Re: ln x = x^2 - 1

David W. Cantrell wrote:
[snip]

The only thing I was guessing was that your past claim that HW
functions solve Kelper's equation was correct. The equation cos(x) =
x is a special case of Kepler's equation, after all.

Therefore you can expect the unexpected ;o)

The HW functions are an _uncountable_ collection of functions, and
not "a function", therefore they solve in closed form an
_uncountable_ number of transcendental equations and an
_uncountable_ number of differential equations. W belongs to this
collection.

The HW functions _cannot_ solve the equation 'cos(x)=x' exactly.

I see that you later recanted.

That's right. I wasn't expecting that the HW functions could solve cos(x)=x.
Basically I didn't bother to look, that's why I didn't make any claims and when
I made the (wrong) claim (after your response), I still hadn't bothered to look.
WHen I _did_ look, it was already too late :-)

[snip]

And then most recently you wrote:

I'll be damned! Unexpected for me as well. They _do_ solve it.

Well, of course. It was _not_ unexpected for me! I had expected that
your claim that HW functions solve Kepler's equation was correct,
trusting soul that I am.
The HW functions _cannot_ solve the equation 'cos(x)=x' exactly.
(Hint: Perhaps that's good reason why I didn't make this claim?)

cos(x) = x =>
cos(x)/x = 1 =>
x*cos(x)/x^2 = 1 =>
x*exp(ln(cos(x)/x^2)) = 1 =>
x = HW(ln(cos(x)/x^2),1)

Let's see what my Maple implementation sez for this HW:

HW(ln(cos(z)/z^2),1,10);
.7390851332

Therefore, what remains of my previous request is that you show us
what HW(a,b,c) does exactly.

The exact mechanics applied with the HW functions to solve various
transcendental equations exactly is an extended subject and I have devoted at
least two papers on it. I think the summary presented above is sufficient for
the purposes of this thread. Anyone interested on further details can look up my
papers on these functions.

I do, however, want to stress that my statement "David is a bit verbal with W"
was not intended to be derrogatory in any way. Perhaps there was a
misunderstanding there which caused you to react.

I certainly am glad that you keep the fortress of W alive. Somebody has to do
it, and since you have volunteered, all the better for us. The only reason I
backed off and stopped jumping in giving solutions in terms of W, was to avoid
potential conflicts with you, since many regulars consider you an expert on this
function.

The Wiki page for W has enough references to my web pages to last me a life
time, so I am very glad that you get a chance to push W either way.

Hopefully this will clear the confusion ;o)

David

fsolve(cos(x)=x,x=0..1);

.7390851332
--
I.N. Galidakis

.

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