Ping: Jose Carlos Santos
Dave L. Renfro
across your paper, co-authored with Gabriela Chaves,
"Why some elementary functions are not rational"
(Mathematics Magazine 77, 2004, 225-226). If you're
interested, the following references also deal with
various aspects of this topic. In particular, Speck
proves that each of the elementary transcendental
functions (trigonometric, exponential, and their
inverses) differs from every algebraic function on
every open interval.
George P. Speck, "Elementary transcendental functions",
Mathematics Magazine 42 (1969), 200-202.
Reprinted on pp. 80-82 of "A Century of Calculus: Part II
1969-1991", The Mathematical Association of America, 1992.
Richard Wesley Hamming, "An elementary discussion of the
transcendental nature of the elementary functions", American
Mathematical Monthly 77 (1970), 294-297.
Godfrey Harold Hardy, A COURSE OF PURE MATHEMATICS,
9'th edition, Cambridge University Press, 1947. [see pp. 52-57]
James Pierpont, THE THEORY OF FUNCTIONS OF REAL VARIABLES,
Volume 1, Ginn and Company, 1905. [see pp. 123-137]
http://historical.library.cornell.edu/math/math_P.html
Dave L. Renfro
José Carlos Santos
> I was sorting through some papers this morning and came
> across your paper, co-authored with Gabriela Chaves,
> "Why some elementary functions are not rational"
> (Mathematics Magazine 77, 2004, 225-226). If you're
> interested, the following references also deal with
> various aspects of this topic. In particular, Speck
> proves that each of the elementary transcendental
> functions (trigonometric, exponential, and their
> inverses) differs from every algebraic function on
> every open interval.
Thanks. I will look at them. The paper by Speck should be interesting; I
certainly would like to see how he deals with the logarithm.
> Godfrey Harold Hardy, A COURSE OF PURE MATHEMATICS,
> 9'th edition, Cambridge University Press, 1947. [see pp. 52-57]
Concerning this one, since I owe a copy of the 10th edition, could you
please provide the section number(s) instead?
> James Pierpont, THE THEORY OF FUNCTIONS OF REAL VARIABLES,
> Volume 1, Ginn and Company, 1905. [see pp. 123-137]
> http://historical.library.cornell.edu/math/math_P.html
This link doesn't seem to be working.
Thanks for your interest.
Best regards,
Jose Carlos Santos
Dave L. Renfro
>> Godfrey Harold Hardy, A COURSE OF PURE MATHEMATICS,
>> 9'th edition, Cambridge University Press, 1947. [see pp. 52-57]
José Carlos Santos wrote:
> Concerning this one, since I owe a copy of the 10th edition,
> could you please provide the section number(s) instead?
Hummm...I have the 10'th edition also. However, I copied
this particular reference from an old post of mine, which
I suppose was made before I bought a copy of Hardy's
book myself (about 2 years ago) and had to use a library's
copy.
See Sections 26-28. In particular, Exercises 14-16 in
Section 28 give the same idea that Pierpont used
to show that no function with a positive period can be
periodic (via a contradiction to the existence of a
minimal monic polynomial for the function). Of course,
this doesn't give the result for any open interval, but
Speck manages to get the stronger open interval
version in an elementary and relatively simple way.
("Elementary" refers to the level of mathematics used,
and "simple" refers to the complexity of the proofs.)
Dave L. Renfro wrote:
> James Pierpont, THE THEORY OF FUNCTIONS OF REAL VARIABLES,
> Volume 1, Ginn and Company, 1905. [see pp. 123-137]
> http://historical.library.cornell.edu/math/math_P.html
José Carlos Santos wrote:
> This link doesn't seem to be working.
You might have to copy and paste the URL into another
internet browser window, although it works for me
by just clicking on the link in my post at
http://groups.google.com/group/sci.math/msg/92152c8b1df157cb
If this still doesn't work, go to
http://historical.library.cornell.edu/
Click on the picture above "Historic Math Book Collection",
then click "Browse", then (making sure you're in the
"Sorted by Author" format) click on "PQ", then look for
Pierpont.
By the way, you might also be interested in a more extreme
type of transcendental function. A function y of x is said
to be "transcendentally transcendental" on an interval (a,b)
if P(x, y, y', y'', ..., y^(n)) is not identically zero on
(a,b) for every positive integer n and every nonzero polynomial P
of n+1 variables with rational function coefficients. In
other words, y doesn't satisfy any algebraic differential
equation (even non-linear). None of the elementary transcendental
functions have this property, and most of the higher functions
in mathematical physics don't either. However, in 1887 Holder
proved that the gamma function is transcendentally transcendental.
(I'm not sure, but I think Holder was also the first to
formulate this property.) A nice survey paper of this topic is:
Lee Albert Rubel, "A survey of transcendentally transcendental
functions", American Mathematically Monthly 96 (1989), 777-788.
Some of the older papers, including Holder's original paper
and some by E. H. Moore (1897 -- this is where the term 'TT'
originates) and J. F. Ritt (1923, 1926), are in Math. Annalen,
and thus are available on the internet. There's also a 1902
paper by Edmond Maillet in Bulletin de la Societe Mathematique
de France (Vol. 30, pp. 195-201) that's on the internet.
Mathematische Annalen
http://dz-srv1.sub.uni-goettingen.de/cache/toc/D25917.html
Bulletin de la Société Mathématique de France
http://www.numdam.org/numdam-bin/feuilleter?j=BSMF
http://www.google.com/search?as_epq=transcendentally+transcendental
http://books.google.com/books?as_epq=transcendentally+transcendental
http://scholar.google.com/scholar?as_epq=transcendentally+transcendental
Dave L. Renfro
Dave L. Renfro
> A function y of x is said to be "transcendentally transcendental"
> on an interval (a,b) if P(x, y, y', y'', ..., y^(n)) is not identically
> zero on (a,b) for every positive integer n and every polynomial P
> of n+1 variables with rational function coefficients.
There might be some additional hypotheses involved, but this
is the general idea. I don't have a copy of Rubel's paper with
me right now to see how he defines the notion. Also, besides
the google web, scholar, and book searches I gave for
"transcendentally transcendental", don't forget to do a usenet
search (if you're interested in reading up on this notion).
Dave L. Renfro
Bill Dubuque
>> "Why some elementary functions are not rational"
>> (Mathematics Magazine 77, 2004, 225-226).
> Thanks for your interest.
With d=deg, your proof H' != 0 -> dH' < dH
is clearer as follows
dH' = d(F/G)' = d(F'/G - F/G G'/G )
<= max(dH-1, dH-1)
I.e. use the prior-proved formula for d(f+g)
rather than inlining it into the above proof
(as you do in the paper).
--Bill Dubuque
José Carlos Santos
>> Concerning this one, since I owe a copy of the 10th edition,
>> could you please provide the section number(s) instead?
>
> Hummm...I have the 10'th edition also. However, I copied
> this particular reference from an old post of mine, which
> I suppose was made before I bought a copy of Hardy's
> book myself (about 2 years ago) and had to use a library's
> copy.
>
> See Sections 26-28. In particular, Exercises 14-16 in
> Section 28 give the same idea that Pierpont used
> to show that no function with a positive period can be
> periodic (via a contradiction to the existence of a
> minimal monic polynomial for the function).
OK. Thanks for the reference. Exercise 16 is funny; it says that
arcsin(x) and arccos(x) have infinitely many values when _x_ is
between -1 and 1.
>> James Pierpont, THE THEORY OF FUNCTIONS OF REAL VARIABLES,
>> Volume 1, Ginn and Company, 1905. [see pp. 123-137]
>> http://historical.library.cornell.edu/math/math_P.html
>
> José Carlos Santos wrote:
>
>> This link doesn't seem to be working.
*Now* it is working, but it wasn't the first time I tried.
> By the way, you might also be interested in a more extreme
> type of transcendental function. A function y of x is said
> to be "transcendentally transcendental" on an interval (a,b)
> if P(x, y, y', y'', ..., y^(n)) is not identically zero on
> (a,b) for every positive integer n and every nonzero polynomial P
> of n+1 variables with rational function coefficients. In
> other words, y doesn't satisfy any algebraic differential
> equation (even non-linear). None of the elementary transcendental
> functions have this property, and most of the higher functions
> in mathematical physics don't either. However, in 1887 Holder
> proved that the gamma function is transcendentally transcendental.
> (I'm not sure, but I think Holder was also the first to
> formulate this property.)
That's what I think too.
José Carlos Santos
>>> I came across your paper, co-authored with Gabriela Chaves,
>>> "Why some elementary functions are not rational"
>>> (Mathematics Magazine 77, 2004, 225-226).
> http://www.fc.up.pt/mp/jcsantos/PDF/artigos/rational.pdf
>> Thanks for your interest.
>
> With d=deg, your proof H' != 0 -> dH' < dH
> is clearer as follows
>
> dH' = d(F/G)' = d(F'/G - F/G G'/G )
>
> <= max(dH-1, dH-1)
Cute!
Virgil
José Carlos Santos <jcsa...@fc.up.pt> wrote:
> Dave L. Renfro wrote:
>
> >> Concerning this one, since I owe a copy of the 10th edition,
> >> could you please provide the section number(s) instead?
> >
> > Hummm...I have the 10'th edition also. However, I copied
> > this particular reference from an old post of mine, which
> > I suppose was made before I bought a copy of Hardy's
> > book myself (about 2 years ago) and had to use a library's
> > copy.
> >
> > See Sections 26-28. In particular, Exercises 14-16 in
> > Section 28 give the same idea that Pierpont used
> > to show that no function with a positive period can be
> > periodic (via a contradiction to the existence of a
> > minimal monic polynomial for the function).
>
> OK. Thanks for the reference. Exercise 16 is funny; it says that
> arcsin(x) and arccos(x) have infinitely many values when _x_ is
> between -1 and 1.
Does it distinguish between the "complete" arc functions and the
"principal" arc functions in the same sense as that the "complete"
square root of 4 is {-2, 2} whereas the principle square root of 4 is 2?
José Carlos Santos
>>> See Sections 26-28. In particular, Exercises 14-16 in
>>> Section 28 give the same idea that Pierpont used
>>> to show that no function with a positive period can be
>>> periodic (via a contradiction to the existence of a
>>> minimal monic polynomial for the function).
>> OK. Thanks for the reference. Exercise 16 is funny; it says that
>> arcsin(x) and arccos(x) have infinitely many values when _x_ is
>> between -1 and 1.
>
> Does it distinguish between the "complete" arc functions and the
> "principal" arc functions in the same sense as that the "complete"
> square root of 4 is {-2, 2} whereas the principle square root of 4 is 2?
In this part of the book, the circular functions and their inverses
haven't been defined rigorously yet. All that Hardy says is "We may
assume that the reader is familiar with their most important
properties." He writes this at the start of the section 28. I saw no
reference to a "principal part". See
(This is supposed to be a single line.)
