Re: Min and max are elementary functions?!

Dave L. Renfro wrote:

The following quote is from p. 10, at the end of the section
"Functions of First Order", in Chapter I: "Elementary Functions
of One Variable" of

Joseph Fels Ritt, "Integration in Finite Terms: Liouville's
Theory of Elementary Methods", Columbia University Press, 1948.

"For instance, u^2 - e^x = 0 gives two distinct analytic
functions. There is no point of difficulty here. A function
u of the first order is a definite function, for which a
scheme of construction can be given as above. There may be
other functions whose schemes of construction employ the
same material which appear in the scheme for u."

David W. Cantrell wrote:

Thanks. (Unfortunately, I don't have ready access to the
Ritt text.

I got it through interlibary loan a few years ago and
made a copy of it because it kept coming up in posts
that I found myself interested in. This is a book that
I'm surprised Dover Publications hasn't reprinted,
by the way.

Dave L. Renfro wrote:

If P(y) is a polynomial in the variable y with elementary
function coefficients, then any solution to P(y) = 0 is
an elementary function. The fact that, for any specific
polynomial P used to witness a certain function being
elementary also witnesses other functions being
elementary, is not excluded by the definition.

David W. Cantrell wrote:

I had been under the impression that x = y^2 gives us
_only one_ algebraic function of y in terms of x. That
function is bivalued.

Let's make up a crazy single-valued square root function:

f(x) = sqrt(x) if x is rational, -sqrt(x) if x is irrational

where sqrt(x) denotes the principal-valued square root function.

Of course, y = f(x) satisfies x = y^2. So then is f(x)
algebraic and hence elementary?

Ritt's definition of "elementary function" is quite precise
(I don't have Ritt with me where I'm at now, however) and
it would certainly exclude the function you defined.

You might want to look up some of these references at
whatever college or university is near you, at least
those that aren't available on-line. Most of these journals
should be at most U.S. colleges and universities (4-year
colleges, that is).

http://scholar.google.com/scholar?q=elementary-function+Ritt+analytic

You might also find something useful looking through these
search hits:

http://www.google.com/search?q=Integration-in-finite-terms+Ritt+analytic

http://scholar.google.com/scholar?q=Integration-in-finite-terms+Ritt+analytic

Finally, searching for the phrases "elementary function"
and (a separate search) "elementary functions" in this
book might be of use:

"Symbolic Integration I: Transcendental Functions"
by Manuel Bronstein, 2004
http://books.google.com/books?vid=ISBN3540214933

Dave L. Renfro

.

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