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Re: Error correction: re my improper use of the term "polynomial"
Posted:
Dec 3, 2000 4:58 PM
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David C. Ullrich, dans le message (sci.math:379515), a ÃÂécrit :
> On Sat, 02 Dec 2000 14:42:35 GMT, somorel@my-deja.com wrote:
>
>>In article <3a1d40fb.934807618@nntp.sprynet.com>,
>> ullrich@math.okstate.edu wrote:
>>
>>
>>> _If_ you can show me what that power series
>>> representation for sqrt(x^2 + y^2) is that will
>>change
>>> really a lot of things (like it will change the
>>fact that
>>> real-analytic functions are differentiable, for
>>one thing.)
>>
>>I'm wondering what you mean exactly by power
>>series; is it formal power series or power series
>>that converge in a small neighbourhood of the
>>origin?
>>If it's the last, I have no question, but if it's
>>the former, how do you prove that sqrt(x^2+y^2) is
>>real analytic ?
>
> Not sure I understand the question. First,
> I was talking only about power series centered at
> the origin - the thing _does_ have power series
> about other points. And yes, I meant "convergent
> power series". (Not that I see that that matters -
> the coefficients in some formal power series would
> be given by certain non-existent derivatives.)
I think the problem is that M. Harris wrote "power series" I understood
"formal power series" and you "convergent power series"
(perhaps it has something to do with my english; if I was mistaken, please
remember that it is not my first language).
I too was talking only about power series centered at the origin. I don't
deny the fact that sqrt(X^2+Y^2) has no power series representation, or
that a power series that converges in a neighborhood of 0 defines an
analytic function in this neighborhood.
What I meant is: let's just suppose that sqrt(X^2+Y^2) has a formal power
series representation; is this representation convergent near the origin?
More generally, if f is a convergent power series and g is a formal power
series such that g^2=f, is g also convergent? (Or still more generally, I
was beginning to ask myself if the ring of convergent power series was
integrally closed in the ring of formal power series (the coefficients
being R or C, or maybe another topological field, a local or global
field for example)).
I'm sorry that I didn't make myself clear the first time, and I hope that
this time I succeeded.
--
Sophie Morel
smorel@clipper.ens.fr
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