Talk:Formal power series

message: I think this article is quite well written,
although I don't know if the introduction of the metric d() is standard
and/or necessary for the notation \sum a_kX^K,
which I was taught to be a mere notation for the sequence (a_k).
I like this approach, but it may lead to confusion. Indeed, this
metric is the only one introduced on this page, while the section
"Formal power series as functions", starting with
 In mathematical analysis, every convergent power series
 defines a function with values in the real or complex numbers.
clearly does not refer to this metric (for "convergent"),
but to the topology in C (or in some complex function space),
for the series of partial sums (of numbers or functions?)
associated to the power series in the "obvious" way
(but a precise define is not so immediate at all...).
As it stands, this phrase is meaningless, because any power series is convergent according to what precedes, even if it has convergence radius of zero. So I think we should change it, so that this subsection becomes as "perfect" as the rest of the article.
If the author of this article can find a more precise introduction for
this subsection in the same style as what precedes, I would appreciate.
(I don't know what is the most "gentle" way to go from a power series to
the associated analytic function.)
— MFH: Talk 12:53, 26 Apr 2005 (UTC)
thread_title:  Formal power series as functions 

message: Would someone add a section on the coefficients of two multiplied power series? I.e., if we have f(z) := \sum_{k=0}^{\infty}{a_kz^k} and g(z) := \sum_{k=0}^{\infty}{b_kz^k}, then what is f(z)g(z)?
--Rocketman768 14:49, 30 April2007 (UTC)
::I find the answer to that question appearing in this article TWICE in its present form (although it does not have an entire section devoted to it). Michael Hardy 16:44, 30 April 2007 (UTC)
thread_title:  Multiplication of power series 

message: There's something a little annoying on this page. In several places it refers to completions with respect to I-adic topologies. Unfortunately, the article completion discusses only metric completion, whereas the most natural way to express I-adic completion is not via a metric. (I'm not even sure if it can always be described in terms of a metric.) A more natural way instead is to use sequences that are coherent with respect to a nested sequence of subgroups (i.e. the ideals I, I^2, I^3 etc). Any opinions on what to do? Dmharvey Talk 01:09, 26 Jun 2005 (UTC)
replies: text: It seems that completion in the sense of topology is most generally defined for uniform spaces using Cauchy nets. Ring-theoretic completion (and in particular l-adic completion) seems to be a separate concept. The link, therefore, may be inappropriate in this case. Maybe someone will get around to writing an article on this other kind of completion. - Gauge 07:26, 26 Jun 2005 (UTC)
replies: text: Thanks for the info. Unfortunately my topology is a bit weak to write this up. Maybe someone else will. Dmharvey Talk 12:24, 26 Jun 2005 (UTC)
thread_title:  completion 

message: The article mentions that the two are not the same topologically, because while associating a topology with  Rx, one takes discrete toplogy on R and then the product topology on Rx. However, if R is itself a topological ring, one could avoid this problem by considering the topology of Rx to be the product of the topology of the topological ring R. Also, if R is not a topological ring, then associate discrete topology with it to make it a topological ring? Any thoughts? Ustad NY (talk) 05:44, 12 December 2007 (UTC)
thread_title:  R<nowiki>[[x]][[y]]</nowiki> Versus R<nowiki>[[x,y]]</nowiki> 

message: The article currently states the following in the section on "Composition of series":
:The critical point here is that this operation is only valid when f(X) has no constant term, so that the series for g(f(X)) converges in the topology of RX. In other words, each cn depends on only a finite number of coefficients of f(X) and g(X). 
The word "only" here is inaccurate, as the operation is also valid when g is finite, i.e. a polynomial.  I'm not sure right now if these are the only two cases in which it's valid; if I did, I'd change the article ... Tsoeto (talk) 03:37, 15 March 2009 (UTC)
replies: text: Right, but if you interpret the term operation in that sentence as an operation that has to be defined on the whole RX the restriction is necessary e.g if the constant term of f  is 1: then g(f(X)) can't be defined for all g∈RX. Certainly there are cases of g(f(X)) being well defined as a composition of formal power series  when f does have a nonzero constant term: for instance when g is a polynomial  as you remark. Also, if g is any formal series, and the constant term of f is nilpotent in the (commutative) ring R, we still have a well defined g(f(X)) with coefficients defined by means of finite sums. I'm not sure if these details may be of interest for the article though. --pma 10:12, 19 January 2010 (UTC)
thread_title:  Composition of series 

message: Some talk about the functional inverse would be an interesting addition to this article:
http://www.mathpages.com/home/kmath625/kmath625.htm
S243a (talk) 22:09, 4 August 2009 (UTC)
replies: text: What exactly from that talk may be of any interest for this article, in your opinion?--pma 10:17, 19 January 2010 (UTC)
thread_title:  Functional Inverse 

message: Is "formal Laurent series" the same as Laurent polynomial?  If so should the article link to it? Thanks! ciphergoth (talk) 08:03, 15 October 2012 (UTC)
replies: text: No, a Laurent polynomial has only finitely many terms (of positive or negative degree), whereas a formal Laurent series may have infinitely many terms of positive degree.  (The difference is similar to the difference between polynomials and formal power series.)  However, it might still be nice to have a link to Laurent polynomial if there's an appropriate place for one.  --JBL (talk) 19:06, 15 October 2012 (UTC)
thread_title:  [[Laurent polynomial]] 

message: I've read quite carefully the whole article, and it seems to me that it has improved a lot in the past years. I found everything correct; the only section I did not check carefully, though it is well written, is Algebraic properties of the formal power series ring. It contains interesting material and maybe could be expanded, and it is the main place still wanting references. Up to this point, it seems to me that we may be close to remove the initial tag "Refimprove", dating back to 2009. pma 00:06, 28 November 2012 (UTC)
thread_title: Towards a final form?

message: In the "very short computation" I don't understand the second equals sign. Maybe the derivation is too short, a hint might help. It somehow enters that f(g(X)) = X, and powers of X, f, g and f' get mixed up in a fascinating way. radical_in_all_things (talk) 08:42, 15 June 2013 (UTC)
replies: text: Actually the second equal sign is a plain application of the above rule iv. Should you need more explanations, you may like to post further questions to the math reference desk --pma 14:18, 3 May 2014 (UTC)
replies: text: Actually, this talk page is exactly the place to discuss the article and in particular whether the article explains a point adequately.  And I tend to agree that the second equality as currently written is not adequately explained.  Deltahedron (talk) 16:03, 3 May 2014 (UTC)
thread_title: Lagrange inversion formula

message: The ring of formal power series is defined here over a commutative ring. Why do we assume the coefficient ring to be commutative? I suggest do take any coefficient ring (maybe unital).
Spaetzle (talk) 13:43, 20 March 2015 (UTC)
thread_title:  commutativity 

message: I have expected some words on division theorem: for a formal power series f and a polynomial P there exist unique power series q and polynomial R with deg(R)<deg(P) such that f=qP+R. Is this not true or of no use? I thought this could help defining f(A) for a square matrix A, in particular exp(A). And lots of nice functions after, including log and Baker-Champbell-Hausdorff formula... I hope this will trigger an add-on to this article... if so, PLEASE signal to me 0ctavte0 (talk) 07:06, 30 May 2023 (UTC)
replies: text: This is just false.  --JBL (talk) 20:41, 30 May 2023 (UTC)
thread_title: QUESTION: division theorem