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From: Chip Eastham on 27 Mar 2010 14:02
On Mar 18, 1:18 pm, "Dave L. Renfro" <renfr...(a)cmich.edu> wrote:
> Chip Eastham wrote:
> > Yes, that's a good point. If we could demonstrate that
> > the only roots of x = tan x are on the real line, then
> > we'd have something close to a rigorous proof.
>
> For a "precalculus level" proof taken from Hardy's book
> "A Course of Pure Mathematics", see pp. 9-10 of the .pdf
> file "tan(x) = x" I just archived in the following Math
> Forum sci.math post:
>
> http://mathforum.org/kb/message.jspa?messageID=7014308
>
> Note that I also give 2 references where the result can
> be proved using Rouché's theorem and 2 references where
> the result can be proved using theorems about eigenvalues
> for certain Sturm-Liouville problems.
>
> Dave L. Renfro

Thanks, Dave, I missed your post for about a week. But
your elementary proof (from Hardy) that tan x = x has
only real roots is satisfying, scarcely more difficult
than a similar proof for sin x = 0.

regards, chip

From: Dave L. Renfro on 30 Mar 2010 16:34
AP wrote (in part):

> My pb is : how find Green's function to obtain
> the 1/h^2 with h>0 and tanh-h=0
> for eigenvalues

Unfortunately, I haven't done anything with Green's functions
since 1983, so I'd have to review some stuff (which I don't
have with me now anyway) before I could begin thinking about
what you're asking. Maybe someone else who has studied this
more recently can help.

Dave L. Renfro
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