Min and max are elementary functions?!
David W. Cantrell
Before trying to explain to them why their classification of min and max as
elementary was wrong, I decided to ask here:
How would you explain to them what's wrong with that classification?
Of course, I'm assuming that you agree that min and max are not elementary.
But if you think that, somehow, they can legitimately be called elementary,
please tell me why. I expect that the people running the web site will say
that
min(a, b) = 1/2 (a + b - sqrt((a - b)^2))
and
max(a, b) = 1/2 (a + b + sqrt((a - b)^2))
justify classifying min and max as elementary. But of course, for those two
identities to be correct, "sqrt" must be the _principal_ branch. And
there's the rub...
David
N. Silver
> I expect that the people running the web site will say
> that min(a, b) = 1/2 (a + b - sqrt((a - b)^2))
> and max(a, b) = 1/2 (a + b + sqrt((a - b)^2))
> justify classifying min and max as elementary. But of
> course, for those two identities to be correct, "sqrt"
> must be the _principal_ branch. And there's the rub...
Is abs(x) elementary? As you know, abs(x) = sqrt(x^2).
David W. Cantrell
No.
> As you know, abs(x) = sqrt(x^2).
To paraphrase myself:
For that identity to be correct, "sqrt" must be the _principal_ branch. And
there's the rub...
Of course, the bivalued square root "function" is algebraic, and hence
elementary as well. But surely we can't build a function out of just bits
and pieces of elementary (multi)functions and then always legitimately call
the result elementary.
David
Dirk Van de moortel
"David W. Cantrell" <DWCan...@sigmaxi.org> wrote in message news:20060604081304.135$l...@newsreader.com...
In the above identities sqrt *is* the principal branch:
sqrt(x^2) = x for x >= 0
sqrt(x^2) = -x for x <= 0
and therefore
sqrt((a-b)^2) = a - b for a - b >= 0 i.o.w. for a >= b
sqrt((a-b)^2) = - a + b for a - b <= 0 i.o.w. for a <= b
producing a non-negative number in all cases
Dirk Vdm
matt271...@yahoo.co.uk
But wouldn't the "elementary function" sqrt consist only of the
principal branch? Otherwise sqrt would be multi-valued, and therefore,
according to many, not even a "proper" function?
Virgil
Is the absolute value function an "elementary function"?
If not, what is your definition of "elementary function"?
Stephen J. Herschkorn
And if absolute value is not an elementary function, then the
indefinite integral of 1/x (w.r.t x) cannot be expressed in terms
of elementary functions.
--
Stephen J. Herschkorn sjher...@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan
G. A. Edgar
sub-classification of "analytic function on a Riemann surface".
So: in one sense, since "max" is defined only for reals, it cannot
qualify. On the other hand, the expression:
max(a, b) = 1/2 (a + b + sqrt((a - b)^2))
two-fold cover of C^2 minus the complex line {a=b}). So you can say
it *is* elementary, being a real restriction of a branch of
that function.
So, what is the very technical definition of "elementary function" used
on that web site? If there is none, then there is no point arguing
whether something does or does not satisfy it.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Dave L. Renfro
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."
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.
Dave L. Renfro
The World Wide Wade
David W. Cantrell <DWCan...@sigmaxi.org> wrote:
> But of course, for those two
> identities to be correct, "sqrt" must be the _principal_ branch. And
> there's the rub...
And why is that the rub? You'll have to throw out ln(x), as it's
the principal branch, most algebraic functions, ... this looks
like a catastrophic restriction; what definition of "elementary"
are you using?
The World Wide Wade
<1149451303.7...@u72g2000cwu.googlegroups.com>,
"Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
> And if absolute value is not an elementary function, then the
> indefinite integral of 1/x (w.r.t x) cannot be expressed in terms
> of elementary functions.
Well ln(x^2)/2 will work, so if ln(x) for x > 0 is elementary,
there is no problem.
David W. Cantrell
> I would say, in the classical sense, "elementary function" is a
> sub-classification of "analytic function on a Riemann surface".
> So: in one sense, since "max" is defined only for reals, it cannot
> qualify. On the other hand, the expression:
> max(a, b) = 1/2 (a + b + sqrt((a - b)^2))
> does define an elementary function on a certain domain (perhaps a
> two-fold cover of C^2 minus the complex line {a=b}). So you can say
> it *is* elementary, being a real restriction of a branch of
> that function.
>
> So, what is the very technical definition of "elementary function" used
> on that web site?
There is none.
> If there is none, then there is no point arguing
> whether something does or does not satisfy it.
That's excellent advice. Perhaps I was just being silly in thinking that
there would be some generally accepted definition of "elementary function"
(or "algebraic function" or ...) so that one could say with assurance that
something is clearly right or wrong in that regard.
Regards,
David
David W. Cantrell
> > In article <20060604081304.135$l...@newsreader.com>,
> > David W. Cantrell <DWCan...@sigmaxi.org> wrote:
> >
> > > A certain web site lists min and max as being elementary functions.
> > >
> > > Before trying to explain to them why their classification of min and
> > > max as elementary was wrong, I decided to ask here:
> > >
> > > How would you explain to them what's wrong with that classification?
> > >
> > > Of course, I'm assuming that you agree that min and max are not
> > > elementary. But if you think that, somehow, they can legitimately be
> > > called elementary, please tell me why. I expect that the people
> > > running the web site will say that
> > >
> > > min(a, b) = 1/2 (a + b - sqrt((a - b)^2))
> > >
> > > and
> > >
> > > max(a, b) = 1/2 (a + b + sqrt((a - b)^2))
> > >
> > > justify classifying min and max as elementary. But of course, for
> > > those two identities to be correct, "sqrt" must be the _principal_
> > > branch. And there's the rub...
> > >
> > > David
> >
> > Is the absolute value function an "elementary function"?
I said "No" to that same question from N. Silver.
> > If not, what is your definition of "elementary function"?
I started this thread, in part, to try to ascertain whether there was a
generally accepted definition of elementary function. I believe that
everyone agrees that algebraic functions are elementary. But, at least
according to some authors, the principal-valued square root function is not
algebraic, while the bivalued function is. (Of course, I'm not disputing
the fact that, _loosely speaking_, the principal-valued square root
function is "algebraic" and hence "elementary". But I wanted to know what
was technically correct in this regard.)
If the principal-valued square root function is not algebraic, then I
should think it would not be elementary either. Hence my answer of "No" to
N. Silver about the absolute value function as well.
> And if absolute value is not an elementary function, then the
> indefinite integral of 1/x (w.r.t x) cannot be expressed in terms
> of elementary functions.
Sure it can. It's log(x) + C. Absolute value is not needed. (If you want a
real answer when x < 0, just use an appropriate imaginary constant as part
of C.)
David
David W. Cantrell
Thanks. (Unfortunately, I don't have ready access to the Ritt text.)
> 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.
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?
Regards,
David
David C. Ullrich
<DWCan...@sigmaxi.org> wrote:
That was my opinion when I read your post (didn't see much
point in saying so, but now that I can _agree_ with you
by saying you were being silly I may as well. heh-heh.)
>Regards,
>David
************************
David C. Ullrich
David W. Cantrell
OK. So we're all agreed then that it's fine to call things like
f(x) = sqrt(x) if x is rational, -sqrt(x) if x is irrational
an algebraic function because it satifies x = y^2
(and an elementary function because it's algebraic).
David Cantrell
Rainer Rosenthal
> OK. So we're all agreed then that it's fine to call things like
>
> f(x) = sqrt(x) if x is rational, -sqrt(x) if x is irrational
>
> an algebraic function because it satifies x = y^2
> (and an elementary function because it's algebraic).
Surely You're Joking, Mr. Cantrell.
Why should a function be called "elementary", if
you have to define it piecewise? The pretty and
innocent function sqrt, whose only fault is its
being undefined for negative numbers, has nothing
to do with the "un-elementaryness" of functions
of the kind you gave above.
Or would you call this function f elementary:
f(x) = 1 if x is rational
f(x) = -1 if x is irrational.
I wouldn't. And I think that g(x) = 1 is quite
elementary.
I remember fascinating contributions of you to
packings of squares and to approximating circum-
ferences of ellipses. Thanks for that!
Best regards and good health,
Rainer Rosenthal
r.ros...@web.de
Dave Seaman
> David W. Cantrell schrieb:
> Or would you call this function f elementary:
> f(x) = 1 if x is rational
> f(x) = -1 if x is irrational.
It may not be an elementary function, but it is definitely a simple function.
On the other hand, the identity function is elementary but is not simple.
--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
David W. Cantrell
> David W. Cantrell schrieb:
>
> > OK. So we're all agreed then that it's fine to call things like
> >
> > f(x) = sqrt(x) if x is rational, -sqrt(x) if x is irrational
> >
> > an algebraic function because it satifies x = y^2
> > (and an elementary function because it's algebraic).
>
> Surely You're Joking, Mr. Cantrell.
Indeed. I'm glad that my sarcasm was not lost in translation.
> Why should a function be called "elementary", if
> you have to define it piecewise? The pretty and
> innocent function sqrt, whose only fault is its
> being undefined for negative numbers, has nothing
> to do with the "un-elementaryness" of functions
> of the kind you gave above.
I assume your "sqrt" refers, as did mine, to the principal-valued square
root function. If so, then it's not entirely "innocent" since it's the
result of an _arbitrary_ choice as to what we call the principal branch.
You seem to feel strongly, as do I, that my f(x) above should not be
considered an elementary function. But look at, say, the definition of
algebraic function as given at PlanetMath:
<http://planetmath.org/encyclopedia/AlgebraicFunction.html>
My f(x) seems to be an algebraic function according to that definition. [If
you think f(x) doesn't fit their definiton, please tell me why it fails to
do so!]
And then look at their definition of elementary function:
<http://planetmath.org/encyclopedia/ElementaryFunction.html>
Just as it should (at least in my opinion), it specifically includes all
algebraic functions as being elementary. Therefore, as best I can tell, if
we use the definitions at PlanetMath, my f(x) is elementary.
Please note that I am not trying to "poke fun" at PlanetMath! (And BTW,
their site is not the site to which I alluded at the beginning of this
thread.)
> I remember fascinating contributions of you to
> packings of squares and to approximating circum-
> ferences of ellipses. Thanks for that!
>
> Best regards and good health,
> Rainer Rosenthal
> r.ros...@web.de
Are you acting as though I have lost my mind??? I assure you that I have
not. (Yeah, I know that's what _all_ the looneys say!)
I never think of any thread in which we're merely trying to clarify
terminology as being a "fascinating contribution". Unfortunately, all too
often, such threads turn out to be mere exercises in futility. I suspect
this thread may turn out that way too.
Regards,
David
Rainer Rosenthal
>
> Indeed. I'm glad that my sarcasm was not lost in translation.
>
:-)
> [... lots of words showing you didn't loose your mind ;-) ...]
> ... trying to clarify terminology ...
Good luck, sounds honest.
Best regards,
Rainer Rosenthal
r.ros...@web.de
David C. Ullrich
<DWCan...@sigmaxi.org> wrote:
Um. Yes, precisely, I can't imagine anyone arguing with that.
>David Cantrell
************************
David C. Ullrich
Dave L. Renfro
>> 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
David W. Cantrell
> Dave L. Renfro wrote:
> 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.
That is comforting (and not surprising, of course). And thanks for the
suggested references, Dave.
Here is some information from two general references.
---------------------------------
Ency. Dict. of Math., 2nd ed. (Math. Soc. of Japan)
Elementary Functions
A function of a finite number of real or complex variables that is
algebraic, exponential, logarithmic, trigonometric, or inverse
trigonometric, or the composite of a finite number of these, is called an
elementary function... [Their next paragraph summarizes Liouville's
construction of elementary functions in classes.]
Algebraic Functions
An algebraic function is a multiple-valued analytic function w = w(z)
defined by an irreducible algebraic equation P(z, w) = 0 with complex
coefficients.
---------------------------------
Comments on the above:
As expected, their definition of algebraic function, which I think is quite
standard, excludes my f(x) from being algebraic. I mentioned earlier that I
was not trying to poke fun at the PlanetMath definition (or similar ones
found on the web). Nonetheless, those responsible for the web definitions
need to pay attention to little things like "multiple-valued analytic" and
"irreducible".
Also, to the several people who asked me if I thought |x| were elementary,
based on |x| = sqrt(x^2), where sqrt denotes the principal-valued square
root function: Note that |x| -- and, for that matter, sqrt -- are not
algebraic according to the definition above. (Of course, sqrt is a _branch_
of an algebraic function, but it is not an algebraic function _per se_.)
I believe that I have seen (but do not recall where) even more liberal
definitions of elementary function, according to which inversion after
composition is allowed. Then, for example, according to such a definition,
the Lambert W function would be elementary.
---------------------------------
Ency. of Math. (Hazewinkel, managing. ed.)
Elementary functions - The class of functions consisting of the
polynomials, the exponential functions, the logarithmic functions, the
trigonometric functions, the inverse trigonometric functions, and the
functions obtained from those listed by the four arithmetic operations and
by superposition (formation of a composite function), applied finitely many
times. ...
---------------------------------
Comments on the above:
This differs in an important way from the EDM definition: General algebraic
functions are not included. That surprised me. (To get the square root, one
must use exponential and logarithmic functions.) I must wonder if this was
really intended.
David W. Cantrell
Dave L. Renfro
> I believe that I have seen (but do not recall where) even
> more liberal definitions of elementary function, according
> to which inversion after composition is allowed. Then, for
> example, according to such a definition, the Lambert W function
> would be elementary.
The inverse of an elementary function does not have to be
an elementary function. Ritt proves this in the case of
the Kepler equation x = y - c*sin(y) on p. 56 of his
1948 book. However, there is the notion of an "implicit
elementary function" that is much more general (basically,
any function of one or more variables can be implicitly
defined by specifying the values of some of the variables
of a multi-variable elementary function), which I discuss
at the end of this post (July 2, 2002):
http://groups.google.com/group/sci.math/msg/d2212e1d42119c51
Here's an excerpt that might be of interest:
Regarding integration of implicitly elementary functions, the
following paper may be of interest:
Robert H. Risch, "Implicitly elementary integrals", Proc. Amer.
Math. Soc. 57 (1976), 1-7.
(from Risch's abstract) "Here we prove a 1923 conjecture of
J. F. Ritt to the effect that if the indefinite integral of an
explicitly elementary function is implicitly elementary, then
it is explicitly elementary."
[I think by "explicit elementary" Risch means "elementary
in the sense that we've been using in this thread, which
includes solving algebraic equations whose solutions can't
be expressed in radical form.]
Dave L. Renfro