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From: Fred on 19 Jul 2010 20:59
There's a paper of Hamming in the Monthly in 1970 in which certain
functions, like tan x, are proved to be transcendental. There seem to
be a few gaps in these proofs. I've been told there is a more recent
paper by someone on the same topic in which errors in previous papers
are corrected. Does anyone know of such a paper?

--Fred
From: Gerry on 19 Jul 2010 23:58
On Jul 20, 10:59 am, Fred <f.rich...(a)comcast.net> wrote:
> There's a paper of Hamming in the Monthly in 1970 in which certain
> functions, like tan x, are proved to be transcendental. There seem to
> be a few gaps in these proofs. I've been told there is a more recent
> paper by someone on the same topic in which errors in previous papers
> are corrected. Does anyone know of such a paper?

There was a thread here just a few days ago
with references to papers discussing transcendence
results for elementary functions. It shouldn't be
too hard to find. I don't remember whether the
Hamming paper was mentioned.
--
GM
From: Gerry Myerson on 20 Jul 2010 03:12
In article
<96e87d21-6285-4220-9a9d-67b62f80d616(a)x20g2000pro.googlegroups.com>,
Gerry <gerry(a)math.mq.edu.au> wrote:

> On Jul 20, 10:59�am, Fred <f.rich...(a)comcast.net> wrote:
> > There's a paper of Hamming in the Monthly in 1970 in which certain
> > functions, like tan x, are proved to be transcendental. There seem to
> > be a few gaps in these proofs. I've been told there is a more recent
> > paper by someone on the same topic in which errors in previous papers
> > are corrected. Does anyone know of such a paper?
>
> There was a thread here just a few days ago
> with references to papers discussing transcendence
> results for elementary functions. It shouldn't be
> too hard to find. I don't remember whether the
> Hamming paper was mentioned.

The subject header was, Why is arctan a transcendental function?

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: José Carlos Santos on 20 Jul 2010 04:29
On 20-07-2010 1:59, Fred wrote:

> There's a paper of Hamming in the Monthly in 1970 in which certain
> functions, like tan x, are proved to be transcendental. There seem to
> be a few gaps in these proofs. I've been told there is a more recent
> paper by someone on the same topic in which errors in previous papers
> are corrected. Does anyone know of such a paper?

I don't know about such paper, but you might be interested in these
ones:

Jose Carlos Santos and Gabriela Chaves, "Why some elementary functions
are not rational" (Mathematics Magazine 77, 2004, 225-226).

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).

Best regards,

Jose Carlos Santos
From: Fred on 20 Jul 2010 13:09
On Jul 20, 4:29 am, José Carlos Santos <jcsan...(a)fc.up.pt> wrote:
> On 20-07-2010 1:59,Fredwrote:
>
> > There's a paper of Hamming in the Monthly in 1970 in which certain
> > functions, like tan x, are proved to be transcendental. There seem to
> > be a few gaps in these proofs. I've been told there is a more recent
> > paper by someone on the same topic in which errors in previous papers
> > are corrected. Does anyone know of such a paper?
>
> I don't know about such paper, but you might be interested in these
> ones:
>
> Jose Carlos Santos and Gabriela Chaves, "Why some elementary functions
> are not rational" (Mathematics Magazine 77, 2004, 225-226).
>
> 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).
>
> Best regards,
>
> Jose Carlos Santos

Thanks for the references. The seoond one looks quite relevant. The
treatment there seems a lot more careful than Hamming's.

Your paper looks interesting also. It deals with a problem that
Hamming doesn't address, namely whether something can be rational (or
algebraic) on some small interval. For example, the fact that tan x is
not algebraic on its whole domain does not say that it couldn't be
algebraic on some nonempty open interval (although it is not).

--Fred
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