Does anyone know of any literature I should read that is relevant
to these functions?
Thanks,
Mike
Robert Israel
See Harold Boas's article from 1996 on the subject
"Entire solutions of f^2+g^2=1"
<http://groups.google.com/group/sci.math.research/msg/bbae0c865fc7e114>
and references given there.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Dave L. Renfro
http://groups.google.com/group/sci.math.research/msg/740f98b5f7c601b9
> I have been studying some very interesting functions
> that arise as higher order analogues of the trig functions.
> The idea is quite simple. Consider the Fermat curve
> x^n + y^n = 1. For any parametrization we must have
> x^{n-1}x' + y^{n-1}y' = 0. Surely the simplest and
> most natural possibility is x' = -y^{n-1} and y' = x^ {n-1}
> together with initial conditions x(0) = 1 and y(0) = 0.
> For n=2 this is how we get the ordinary cosine and sine.
>
> I find it strange that these functions are not better
> known. Any bright calculus student could think of the idea.
> There is almost NOTHING in the literature on these functions,
> which is quite surprising given the obvious connection with
> the Fermat-Wiles theorem. I know of only one thorough study
> of these functions. In 1890 Alfred Cardrew Dixon wrote a
> long rather thorough analysis of a slight generalization of
> the case n=3.
Kaufman [1] gives a survey of the literature and has 106 references.
Incidentally, the submission date for Kaufman's paper is July 1955.
I figured this was a typo for July 1965 until I looked at the dates
of the papers in his bibliography. The latest date for an item I
saw (in an admittedly very quick scan) was 1954 -- I saw 2 or 3 with
this date. I saw 1 or 2 with a 1953 date, and a few more (2 to 4)
dated 1950, 1951, or 1952. I wonder if this paper is a modern
record for the gap between a published submission date and the
actual publication date?
[1] H. Kaufman, "A bibliographical note on higher order sine
functions", Scripta Mathematica 28 #1 (May 1967), 29-36.
I haven't done any searches for papers on this topic (JSTOR, MR,
JFM, google-scholar, etc.), but it does happen to be one of the
topics I've been collecting papers on when I come across them.
What follows are papers I have that I didn't see in Kaufman's
bibliography. Indeed, this is over half of the papers in my
notebook on this topic, and I should also point out that no
paper by Dixon is listed in Kaufman's bibliography.
[2] Russell Euler and Jawad Sadek, "The $\pi$s go full circle",
Mathematics Magazine 72 #1 (February 1999), 59-63.
[3] A. F. Frumveller, "A theory and generalization of the
circular and hyperbolic functions", American Mathematical
Monthly 26 #7 (September 1919), 280-288.
[4] H. Kaufman, "Remarks on a generalization of the
trigonometric functions", American Mathematical Monthly
67 #8 (October 1960), 753.
[5] David Shelupsky, "A generalization of the trigonometric
functions", American Mathematical Monthly 66 #10
(December 1959), 879-884.
[6] H. P. Thielman, "A generalization of trigonometry",
(National) Mathematics Magazine 11 #8 (May 1937),
349-351.
[7] H. P. Thielman, "On generalized Cauchy functional equations",
American Mathematical Monthly 56 #7 (Aug.-Sept. 1949),
452-457.
[8] William H. Young, "On infinite integrals involving a
generalisation of the sine and cosine functions",
Quarterly Journal of Pure and Applied Mathematics
43 (1912), 161-177.
Various conference talks: V. B. Temple (AMM 45, 1937, p. 569),
G. E. Alrich (AMM 47, 1940, p. 413), Mildred Nelson (BAMS 34,
1928, p. 548), H. P. Thielman (BAMS 43, 1937, pp. 195-196),
Dorothory Pennock (BAMS 33, 1927, p. 518), R. Grammel (MTOAC
5 #35, July 1951, p. 155).
AMM = American Mathematical Monthly
BAMS = Bulletin of the American Mathematical Society
MTOAC = Mathematical Tables and Other Aids to Computation
Dave L. Renfro
dro...@gmail.com
In addition to the references already given, there is a couple of
papers by J. Mikusinski,
titled "On the trigonometry of the equation x^(n) + x = 0", where x^
(n) refers to the
nth derivative. This is not exactly what you ask for, but these might
have further pointers.
The Math Reviews numbers are MR114933 and MR49334. There might be also
others, I sort of remember him publishing on the subject occasionally,
As ever,
Vladimir Drobot
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
* Vladimir Drobot
* Retired and gainfully unemployed
* http://www.vdrobot.com
* mailto:dro...@pacbell.net
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
Mike
> I haven't done any searches for papers on this topic (JSTOR, MR,
> JFM, google-scholar, etc.), but it does happen to be one of the
> topics I've been collecting papers on when I come across them.
> What follows are papers I have that I didn't see in Kaufman's
> bibliography. Indeed, this is over half of the papers in my
> notebook on this topic, and I should also point out that no
> paper by Dixon is listed in Kaufman's bibliography.
Many thanks to everyone for the responses. It is strange that
Dixon's paper is not listed. The paper is
A.C. Dixon. On the doubly periodic functions arising out of the curve
x^3 + y^3 - 3axy = 1. Quart. J. 24 (1890), pp. 167-233.
Dixon's analysis is quite thorough. It turns out that the extra term
-3axy (a is an arbitrary parameter) does not make much difference. I
am mostly interested in the case a=0.
I am in the process of typing up what I know about n=3 and the
(much less) I know about n=4. I know very little about n>4 and I'm
not aware that anyone else knows much either.
Besides finding these functions interesting, I think they make
great projects for students. If you ever get an unusual student who
is very bright and has the exploratory instincts of a mathematician,
but who knows nothing beyond a couple semesters of calculus, these
functions are an ideal project. I have worked out that almost all of
Dixon's results can be reasoned out using only the most elementary
calculus techniques.
Regards,
Mike Cole
Dave L. Renfro
> Many thanks to everyone for the responses. It is strange that
> Dixon's paper is not listed. The paper is
>
> A.C. Dixon. On the doubly periodic functions arising out
> of the curve x^3 + y^3 - 3axy = 1. Quart. J. 24 (1890),
> pp. 167-233.
On the off-chance that someone interested in this paper is not
yet aware that virtually every volume of every 1800s era math
journal has been digitized by google and is freely available
on the internet, here's a URL that takes you to Dixon's paper:
http://books.google.com/books?id=R8EKAAAAIAAJ&pg=PA167
Dave L. Renfro
Dave L. Renfro
> Besides finding these functions interesting, I think they
> make great projects for students. If you ever get an unusual
> student who is very bright and has the exploratory instincts
> of a mathematician, but who knows nothing beyond a couple
> semesters of calculus, these functions are an ideal project.
> I have worked out that almost all of Dixon's results can be
> reasoned out using only the most elementary calculus techniques.
Here's a fairly recent (and thus perhaps better known at this
time) paper that I somehow overlooked earlier, a paper that
would serve nicely as warm-up reading for such a student.
Margen Cuko and Paul Belcher, "We need some new functions",
Mathematical Spectrum 36 #3 (2003/2004), 55-60.
Note: This paper is a more concise version of an essay that
Margen Cuko wrote at age 18 (under the guidance of
Paul Belcher) for the International Baccalaureate program.
Dave L. Renfro