Regarding tan(x) = x
Dave L. Renfro
equation tan(x) = bx for b > 0 in connection with his
researches in the differential equations of heat flow.
Does anyone know whether the equation tan(x) = x
had been previously studied, or even just brought
up as an example of how graphing (in this case,
y = tan(x) together with y = x) can sometimes
give a lot of information about the nature of
the solutions to an equation?
In addition, is it known whether some or all of the
nonzero solutions to tan(x) = x are irrational?
Sure, they're probably all transcendental, but
have any of these solutions even been proved
irrational?
Finally, besides heat flow, I know this equation
arises in the solution to the Schrödinger equation
for a finite square well potential and as the zeros
of the spherical Bessel function j_1(x). Does anyone
know of other applications where this equation appears?
Dave L. Renfro
David Moran
"Dave L. Renfro" <renf...@cmich.edu> wrote in message
news:1137711971.7...@g49g2000cwa.googlegroups.com...
Dave L. Renfro
The only place I remember seeing it is in Partial Differential Equations.
I'm not sure what the context was, but I remember seeing it in that course.
Dave
Bleem Bleom
"Dave L. Renfro" <renf...@cmich.edu> wrote in message
news:1137711971.7...@g49g2000cwa.googlegroups.com...
equation tan(x) = bx for b > 0 in connection with his
researches in the differential equations of heat flow.
Does anyone know whether the equation tan(x) = x
had been previously studied, or even just brought
up as an example of how graphing (in this case,
y = tan(x) together with y = x) can sometimes
give a lot of information about the nature of
the solutions to an equation?
........................................................
Try the expansions, google it,
tan(x) = x - x^3/3 + 2x^5/15 - 17x^7/105 + ...
David W. Cantrell
> Joseph Fourier studied the solutions to the transcendental
> researches in the differential equations of heat flow.
>
> had been previously studied, or even just brought
> up as an example of how graphing (in this case,
> give a lot of information about the nature of
> the solutions to an equation?
Sorry that I don't know the answers to your questions. But you might,
nonetheless, be interested in expressions (2) and (3) at
<http://mathworld.wolfram.com/TancFunction.html>. BTW, do you know who
first came up with that series for the solutions? I should think that
someone (Fourier?) found it long before I did.
David W. Cantrell
> In addition, is it known whether some or all of the
> nonzero solutions to tan(x) =3D x are irrational?
> Sure, they're probably all transcendental, but
> have any of these solutions even been proved
> irrational?
>
> Finally, besides heat flow, I know this equation
> arises in the solution to the Schr=F6dinger equation
Dave L. Renfro
> The only place I remember seeing it is in Partial Differential
> Equations. I'm not sure what the context was, but I remember
> seeing it in that course.
It can be found in ODE and PDE books, because it affords
a nice illustration of the general behavior of eigenvalues in
a Sturm-Liouville problem. For example, y" + (b^2)y = 0
with the boundary conditions y(0) + y'(0) = 0 and y(1) = 0
gives rise to tan(b) = b when solving for the eigenvalues b^2.
To others who have replied so far, you can rest assured
that I've googled rather extensively already -- Usenet,
the web, and google-books. I was hoping someone
might mention something I haven't seen already. (It's
come up in sci.math many times before, for example.)
Dave L. Renfro (posting from Univ. of Iowa's math library)
Robert Israel
Dave L. Renfro <renf...@cmich.edu> wrote:
>In addition, is it known whether some or all of the
>nonzero solutions to tan(x) = x are irrational?
>Sure, they're probably all transcendental, but
>have any of these solutions even been proved
>irrational?
By Lindemann's theorem, if z is algebraic and nonzero
then exp(z) is transcendental.
Now tan(x) = x says exp(2 i x) = (i - x)/(i + x).
If x was algebraic and nonzero, 2 i x and (i-x)/(i+x)
would also be algebraic, and we'd have a contradiction.
So all the nonzero solutions are transcendental.
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Rouben Rostamian
>equation tan(x) = bx for b > 0 in connection with his
>researches in the differential equations of heat flow.
>
>Does anyone know whether the equation tan(x) = x
>had been previously studied, or even just brought
>up as an example of how graphing (in this case,
>y = tan(x) together with y = x) can sometimes
>give a lot of information about the nature of
>the solutions to an equation?
OK, I don't know the answer to your questions, but what
you asked brought back a memory from about 40 years ago,
so I will _have_ to write it down.
I was a freshman engineering student in 1968. That was
way before there were cell phones, iPods, laptops and PCs.
Even hand-held calculators had not been invented yet.
We did our engineering calculations on slide-rules.
I recall an afternoon in the library working feverishly with
a friend on a lab report where we needed to calculate a few
solutions of the equation tan x = x. After struggling with
this a little, I had a brilliant idea!
The university recently had acquired, with much fanfare, the
world's newest and fastest computer: a CDC-6400. It filled
the entire first floor of the Computer Science department.
This is before we had taken a course in Fortran II so we
had no idea about how one uses a computer to solve problems.
But I knew that computers were meant to solve _difficult_
problems and tan x = x was certainly very difficult.
Therefore we wrote the equation neatly on a notepad and
marched to meet the computer guys.
Someone opened the door to the air-conditioned facility that
housed the computer and asked what we wanted. I pointed to
the equation on the notepad and asked if he kindly could
help us solve it with the computer. I don't remember any
words he uttered but I remember his look that basically said:
"You must be out of your minds". Then he closed the door.
So we went back to the library and continued cranking out the
solutions on our trusty slide-rules. I was quite disappointed
that all the hype about the big computer had brought us nothing.
--
Rouben Rostamian
A Aitken
They didn't have Mathematica/Maple back then?
mhi...@netzero.net
finding the modes of vibration of a beam).
-- Michael
Randy Poe
Heh. They DID have FORTRAN. I took my first numerical analysis
course in 1973 and we were using the campus computer to
solve these kinds of problems in FORTRAN. Anybody need
a matrix inversion program on punch card? I'll bet it's still in
a box somewhere...
I don't think the computer operators Rouben talked to were
programmers. Their job was to mount tapes, shuttle printout
to users, and basically keep the beast happy. Probably there
were faculty and grad students using the thing via punch
card to solve numerical problems. And probably they had
to pay money from their grants to get access.
- Randy
Zdislav V. Kovarik
On Thu, 19 Jan 2006, Dave L. Renfro wrote:
> Joseph Fourier studied the solutions to the transcendental
> equation tan(x) = bx for b > 0 in connection with his
> researches in the differential equations of heat flow.
>
> Does anyone know whether the equation tan(x) = x
> had been previously studied, or even just brought
> up as an example of how graphing (in this case,
> y = tan(x) together with y = x) can sometimes
> give a lot of information about the nature of
> the solutions to an equation?
>
> In addition, is it known whether some or all of the
> nonzero solutions to tan(x) = x are irrational?
> Sure, they're probably all transcendental, but
> have any of these solutions even been proved
> irrational?
>
Robert Israel showed that they are all transcendental.
In contrast (denoting the positive solutions x(k)),
the sums of 1/x(k)^2, of 1/x(k)^4, ... , 1/x(k)^(2*j)
are all rational. For j=1, it is 1/10, for j=2, we have
1/350, ..., in general, obtained from Laurent coefficients
of an ugly analytic function g(L)/h(L):
g(L) = (L^2+1) * sinh(L) - L * cosh(L)
h(L) = 2 * L^2 * (L * cosh(L) - sinh(L))
Others found it by contour integration, I used trace class
operators on L^2([0,1]).
> Finally, besides heat flow, I know this equation
> arises in the solution to the Schrödinger equation
> for a finite square well potential and as the zeros
> of the spherical Bessel function j_1(x). Does anyone
> know of other applications where this equation appears?
>
An utterly elementary situation: finding local maxima
and minima of F(x) = sin(x) / x^(1/b), x>0, leads to
the equation tan(x) = b*x.
> Dave L. Renfro
Cheers, ZVK(Slavek).
Dave L. Renfro
> Sorry that I don't know the answers to your
> questions. But you might, nonetheless, be
> interested in expressions (2) and (3) at
> <http://mathworld.wolfram.com/TancFunction.html>.
> BTW, do you know who first came up with that
> series for the solutions? I should think that
> someone (Fourier?) found it long before I did.
Thanks for the web page reference, which I think
I'll cite in the article I'm writing -- an AP calculus
level survey for high school teachers of many of
the features of the graph of y = x / (x - sin x).
I don't know about the series you mentioned, but
I've gathered a few references from the library
(an 1878 English translation of Fourier's 1822
book, Carslaw's "Conduction of Heat in Solids"
which has an appendix with a table of the first
5 solutions to tan(x) = bx for various values
of b, etc.), so I'll keep an eye open.
I've seen quite a number of related transcendental
equations recently, and it's hard to decide where
to draw the line as to whether something should
really be considered as a tan(x) = x application.
For example, the equation tan(ax+b) = x for a < 0
comes up in the molecular field theory of
ferromagnetism (at least I've seen this mentioned;
I haven't found an explicit reference for this yet),
which I'm inclined to include. Many of Charles E.
Siewert's publications deal with applications of
transcendental equations, if you're interested:
Publications by Charles E. Siewert
http://www4.ncsu.edu/~ces/publist.html
Robert Israel's argument that the nonzero solutions
to tan(x) = x are transcendental is so straightforward
that I'm curious as to whether this came up anywhere
shortly after Lindemann's theorem appeared, and I'm
going to keep an eye out for this as well.
Incidentally (for others who might be interested),
here's Robert Israel's argument with a few of the
details filled in:
Lindemann's theorem says that if z is a nonzero
algebraic number, then e^z is transcendental.
Suppose x is a real solution to tan(x) = x. [1] [2]
Then exp(2ix) = exp(ix) / exp(-ix)
= [(cos x) + i*(sin x)] / [(cos x) - i*(sin x)]
{ divide numerator and denominator by cos x }
= [1 + i*(tan x)] / [1 - i*(tan x)]
{ use the fact that tan(x) = x }
[##] exp(2ix) = (1 + ix) / (1 - ix)
If x is nonzero and algebraic, we get
a contradiction as follows. Since the
algebraic numbers form a field, both
2ix and (1 + ix) / (1 - ix) are algebraic.
* The left-hand side of [##] is transcendental,
since 2ix is nonzero and algebraic (Lindemann).
* The right-hand side of [##] is algebraic.
[1] Actually, the only solutions to tan(x) = x are
real, and this issue was important for Fourier's
work. Grattan-Guinness' 1972 book on Fourier
discusses Poisson's concerns about this issue,
for example.
[2] This argument actually doesn't require that x be
real, but high school students who know about
e^(ix) = cos(x) + i*sin(x) will probably follow
better with this restriction, plus I'm only
considering tan(x) = x for real numbers x.
Dave L. Renfro
Dave L. Renfro
> An utterly elementary situation: finding local maxima
> and minima of F(x) = sin(x) / x^(1/b), x>0, leads to
> the equation tan(x) = b*x.
Thanks. FYI, here's an excerpt from my short article that
I mentioned just a moment ago in this thread:
"The nonzero solutions to tan(x) = x give the local extrema
of y = (sin x)/x, a function that is used to define the Sine
Integral Si(x) = [...] (which has a number of applications
in mathematics, physics, and engineering) and a function
that is often given to show that a graph can intersect its
asymptote infinitely many times. [footnote on a couple
of meanings of "asymptote" omitted] The reciprocals of
the nonzero solutions to tan(x) = x give the local extrema
of y = x*sin(1/x), a function that has many uses in mathematics,
one of which is its nearly universal use in calculus texts
to illustrate the squeeze (or sandwich) theorem for limits."
Dave L. Renfro
martin cohen
about the roots of tan(x) = x. iirc, it was written in the 30's or so.
Dave L. Renfro
> I recall having read an article by the English mathematician
> G. H. Hardy about the roots of tan(x) = x. iirc, it was written
> in the 30's or so.
Thanks. I didn't see anything definite at
http://www.emis.de/MATH/JFM/JFM.html
(using author=Hardy), but I only looked very quickly
through a couple of hundred titles in a couple of
minutes. I'll look through Hardy's "Collected Works"
the next time I'm at the library, probably tomorrow.
Dave L. Renfro
Ronald Bruck
<rou...@pc18.math.umbc.edu> wrote:
> In article <1137711971.7...@g49g2000cwa.googlegroups.com>,
> Dave L. Renfro <renf...@cmich.edu> wrote:
> >Joseph Fourier studied the solutions to the transcendental
> >equation tan(x) = bx for b > 0 in connection with his
> >researches in the differential equations of heat flow.
> >
> >Does anyone know whether the equation tan(x) = x
> >had been previously studied, or even just brought
> >up as an example of how graphing (in this case,
> >y = tan(x) together with y = x) can sometimes
> >give a lot of information about the nature of
> >the solutions to an equation?
>
> OK, I don't know the answer to your questions, but what
> you asked brought back a memory from about 40 years ago,
> so I will _have_ to write it down.
>
> I was a freshman engineering student in 1968. That was
> way before there were cell phones, iPods, laptops and PCs.
> Even hand-held calculators had not been invented yet.
> We did our engineering calculations on slide-rules.
No doubt you did your engineering calculations on slide-rules--but
hand-held calculators DID exist, and were even coming down in price by
1968. I remember using one to complete some number-theoretic
calculations I had started in high-school.
--Ron Bruck
Eric Gisse
Dave L. Renfro wrote:
> Joseph Fourier studied the solutions to the transcendental
> equation tan(x) = bx for b > 0 in connection with his
> researches in the differential equations of heat flow.
>
> Does anyone know whether the equation tan(x) = x
> had been previously studied, or even just brought
> up as an example of how graphing (in this case,
> y = tan(x) together with y = x) can sometimes
> give a lot of information about the nature of
> the solutions to an equation?
I believe I saw it used as an example in my numerical analysis class
for something that Newton's method is perfect for.
>
> In addition, is it known whether some or all of the
> nonzero solutions to tan(x) = x are irrational?
> Sure, they're probably all transcendental, but
> have any of these solutions even been proved
> irrational?
>
> Finally, besides heat flow, I know this equation
> arises in the solution to the Schrödinger equation
> for a finite square well potential and as the zeros
> of the spherical Bessel function j_1(x). Does anyone
> know of other applications where this equation appears?
Wave equations and heat equations that are solved via eigenvalue
expansion, which is a rather large swath of equations which includes
your Schrödinger equation and heat equation.
>
> Dave L. Renfro
Rouben Rostamian
Ronald Bruck <br...@imperator.usc.edu> wrote:
>
>No doubt you did your engineering calculations on slide-rules--but
>hand-held calculators DID exist, and were even coming down in price by
>1968. I remember using one to complete some number-theoretic
>calculations I had started in high-school.
I must have missed those :-(
My first encounter with a hand-held calculators was in 1972
when Hewlett-Packard HP-35 and Texas Instruments TI-2500
were introduced. Any calculators I had seen before then were
desktop models.
Rouben Rostamian
Randy Poe
Really? From who? I remember seeing the HP-35 for the first time
in 1972, a top of the line scientific calculator which came out at
something like $800 as I recall. I hadn't even heard of handheld
calculators before that. Wang demonstrated a desktop calculator
at my high school the year before, and I think Olivetti had a
similar machine. I don't remember seeing four-function calculators
(from Texas Instruments) until at least a couple of years after
HP introduced the scientific ones.
- Randy
Dave Seaman
> Ronald Bruck wrote:
>>
>> No doubt you did your engineering calculations on slide-rules--but
>> hand-held calculators DID exist, and were even coming down in price by
>> 1968. I remember using one to complete some number-theoretic
>> calculations I had started in high-school.
> Really? From who? I remember seeing the HP-35 for the first time
> in 1972, a top of the line scientific calculator which came out at
> something like $800 as I recall. I hadn't even heard of handheld
> calculators before that. Wang demonstrated a desktop calculator
> at my high school the year before, and I think Olivetti had a
> similar machine. I don't remember seeing four-function calculators
> (from Texas Instruments) until at least a couple of years after
> HP introduced the scientific ones.
The HP-35, the first "electronic slide rule" (i.e., the first handheld
that could do transcendental functions), originally sold for $395 in
1972. The HP-65 programmable calculator, circa 1974, originally sold for
$795.
--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
Robert Israel
Ronald Bruck <br...@imperator.usc.edu> wrote:
1968? I think you're off by a couple of years.
From <http://www.xnumber.com/xnumber/kilby3.htm>:
...
Together, Kilby, Merryman, and Van Tassel began work on the calculator in October 1965.
On September 29, 1967, the three filed for a U.S. patent for the world's first handheld
calculator. Although the machine did no go into actual production for three years, the prototype
had been made. ...
The model was not mass-produced immediately. Texas Instruments approached Canon Inc. (Tokyo) and
arranged to coproduce a pocket calculator completely built with Texas Instruments parts. In
April 1970, the Pocketronic appeared on the Japanese market; it was a four-function, entirely
electronic calculator that retailed for about $400. The machine was marketed in the United
States in the fall of that year.
Ronald Bruck
<isr...@math.ubc.ca> wrote:
Well, I remember what I did and when. The date is absolutely firm: I
received my PhD in June 1969--you don't forget that--and spent part of
that summer re-doing some number-theoretic calculations from my
high-school days. The calculator had instructions written in English
by an Asian (we're all familiar with the genre), and certainly didn't
cost me $400.
I don't remember where I got it. Some magazine ad, I think, but my
father is another possibility (he worked for the Air Force).
It may have been an unofficial ripoff of the official TI/Canon product.
But it did exist, didn't cost that much, and was in the summer of 1969.
--Ron Bruck
Pubkeybreaker
Dave Seaman wrote:
<snip>
> The HP-35, the first "electronic slide rule" (i.e., the first handheld
> that could do transcendental functions), originally sold for $395 in
> 1972. The HP-65 programmable calculator, circa 1974, originally sold for
> $795.
I was given an HP-65 as a high school graduation present in 1973.
Dave L. Renfro
>> I recall having read an article by the English mathematician
>> G. H. Hardy about the roots of tan(x) = x. iirc, it was written
>> in the 30's or so.
Dave L. Renfro wrote:
> Thanks. I didn't see anything definite at
>
> http://www.emis.de/MATH/JFM/JFM.html
>
> (using author=Hardy), but I only looked very quickly
> through a couple of hundred titles in a couple of
> minutes. I'll look through Hardy's "Collected Works"
> the next time I'm at the library, probably tomorrow.
I found three papers that looked as if they might be what
martin cohen was thinking of, but none of them wound up
being what I was hoping for -- an expository/historical
essay filled with some of Hardy's keen insights.
"On the zeroes of the integral function ...", Messenger
of Mathematics 31 (1901), 161-165. [pp. 7-11 in Volume IV
of "Collected papers of G.H. Hardy"]
http://www.emis.de/cgi-bin/JFM-item?33.0458.02
Studies the asymptotic behavior of the complex
solutions to sin(z) = az, where a is a positive
real number.
"On the zeroes of certain integral functions"
Messenger of Mathematics 32 (1902), 36-45.
[pp. 12-21 in Volume IV of "Collected papers
of G.H. Hardy"]
http://www.emis.de/cgi-bin/JFM-item?33.0458.03
Studies the complex zeros of sin(z) = P(z)
and e^z = P(z), where P is a polynomial.
I didn't take note of whether the coefficients
of P are restricted to real numbers.
"The asymptotic solution of certain transcendental
equations", Quarterly Journal of Mathematics 35 (1903),
261-282. [pp. 22-43 in Volume IV of "Collected papers
of G.H. Hardy"]
http://www.emis.de/cgi-bin/JFM-item?35.0118.02
Studies more general equations, including things
like exp(az)*sin(bz) = P(z) for polynomials P.
I didn't take note of whether b and/or the
coefficients of P are restricted to real numbers.
I've come across some other things as well, and at
some point I might post a summary of the things
I've come across. For instance, the solution to
Monthly Problem #6488 in American Mathematical
Monthly 93 #8 (October 1986), 660-664 includes
a number of references about infinite sums of
zeros of half-order Bessel functions and, in
particular, several proofs of the fact that
the sum of the reciprocals of the squares of
the positive solutions to tan(x) = x is 1/10.
Zdislav V. Kovarik had mentioned this previously
in this thread, and it's been discussed several
times previously in sci.math.
The August 3, 1997 post below, in fact, says this
had recently (circa August 1997) been submitted to
the problems section of the American Mathematical
Monthly, just 11 years after a fairly detailed
publication of its solution in the Amer. Math.
Monthly's problems section!
http://groups.google.com/group/sci.math/msg/6541c99085604cf3
I haven't looked very hard into when this particular
result was first proved (the sum being 1/10), but it
goes back at least to 1874, when Rayleigh published
some more general results about zeros of Bessel
functions on pp. 119-124 of Volume 5 (Series 1)
of The Proceedings of the London Mathematical Society
[pp. 190-195 in Volume I of Rayleigh's collected works].
Dave L. Renfro
Dave L. Renfro
> Robert Israel's argument that the nonzero solutions
> to tan(x) = x are transcendental is so straightforward
> that I'm curious as to whether this came up anywhere
> shortly after Lindemann's theorem appeared, and I'm
> going to keep an eye out for this as well.
The earliest proof I've come across at this point is
in the following:
Milton Brockett Porter, "On the roots of the hypergeometric
and Bessel's functions", American Journal of Mathematics
20 (1898), 193-214. [Received by the editors June 3, 1897.]
http://www.emis.de/cgi-bin/JFM-item?29.0402.01
On p. 202 Porter writes the following, where J_n is
the Bessel function of order n and i is (I believe)
an arbitrary fixed nonzero integer.
"If we knew that the positive roots of J_n(x) were,
when n is rational or algebraically irrational,
transcendental numbers, we could at once conclude
that J_n(x) and J_{n+i}(x) have no positive root
in common. [*] When n is the half of an odd integer,
it is easy to show that the roots of J_n(x) are
transcendental numbers."
[*] "Bourget, Ann. de l'Ecole Normale, 1866, p. 66
stated such a theorem when n is integral
without proof."
Recall that the positive roots of J_n for n = 3/2
are the positive solutions to tan(x) = x.
The transcendence proof that Porter gives is at the
bottom third of p. 202 and the top third of p. 203.
The proof is essentially the same that Robert Israel
gave ... Porter shows that if x is a root of J_n, where
n is half of an odd integer, then one can easily show
that exp(2ix) = (1 + bi)/(1 - bi), where b is a rational
function (with integer coefficients) of sqrt(x), and
hence a contradiction follows from a special case of
Lindemann's theorem.
Dave L. Renfro
Dave L. Renfro
>>> I recall having read an article by the English mathematician
>>> G. H. Hardy about the roots of tan(x) = x. iirc, it was
>>> written in the 30's or so.
Dave L. Renfro wrote:
>> Thanks. I didn't see anything definite at
>>
>> http://www.emis.de/MATH/JFM/JFM.html
>>
>> (using author=Hardy), but I only looked very quickly
>> through a couple of hundred titles in a couple of
>> minutes. I'll look through Hardy's "Collected Works"
>> the next time I'm at the library, probably tomorrow.
Dave L. Renfro wrote (in part):
> I found three papers that looked as if they might be what
> martin cohen was thinking of, but none of them wound up
> being what I was hoping for -- an expository/historical
> essay filled with some of Hardy's keen insights.
I just discovered something while searching the
Jahrbuch Database (JFM) that might be what martin
cohen was thinking of.
George Pólya, "Some problems connected with Fourier's
work on transcendental equations", Quarterly Journal
of Mathematics (Oxford) 1 (1930), 21-34.
http://www.emis.de/cgi-bin/JFM-item?56.0283.01
I'll look this up in the library sometime tomorrow
and post what I find (maybe not tomorrow, but soon).
Dave L. Renfro
Dave L. Renfro
> I just discovered something while searching the
> Jahrbuch Database (JFM) that might be what martin
> cohen was thinking of.
>
> George Pólya, "Some problems connected with Fourier's
> work on transcendental equations", Quarterly Journal
> of Mathematics (Oxford) 1 (1930), 21-34.
> http://www.emis.de/cgi-bin/JFM-item?56.0283.01
>
> I'll look this up in the library sometime tomorrow
> and post what I find (maybe not tomorrow, but soon).
Well, I did look this up the next day, but it wasn't
what I was hoping for. It does have some connections
with proving that the (complex) zeros of tan(z) = z are
real, which came up in a couple of recent posts of mine:
January 20, 2006 [Incidentally, the date for the
Grattan-Guinness book I mentioned should have been
1978, not 1972.]
http://groups.google.com/group/sci.math/msg/8f25a96ef8a71a12
February 4, 2006
http://groups.google.com/group/sci.math/msg/2f8da6313e8e7c65
However, I was more interested in early speculations
about the irrationality/transcendentality of the nonzero
solutions to tan(x) = x, and the early use of basic graph
properties and continuity considerations to conclude,
for example, that there are infinitely many solutions
to tan(x) = x and that these solutions are close to
half-odd-integer multiples of Pi (or similar such
results obtained in the same way about other
transcendental equations).
Pólya's paper begins with:
"This paper is concerned with a very bold argument
by which Fourier tried to prove that a certain
integral function has only real zeros: Fourier
applied a rule which was, then and now, proved
only for polynomials, to a transcendental function.
The question arises whether this application can be
justified by some general theorem. This question
was much discussed between Fourier and Poisson,
mentioned by some contemporary writers, and quite
forgotten afterwards. I resume the question in
the present paper. I have not been able to solve
it completely, though I have been led by it to
some general theorems which may be interesting
in themselves (section 3). All I can show is
that we may draw from Fourier's argument _some_
definite conclusion, namely, that the function
under consideration has _either_ only real zeros
_or_ an _infinity_ of imaginary zeros."
If anyone is interested in more details (I'm not),
p. 71 of the following book might be worth a look.
"Complex Analysis and Its Applications" by Chung-Chun Yang
http://tinyurl.com/bx3ru [google digital copy of p. 71]
Dave L. Renfro
Dave L. Renfro
> Sorry that I don't know the answers to your questions.
> But you might, nonetheless, be interested in expressions
> (2) and (3) at <http://mathworld.wolfram.com/TancFunction.html>.
> BTW, do you know who first came up with that series for
> the solutions? I should think that someone (Fourier?)
> found it long before I did.
I'm replying to my most recent post in this thread because
google isn't allowing me to reply directly to your post.
I finally have some information about the asymptotic series
for the zeros of the solutions to tan(x) = x:
q - (1/q) - (2/3)*(1/q^3) - (13/15)*(1/q^5) - (146/105)*(1/q^7) - ...
where q = (2k+1)*Pi/2.
Apparently, this series was independently obtained by
Euler [1] (1748), Cauchy [2] (1827), and Rayleigh [3] (1877).
[1] Leonhard Euler, "Introductio in Analysin Infinitorum",
Volume 2, 1748. [Reprinted in Euler's OPERA OMNIA
Series 1, Volume 9.]
See pp. 318-320.
This also appears on pp. 323-324 of the French
translation "Introduction à l'Analyse Infinitésimale"
that is on the internet at
http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-3885
See two-thirds down p. 323 for the series.
[2] Augustin-Louis Cauchy, "Théorie de la Propagation
des Ondes à la Surface d'un Fluide Pesant d'une
Profondeur Indéfinie", 1827.
Oeuvres complètes, Series 1, Volume 1, pp. 5-318.
http://math-doc.ujf-grenoble.fr/cgi-bin/oetoc?id=OE_CAUCHY_1_1
According to [5] (p. 273 & 275), the series is developed
on pp. 277-278 (p. 272 of the original 1827 publication).
[3] John William Strutt Rayleigh, "Theory of Sound",
1877-1878. [Reprinted by Dover in 1945 and 1976.]
http://tinyurl.com/n9n54
See p. 334, which the URL above will take you to.
[4] Raymond Clare Archibald and Henry Bateman, "Roots
of the equation tan x = cx", Note #8, Mathematical
Tables and Other Aids to Computation (= Mathematics
of Computation) 1 #6 (April 1944), 203.
See the follow-ups by Henry Bateman (Vol. 1 #8,
October 1944, p. 336), Raymond Clare Archibald
(Vol. 1 #12, October 1945, p. 459), Abraham P.
Hillman and Herbert E. Salzer (Vol. 2 #14,
April 1946, p. 95), and L. G. Pooler
(Vol. 3 #27, July 1949, pp. 495-496).
[5] Raymond Clare Archibald and Henry Bateman, "A guide
to tables of Bessel functions", Mathematical Tables
and Other Aids to Computation (= Mathematics of
Computation) 1 #7 (July 1944), 205-308.
See Section F: "Series for the zeros of
Bessel functions", pp. 271-275.
Dave L. Renfro
David W. Cantrell
>
> > Sorry that I don't know the answers to your questions.
> > But you might, nonetheless, be interested in expressions
> > (2) and (3) at <http://mathworld.wolfram.com/TancFunction.html>.
> > BTW, do you know who first came up with that series for
> > the solutions? I should think that someone (Fourier?)
> > found it long before I did.
>
> I'm replying to my most recent post in this thread because
> google isn't allowing me to reply directly to your post.
>
> I finally have some information about the asymptotic series
> for the zeros of the solutions to tan(x) = x:
>
> q - (1/q) - (2/3)*(1/q^3) - (13/15)*(1/q^5) - (146/105)*(1/q^7) - ...
>
> where q = (2k+1)*Pi/2.
>
> Apparently, this series was independently obtained by
> Euler [1] (1748), Cauchy [2] (1827), and Rayleigh [3] (1877).
Excellent! Many thanks, Dave!!
I'll also take this opportunity to note that, as some of the references
below indicate, in the same way that we can get an asymptotic series for
the roots of tan(x) = x, we can also get asymptotic series for the roots of
various closely related equations, such as cot(x) = c*x or tanh(x) = x.
David W. Cantrell
> [1] Leonhard Euler, "Introductio in Analysin Infinitorum",
> Volume 2, 1748. [Reprinted in Euler's OPERA OMNIA
> Series 1, Volume 9.]
>
> See pp. 318-320.
>
> This also appears on pp. 323-324 of the French
> translation "Introduction =E0 l'Analyse Infinit=E9simale"
> that is on the internet at
>
> http://visualiseur.bnf.fr/Visualiseur?Destination=3DGallica&O=3DNUMM-3885
>
> See two-thirds down p. 323 for the series.
>
> [2] Augustin-Louis Cauchy, "Th=E9orie de la Propagation
> des Ondes =E0 la Surface d'un Fluide Pesant d'une
> Profondeur Ind=E9finie", 1827.
>
> Oeuvres compl=E8tes, Series 1, Volume 1, pp. 5-318.
> http://math-doc.ujf-grenoble.fr/cgi-bin/oetoc?id=3DOE_CAUCHY_1_1
>
> According to [5] (p. 273 & 275), the series is developed
> on pp. 277-278 (p. 272 of the original 1827 publication).
>
> [3] John William Strutt Rayleigh, "Theory of Sound",
> 1877-1878. [Reprinted by Dover in 1945 and 1976.]
> http://tinyurl.com/n9n54
>
> See p. 334, which the URL above will take you to.
>
> [4] Raymond Clare Archibald and Henry Bateman, "Roots
> of the equation tan x =3D cx", Note #8, Mathematical
> Tables and Other Aids to Computation (=3D Mathematics
> of Computation) 1 #6 (April 1944), 203.
>
> See the follow-ups by Henry Bateman (Vol. 1 #8,
> October 1944, p. 336), Raymond Clare Archibald
> (Vol. 1 #12, October 1945, p. 459), Abraham P.
> Hillman and Herbert E. Salzer (Vol. 2 #14,
> April 1946, p. 95), and L. G. Pooler
> (Vol. 3 #27, July 1949, pp. 495-496).
>
> [5] Raymond Clare Archibald and Henry Bateman, "A guide
> to tables of Bessel functions", Mathematical Tables
> and Other Aids to Computation (=3D Mathematics of
Dave L. Renfro
> I'll also take this opportunity to note that, as some
> of the references below indicate, in the same way that
> we can get an asymptotic series for the roots of
> tan(x) = x, we can also get asymptotic series for
> the roots of various closely related equations,
> such as cot(x) = c*x or tanh(x) = x.
Here's another reference for the asymptotic series
of the solutions to tan(x) = x:
George Neville Watson, A TREATISE ON THE THEORY
OF BESSEL FUNCTIONS, 2'nd edition, Cambridge
University Press, 1944, viii + 804 pages.
At the bottom of p. 506 there is an asymptotic
formula for the zeros of the cylinder function
(J_nu)(z)*cos(alpha) - (Y_nu)(z)*sin(alpha)
If you consider the special case where nu = 3/2
and alpha = 0, you'll get the asymptotic expansion
for the solutions to tan(x) = x. [Recall that the
zeros of the Bessel function of order 3/2, J_{3/2},
are the solutions to tan(x) = x.]
http://tinyurl.com/ohdpy [p. 506 of Watson's book]
By the way, I haven't figured out why Cauchy should
get credit for this asymptotic series yet. What are
your thoughts? See my previous post in this thread
for the details.
Dave L. Renfro