24.02.2011 14:33, Alexander Eberspächer пишет:
Hello,
On Thu, 24 Feb 2011 09:58:59 +0300
"Alexey U. Gudchenko"<[email protected]> wrote:
And I have some little question about Laurent.
exp(1/x).series(x, oo) = 1 + 1/x + 1/(2*x**2) ...
Is it Laurent or power series?
In my use of maths language, this is a Laurent series as negative
powers of x contribute. Power series (or, in this case, Taylor series)
contain only exponents in [0; oo). So I would call this expansion a
Laurent series about the point x=oo.
I think so.
In [1] there is defined only "formal Laurent series" in which only
finite number of negative powers is allowed.
But generally "Laurent series" can contain infinitely number of terms
with negative powers. So it must be reflected in [1], and distinguished.
If there are no objections.
In this case (opposite to "formal Laurent series") there are not general
rules how to multiplicate series with each other.
By the way in [1]:
"K[[X]] and has a ring structure."
"K((X)) and has a field structure."
I do not understand the difference about ring structure or field
structure. (Is it misprint?)
Also, as number of terms with positive powers in "Laurent series" can be
infinitely too with the same time as negative ones, it's interesting in
this case what to do with O() when consider some asymptotic expansion
from this series or does sense of it exists, and what sort of if it
does. I do not know exactly the answer now.
[1] https://github.com/sympy/sympy/wiki/Function-expansions
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