Regarding tan(x) = x

[David W. Cantrell]

Way back on Feb. 28, 2006, Dave L. Renfro wrote:

> 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.

Sorry for the delay in responding, but I can't see why Cauchy should get
credit either.

We mathematicians aren't always good at giving credit where it's most
properly due. The Pythagorean Theorem, L'Hopital's Rule, ... There's the
joke that "You might be a mathematician if your major result will be
named for someone else."

But of course mathematics is not the only field with poor attributions.
Most of us have heard of Maxwell's equations. But I didn't realize until
recently that Maxwell himself had 20 equations in 20 variables! It was
Heaviside who reduced them to the famous four equations in two variables.

David Cantrell

[Dave L. Renfro]

David W. Cantrell wrote (in part):

> Sorry for the delay in responding, but I can't see why
> Cauchy should get credit either.
>
> We mathematicians aren't always good at giving credit
> where it's most properly due. The Pythagorean Theorem,
> L'Hopital's Rule, ... There's the joke that "You might
> be a mathematician if your major result will be named
> for someone else."

I don't think this is a case of what you're talking
about. The Cauchy reference was in the Archibald/Bateman
reference I gave at

http://groups.google.com/group/sci.math/msg/5d54b3a7052978cf

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.

Archibald is well known (to me, at least) for the
many very precise historical articles he's written.
See <http://tinyurl.com/m2xjy>, for example. I cited
his historical survey article on i^i back on March 18.

However, I checked the appropriate volume of Cauchy's
collected works out of the local university library
(easier to flip through than the internet version I
gave a link to), and as I said, it's not clear to
me whether Cauchy obtained the asymptotic series
representation for the solutions to tan(x) = x.
I think it's probably implicit from the more general
expansions at the bottom of p. 277 and the top of
p. 278, which are apparently specialized to

tan(x - pi/4) = (8/15)*x - (63/80)*(1/x) + ...

tan(x + pi/4) = (8/23)*x - (15/23)*(39/16)*(1/x) + ...

at the bottom of p. 278 (but I haven't looked into this),
and Cauchy explicitly mentions tan(x) = x on p. 283.
(All pages refer to the Cauchy reference I gave in
my February 21 post that I cited above.)

Incidentally, I'm giving a talk on tan(x) = x at
the 2006 meeting of the Iowa Section of The Mathematical
Association of America, April 7-8, Iowa State University.

Title: "The Remarkable Equation tan(x) = x"

Time: 10:15-10:35 A.M. Saturday April 8 in Concurrent Session 6

http://www.central.edu/maa/Meetings/

As I promised in the other thread, at some point
(probably within the next 3 weeks) I'll write a
fairly comprehensive summary of everything I've
found, in particular all the numerous references
I've collected on tan(x) = x.

Dave L. Renfro

[Zdislav V. Kovarik]

On Wed, 5 Apr 2006, Dave L. Renfro wrote:

> David W. Cantrell wrote (in part):
>
> > Sorry for the delay in responding, but I can't see why
> > Cauchy should get credit either.
> >
> > We mathematicians aren't always good at giving credit
> > where it's most properly due. The Pythagorean Theorem,
> > L'Hopital's Rule, ... There's the joke that "You might
> > be a mathematician if your major result will be named
> > for someone else."
>

...And a really great mathematician if the name of the result is written
in lower case (euclidean, abelian,...)
[...]

Cheers, ZVK(Slavek).

[Dave L. Renfro]

David W. Cantrell wrote (in part):

>> There's the joke that "You might be a mathematician if

>> your major result will be named for someone else."

Zdislav V. Kovarik wrote:

> ...And a really great mathematician if the name of the
> result is written in lower case (euclidean, abelian,...)

Let's hope that someone with the last name "Normal" [1]
never becomes a great mathematician!

[1] normal subgroup, normal topological space, normal
operator, normal as meaning perpendicular, normal
probability distribution, normal form of a matrix,
normal extension of a field, normal bundles in
differential geometry, ...

Dave L. Renfro

[Robert Israel]

In article <1144262321....@t31g2000cwb.googlegroups.com>,

Dave L. Renfro <renf...@cmich.edu> wrote:

>Let's hope that someone with the last name "Normal" [1]
>never becomes a great mathematician!

The MathSciNet database doesn't seem to list anybody with that name.
On the other hand, there are mathematicians named E, I, Pi, Root,
Sine, Grad, Curl, Basis, Eigen, Lemma, Converse, Real, Field, Ring,
Plane, Line, and Series (in fact, there's a "Laurent Series").

Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

[Larry Lard]

Interesting you should say that. I would regard it at as a spelling
error to fail to capitalize such adjectives. Have you got any reputable
references showing lower case spelling?

--
Larry Lard
Replies to group please

[Robert Israel]

I think the vast majority of modern algebra texts have abelian in lower
case.
It's less common to have euclidean in lower case, but hardly unknown.
Taking a sampling from my bookshelves:

Zariski and Samuel, Commutative Algebra: abelian, euclidean
Herstein, Topics in Algebra: abelian, Euclidean
Cohn, Algebra: abelian, Euclidean
McCoy, Introduction to Modern Algebra: abelian, Euclidean
Lang, Algebra: abelian
Albert, Fundamental Concepts of Higher Algebra: abelian
Moore, Elements of Linear Algebra and Matrix Theory: euclidean
Birkhoff & MacLane, A Survey of Modern Algebra: Abelian and Euclidean
Burnside, Theory of Groups of Finite Order: Abelian