23.02.2011 22:21, Ondrej Certik пишет:
Hi Ronan!
On Tue, Feb 22, 2011 at 7:12 PM, Ronan Lamy<[email protected]> wrote:
I think that a significant part of the difficulties in the discussions
about Taylor series and the like has been the lack of a common and
unambiguous vocabulary. So, I've started a wiki page defining some
concepts in order to clarify things:
https://github.com/sympy/sympy/wiki/Function-expansions
Feel free to expand or amend it, add links to and from it, etc.
NB: I had to use Markdown syntax because I couldn't figure out how to
get LaTeX to work in rST.
Thanks for writing this up, it will greatly help, and then we can
refactor it into sphinx later on (don't worry about using Markdown,
that's ok).
I agree with your definition of formal power series. Also with Laurent
+ Taylor series. So in other words, for analytic functions, Laurent
series is a Taylor series divided by some x^m (where m is an integer).
And today's remarks about difference between "Laurent series" and
"formal Laurent series" must be taken into consideration.
I agree with your O() definitions. Should we enable O() symbols around
different points than x=0 in sympy (including x=oo)? It now seems to
me we probably should.
Of course we should, (it hang smichr/2084).
As it begin track point x0, so it is assumed that the whole expression
tend to x0 too.
So O((x-x0)**n) must be determ terms (x-x0)**k of it expression to deal
with them. Don't it?
Also operations with the expressions that contains O((x-x0)**n) (so it
tend to x0) on the one hand and O((x-x1)**n) that tend to x1 on the
other hand must be not allowed.
Then, I supposed that various particular class of O can be implemented
(almost accordingly various kinds of series) that derived from O, which
can be defined generally, as Ronan written.
How does asymptotic series fit into this? Series of the type 1 + 1/x +
1/x**2 + 1/x**3 and so on. Can it be viewed as Laurent series around
the point x=oo? This should be clarified and put there as well.
Ondrej
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