Why is ln(x) an elementary function?
mike3
I have a question: Why is ln(x) (the natural logarithm function)
elementary? Is there a closed form formula using e that gives the
natural logarithm?
Robert J. Kolker
mike3 wrote:
There is a power series expansion for ln(x).
BTW, I thought elementary functions are those finitely constructed from
the basic arithmetic operations.
Bob Kolker
Slappy the Squirrel
"mike3" <mike...@yahoo.com> wrote in message
news:1d54b7e4.02063...@posting.google.com...
> Hi.
>
> I have a question: Why is ln(x) (the natural logarithm function)
> elementary?
I believe that "elementary functions," are functions involving
transcendental, arithmetic, and trigonometric functions. In other words,
ln(x) is "elementary" by convention.
> Is there a closed form formula using e that gives the
> natural logarithm?
Nope, but it is possible to form a series expansion which converges to ln(x)
over a certain interval.
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David W. Cantrell
> BTW, I thought elementary functions are those finitely constructed from
> the basic arithmetic operations.
The definition is not that restrictive.
The exponential functions (and so the trigonometric functions too) and
their inverses are always allowed in the construction, AFAIK. And some
definitions, IIRC, also allow the inverse of any elementary function to be
considered elementary as well (in which case, for example, the Lambert W
function would be elementary).
David
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Robert Israel
David W. Cantrell <DWCan...@sigmaxi.org> wrote:
>"Robert J. Kolker" <bobk...@attbi.com> wrote:
>> BTW, I thought elementary functions are those finitely constructed from
>> the basic arithmetic operations.
>The definition is not that restrictive.
>The exponential functions (and so the trigonometric functions too) and
>their inverses are always allowed in the construction, AFAIK. And some
>definitions, IIRC, also allow the inverse of any elementary function to be
>considered elementary as well (in which case, for example, the Lambert W
>function would be elementary).
I've never seen such a definition of "elementary". On the other hand,
algebraic functions are allowed. The definition that is used in the
theory of differential fields (and thus is the criterion for saying
whether a function has an elementary antiderivative) goes something
like this.
Start with F_0 = C(z), the field of rational functions in an indeterminate z.
Extend this by introducing, one by one, a finite number of new functions
t_1, ..., t_n, so that F_j is the field obtained from F_{j-1} by adjoining
t_j, where each t_j can be either:
1) algebraic, i.e. P(t_j) = 0 where P is a polynomial of degree >= 1 with
coefficients in F_{j-1},
2) exponential, i.e. t_j = exp(s) for some s in F_{j-1}, or
3) logarithmic, i.e. t_j = ln(s) for some s in F_{j-1}.
[Really it's done a bit more abstractly - it's algebra rather
than function theory. So the last two are stated as
t'_j = s' t_j and t'_j = s'/s respectively]
Then F_n is an elementary extension of F_0, and the functions in it are
elementary functions.
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
David W. Cantrell
> In article <20020630233743.538$S...@newsreader.com>,
> David W. Cantrell <DWCan...@sigmaxi.org> wrote:
> >"Robert J. Kolker" <bobk...@attbi.com> wrote:
> >> BTW, I thought elementary functions are those finitely constructed
> >> from the basic arithmetic operations.
>
> >The definition is not that restrictive.
> >The exponential functions (and so the trigonometric functions too) and
> >their inverses are always allowed in the construction, AFAIK. And some
> >definitions, IIRC, also allow the inverse of any elementary function to
> >be considered elementary as well (in which case, for example, the
> >Lambert W function would be elementary).
>
> I've never seen such a definition of "elementary".
Then, until such a time as I may find a reference to the contrary, I shall
suspect that my memory was wrong.
> On the other hand, algebraic functions are allowed.
Yes indeed. For example, the inverse of f(x) = x^5 + x, mentioned in the
current thread "QUINTICS", is an elementary function. [And yet I would
guess that most people who use mathematics (but aren't professional
mathematicians) would say that it is not elementary.]
Regards,
José Carlos Santos
> > I have a question: Why is ln(x) (the natural logarithm function)
> > elementary? Is there a closed form formula using e that gives the
> > natural logarithm?
>
>
>
> There is a power series expansion for ln(x).
The gamma function, for instance, also has a power series expansion,
but nobody says that it is elementary.
Best regards
Jose Carlos Santos
mike3
David W. Cantrell
> Could you please tell me what the inverse of f(x) = x^5 + x is?
Tell you what it is _in terms of what precisely_? It can't be expressed in
terms of radicals, for example.
Ronald Bruck
the "To," "Cc," and "Newsgroups" headers for details. ]]
In article <20020701084556.525$W...@newsreader.com>, David W. Cantrell
<DWCan...@sigmaxi.org> wrote:
> isr...@math.ubc.ca (Robert Israel) wrote:
> > In article <20020630233743.538$S...@newsreader.com>,
> > David W. Cantrell <DWCan...@sigmaxi.org> wrote:
> > >"Robert J. Kolker" <bobk...@attbi.com> wrote:
> > >> BTW, I thought elementary functions are those finitely constructed
> > >> from the basic arithmetic operations.
> >
> > >The definition is not that restrictive.
> > >The exponential functions (and so the trigonometric functions too) and
> > >their inverses are always allowed in the construction, AFAIK. And some
> > >definitions, IIRC, also allow the inverse of any elementary function to
> > >be considered elementary as well (in which case, for example, the
> > >Lambert W function would be elementary).
> >
> > I've never seen such a definition of "elementary".
>
> Then, until such a time as I may find a reference to the contrary, I shall
> suspect that my memory was wrong.
>
> > On the other hand, algebraic functions are allowed.
>
> Yes indeed. For example, the inverse of f(x) = x^5 + x, mentioned in the
> current thread "QUINTICS", is an elementary function. [And yet I would
> guess that most people who use mathematics (but aren't professional
> mathematicians) would say that it is not elementary.]
Well, I **am** a professional mathematician, and I don't recall
whether, in this context of elementary functions, "algebraic" means
"solution of an algebraic equation" or "expressible in radicals in a
finite number of steps". I **thought**, in integration theory, it was
the latter.
This can all be made precise, of course; all I have to do is look up
the many posts which have discussed differential fields and how to
prove whether a given function cannot be integrated in terms of
elementary functions. I'm just too lazy.
--Ron Bruck
mike3
functions?
"David W. Cantrell" <DWCan...@sigmaxi.org> wrote in message
news:20020701220102.378$M...@newsreader.com...
Robert Israel
Ronald Bruck <br...@math.usc.edu> wrote:
>In article <20020701084556.525$W...@newsreader.com>, David W. Cantrell
><DWCan...@sigmaxi.org> wrote:
>> isr...@math.ubc.ca (Robert Israel) wrote:
>> > On the other hand, algebraic functions are allowed.
>> Yes indeed. For example, the inverse of f(x) = x^5 + x, mentioned in the
>> current thread "QUINTICS", is an elementary function. [And yet I would
>> guess that most people who use mathematics (but aren't professional
>> mathematicians) would say that it is not elementary.]
>Well, I **am** a professional mathematician, and I don't recall
>whether, in this context of elementary functions, "algebraic" means
>"solution of an algebraic equation" or "expressible in radicals in a
>finite number of steps". I **thought**, in integration theory, it was
>the latter.
It is the former.
>This can all be made precise, of course; all I have to do is look up
>the many posts which have discussed differential fields and how to
>prove whether a given function cannot be integrated in terms of
>elementary functions. I'm just too lazy.
That's what I did. See e.g. Matthew Wiener's article of November 16 1995
on the subject "Preliminary FAQ for elementary integrals".
Dann Corbit
> Would trig work? or do you need something like elliptic or hypergeometric
> functions?
>
> "David W. Cantrell" <DWCan...@sigmaxi.org> wrote in message
> news:20020701220102.378$M...@newsreader.com...
> > mike...@yahoo.com (mike3) wrote:
> > > David W. Cantrell <DWCan...@sigmaxi.org> wrote in message
> > > news:<20020701084556.525$W...@newsreader.com>...
> > > > isr...@math.ubc.ca (Robert Israel) wrote:
> > > > > On the other hand, algebraic functions are allowed.
> > > >
> > > > Yes indeed. For example, the inverse of f(x) = x^5 + x, mentioned in
> > > > the current thread "QUINTICS", is an elementary function. [And yet I
> > > > would guess that most people who use mathematics (but aren't
> > > > professional mathematicians) would say that it is not elementary.]
> >
> > > Could you please tell me what the inverse of f(x) = x^5 + x is?
> >
> > Tell you what it is _in terms of what precisely_? It can't be expressed
in
> > terms of radicals, for example.
You could always write a program to find the answer.
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
/*
One of Schroder's iteration formulae:
*/
double fest(double x, double C)
{
double a = x * x;
double b = a * a;
double q = b * b;
return x - (5.0 * b + 1.0) * (b * x + x - C) /
(5.0 * q - 10.0 * b + 1.0 + 20.0 * a * x * C);
}
/*
If the value is larger than 1, use 5th root as an approximation.
If the value is smaller than 1, use the original value as the guess.
*/
double finit(double x)
{
if (x > 1)
return pow(x, 0.2);
else
return x;
}
char string[32767];
int main(void)
{
puts("enter the number to transform:");
if (fgets(string, sizeof string, stdin)) {
int i = 0;
double x = atof(string);
double y = pow(x, 5.0) + x;
double guess;
printf("x = %g\n", x);
printf("y = x^5 + x = %g\n", y);
guess = finit(y);
printf("initial guess for inverse is %g\n", guess);
for (i = 1; i < 10; i++) {
guess = fest(guess, y);
printf("Current guess for y is %30.20g\n", guess);
}
return EXIT_SUCCESS;
} else {
return EXIT_FAILURE;
}
}
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Dave L. Renfro
[sci.math Mon, 01 Jul 2002 19:21:52 -0700]
http://mathforum.org/epigone/sci.math/bookhyrtrong
wrote
> Well, I **am** a professional mathematician, and I don't recall
> whether, in this context of elementary functions, "algebraic"
> means "solution of an algebraic equation" or "expressible in
> radicals in a finite number of steps". I **thought**, in
> integration theory, it was the latter.
>
> This can all be made precise, of course; all I have to do is
> look up the many posts which have discussed differential fields
> and how to prove whether a given function cannot be integrated
> in terms of elementary functions. I'm just too lazy.
It's the former. Gerald A. Edgar pointed this out to me in a
sci.math post back in Feburary 2000 at
http://mathforum.org/epigone/sci.math/spandbrerglen
Since then I've been careful to mention this whenever I
discuss it. [I also did some library research on the topic
and made a copy of Ritt's well-known book and some articles.]
For instance, in a December 3, 2001 ap-calculus post that I've
cited several times in sci.math, I wrote
> function, it follows that the integral of [sin(x^2)]^2 is also
> a non-elementary function.
>
> The situation is similar to sqrt(2) being irrational, which means
> that sqrt(2) cannot be expressed using a finite sequence of the
> four basic operations (+, -, *, /) applied to the integers.
>
> [[ Actually, the situation is more akin to Pi being
> transcendental, since "elementary function" includes
> functions that cannot be expressed in a finite way
> using "standard calculus functions" along with "standard
> precalculus" operations (in the same way that "algebraic
> number" includes solutions to polynomial equations having
> integer coefficients that cannot be expressed using a
> finite sequence of the four basic arithmetic operations
> along with positive integer root extractions -- for more
> about this, see my two Jan. 14, 2001 posts at
> <http://mathforum.org/epigone/sci.math.symbolic/playzerdblar>). ]]
This post, which also contains several useful on-line references,
can be found at
http://mathforum.org/epigone/ap-calc/fryspencrim
Besides Ritt's book, some useful non-on-line surveys are
Maxwell Rosenlicht, "Integration in finite terms", Amer. Math.
Monthly 79 (1972), 963-972.
A. D. Fitt and G. T. Q. Hoare, "The closed-form integration of
arbitrary functions", The Mathematical Gazette 77(479) (July 1993),
227-236.
Elena A. Marchisotto and Gholam-Ali Zakeri, "An invitation to
integration in finite terms", College Math. Journal 25 (1994),
295-308.
Incidentally, the function y = y(x) defined as the solution to
x = y - c*sin(y) [c is any fixed nonzero complex number]
is not elementary, even in this sense. Ritt proves this on page
56 of his book. [This function arises in some of Kepler's work
and Liouville proved it wasn't elementary back in the 1830's.]
Hence, the inverse of an elementary function is not elementary.
However, if we enlarge our class of functions to include functions
such as these, then we have what are called the "implicit elementary
functions". [More generally, we consider any function obtained
by implicitly solving for some variables in terms of other variables
in a system of equations formed by setting finitely many elementary
functions equal to zero.] To help distinguish these two notions, we
call the usual elementary functions "explicit elementary functions".
Of course, here the term 'explicit' DOESN'T refer to finite
expressibility in terms of radicals, trig. functions, etc.
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."
Dave L. Renfro
David W. Cantrell
> Would trig work?
No.
> or do you need something like elliptic or hypergeometric functions?
Yes. I think that there is an article at MathWorld about solving quintics
which gives some of the details. [Of course, you realize that if you
express the inverse of f(x) = x^5 + x that way, then you've resorted to
expressing an elementary function in terms of nonelementary function(s),
which doesn't sound nice!]
David
> "David W. Cantrell" <DWCan...@sigmaxi.org> wrote in message
> news:20020701220102.378$M...@newsreader.com...
> > mike...@yahoo.com (mike3) wrote:
> > > David W. Cantrell <DWCan...@sigmaxi.org> wrote in message
> > > news:<20020701084556.525$W...@newsreader.com>...
> > > > isr...@math.ubc.ca (Robert Israel) wrote:
> > > > > On the other hand, algebraic functions are allowed.
> > > >
> > > > Yes indeed. For example, the inverse of f(x) = x^5 + x, mentioned
> > > > in the current thread "QUINTICS", is an elementary function. [And
> > > > yet I would guess that most people who use mathematics (but aren't
> > > > professional mathematicians) would say that it is not elementary.]
> >
> > > Could you please tell me what the inverse of f(x) = x^5 + x is?
> >
> > Tell you what it is _in terms of what precisely_? It can't be expressed
> > in terms of radicals, for example.
--
Ronald Bruck
the "To," "Cc," and "Newsgroups" headers for details. ]]
In article <28ae5e5e.02070...@posting.google.com>, Dave L.
Renfro <renf...@cmich.edu> wrote:
Dave,
I accept your explanation. My mnemonic for remembering this is faulty:
I was thinking along the lines, "If we DID mean to include
non-expressible-in-radicals solutions of y^5 + y = c, for example,
then, since this can be solved in terms of elliptic functions, the
elliptic functions would stand a good chance of being elementary--and
nobody considers THEM to be elementary." Of course, recovering
elliptic functions from the particular way they solve this equation may
be nontrivial.
--Ron Bruck
(Also with a tip of the hat to Robert Israel)
mike3
> "mike3" <mike...@yahoo.com> wrote:
> > Would trig work?
>
> No.
>
> > or do you need something like elliptic or hypergeometric functions?
>
> Yes. I think that there is an article at MathWorld about solving quintics
> which gives some of the details. [Of course, you realize that if you
> express the inverse of f(x) = x^5 + x that way, then you've resorted to
> expressing an elementary function in terms of nonelementary function(s),
> which doesn't sound nice!]
>
using hypergeometric functions (That means the the inverse of f(x) =
x^5 - x is expressable that way (using non elementary functions to
describe an "elementary" function). But, how do you solve x^5 + x = a?
Robert Israel
Ronald Bruck <br...@math.usc.edu> wrote:
>I accept your explanation. My mnemonic for remembering this is faulty:
>I was thinking along the lines, "If we DID mean to include
>non-expressible-in-radicals solutions of y^5 + y = c, for example,
>then, since this can be solved in terms of elliptic functions, the
>elliptic functions would stand a good chance of being elementary--and
>nobody considers THEM to be elementary." Of course, recovering
>elliptic functions from the particular way they solve this equation may
>be nontrivial.
Or rather, impossible...
For a simpler example:
3 5
sin(5 x) = 5 sin(x) - 20 sin(x) + 16 sin(x)
so the equation 16 t^5 - 20 t^3 + 5 t = c has solutions
t = sin(1/5 arcsin(c) + 2 n Pi/5), expressed in terms of trig and
inverse-trig functions. But this doesn't mean the trig or inverse-trig
functions are algebraic: it's just that there are algebraic relations
involving the trig functions.
Dave L. Renfro
[sci.math Wed, 03 Jul 2002 09:42:40 -0700]
http://mathforum.org/epigone/sci.math/bookhyrtrong
wrote
> I accept your explanation. My mnemonic for remembering this
> is faulty: I was thinking along the lines, "If we DID mean to
> include non-expressible-in-radicals solutions of y^5 + y = c,
> for example, then, since this can be solved in terms of elliptic
> functions, the elliptic functions would stand a good chance of
> being elementary--and nobody considers THEM to be elementary."
> Of course, recovering elliptic functions from the particular way
> they solve this equation may be nontrivial.
>
> --Ron Bruck
>
> (Also with a tip of the hat to Robert Israel)
I don't know if this is in any way analogous, but I can't help
but notice that ANY real number (transcendental, non-Turing
computable, or whatever) can be used as coefficients of things
when we're talking about elementary functions. I suppose many
things in math simply wind up being defined the way they are because
they're the best (known) compromise between theoretical nice'ness
and closeness of fit with the intuitive notion being modeled.
[[ I wrote this midday July 3 but for some reason
google-groups wouldn't connect me up to post it
until now. (Specifically, clicking on google's
"Post a follow-up to this message" gave no response,
even after waiting over a minute.) I've since seen
Robert Israel's excellent example in response to
Ron Bruck's comments. ]]
Dave L. Renfro
Ray and Lynne Vickson
school. Trigonometric functions, logarithms, etc., are, by this
definition, elementary. That does not mean that
there are simple formulas for them. Back in ancient times when I was in
high school, we used these functions by looking them up in tables in the
back of the book or by using a sliderule.
This answer is not intended as a joke or as sarcasm. It really does seem
that the
distinction between elementary and non-elementary is pretty arbitrary.
Error functions or elliptic entegrals are not "elementary", but they are
no harder to use than logs or cosines if you employ good software.