Re: Are bar{omega} and agM(1, sqrt{2}) transcendental?

On Wed, 16 Apr 2008 23:38:59 -0700 (PDT), galathaea
<galathaea@xxxxxxxxx> wrote:

On Apr 16, 1:42 pm, Angus Rodgers <twir...@xxxxxxxxxxx> wrote:
On Wed, 16 Apr 2008 21:29:56 +0100, I wrote:
On Wed, 16 Apr 2008 12:51:01 -0700 (PDT), galathaea
<galath...@xxxxxxxxx> wrote:

On Apr 16, 12:31 pm, galathaea <galath...@xxxxxxxxx> wrote:
On Apr 16, 10:44 am, Angus Rodgers <twir...@xxxxxxxxxxx> wrote:

In Gauss's relation pi/bar{omega} = agM(1, sqrt{2}), where agM
is the arithmetic-geometric mean, and bar{omega} = 2\int_0^1
dt/sqrt{1 - t^4}, are all three of the numbers transcendental?

yes

it may be expressed in a linear expression of
^^^^^^
algebraic

--- / 1 \
| ' | - |
| \ 4 /

and pi
(as the beta function B(1/4, 1,4))

these two values are linearly independent
and so since pi is transcendental
so is gauss' constant

sorry for the abuse of language

Thanks! I just came across a mention of this amazing equation
today, but it didn't use the phrase "Gauss's constant", and so
I didn't know what to Google for. From Wikipedia or MathWorld:

G := 1/M(1, sqrt{2}) = 0.8346268...

and the other constant, which was denoted by bar{omega} in the
book I was reading, and is the arc length of one of the two loops
of the lemniscate of Bernoulli (polar equation r^2 = cos(2theta)),
is:

L (or L_1) = pi*G

This is called the [first] lemniscate constant, and according to
MathWorld <http://mathworld.wolfram.com/LemniscateConstant.html>
is also transcendental.

Indeed, were you saying that G = f(pi, Gamma(1/4)), where f is
algebraic and pi and Gamma(1/4) are algebraically independent,
so it follows immediately that L is also transcendental?

(I haven't studied any of this stuff properly, but I mean to do
so eventually.)

because pi and gamma(1/4)
are algebraically independent
there are no nontrivial algebraic f(pi, gamma(1/4)) = 0

therefor no algebraic combination of pi and gamma(1/4)
will ever give an algebraic number
(because then we could easily construct an algebraic expression = 0)

including the combo that gives G

I understood that part, but your post referred to "it", which I
gathered must mean G, because you also referred to it as "Gauss's
constant". I had to Google this to find which of the two numbers
you meant. Merely from pi transcendental and G transcendental it
doesn't follow that pi*G is transcendental. However, the argument
you seemed to be giving for G works identically for pi*G, and I
was just checking that this was in fact what you had in mind, as
the wording wasn't very clear.

I'm also not accustomed to working with "algebraic expressions", so
I wanted to check that I hadn't misunderstood, in some ridiculously
fundamental way. Press me to define "f(x, y) is algebraic", and I
might struggle a bit! I suppose I would say that, given a field k
and an extension K of k, a function f: K x K --> K is algebraic if
there exists a polynomial g{X, Y, Z) in k[X, Y, Z] such that the
expression g(x, y, f(x, y)) is identically zero for all x and y in
K? Perhaps this definition is only worth making if K is infinite;
it's something I've never thought about. My first attempt at a
definition this morning (well, I've only just woken up, so there's
still too much blood in my caffeine stream) was complete gibberish
(or even more complete gibberish!). It had only X's, Y's, and Z's
in it, and only a k and no K ... trust me, you don't want to know.

I've now looked for a definition like this in Lang's "Algebra" (a
book I never open unless I have to, like the dreaded Necronomicon
of the mad Arab Abdul Alhazred), but I can't find it even there ...

Ah, here we are: <http://eom.springer.de/A/a011490.htm> I do seem
to have got roughly the right idea. (I won't struggle with it any
more, at least not until I have had breakfast.)

Now (having more or less sorted that out, I hope!), what was the
actual expression for G, in terms of pi and Gamma(1/4)? Looking
at the Wikipedia page for "Gauss's constant" again, it seems to be:

G = (Gamma(1/4)^2)/(2pi)^{3/2}

which is not a field expression, so we do indeed seem to need some
generality in the concept of an "algebraic expression". I suppose
the simplest form of "g(x, y, f(x, y)) = 0", in this case, would be
(straight from the expression given by Wikipedia):

Gamma(1/4)^4 = 8(pi^3)G^2

or, in terms of L:

Gamma(1/4)^4 = 8pi*L^2

--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

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