Talk:Partial fraction decomposition

message: Shouldn't this article include something about how to deal with terms that have complex/imaginary roots?  Damn complex roots screwed me over on the exam.... —Preceding unsigned comment added by 203.128.4.54 (talk • contribs) 
: There is such a section: it's the one on irreducible quadratic factors.  Having non-real complex roots is the same as having irreducible quadratic factors. Michael Hardy (talk) 18:52, 6 November 2008 (UTC)
thread_title: Complex Roots

message: This page needs some talk of PFD of transfer functions in the z-domain—they're hard. jScott 00:01, 2005 Mar 18 (UTC)
I don't know about partial fraction decompositions of, specifically, transfer functions, but I agree that the article in its present form is missing a lot of the standard material. Michael Hardy 01:46, 18 Mar 2005 (UTC)
replies: text: I know my edits today have left the article a bit oddly organized, but I'll be back with much more later. Michael Hardy 19:37, 14 Apr 2005 (UTC)
thread_title: Z-domain

message: This doesn't sound like it belongs in this article. I'm not sure what it means, but if it does belong in the article please put it back in. I'm going to delete it until someone says otherwise.-The Lab Rat
:You're right; the previous editor put in some fanciful stuff and then fixed all of it except that. Michael Hardy 23:49, 18 May 2005 (UTC)
replies: text: That is Malay, which means 'Ahmad Hamidi is a smart person'.  :)  -x42bn6 07:45, 1 September 2005 (UTC)
thread_title:  Ahmad Hamidi seorang yang pandai. 

message: The sentence
:This article or section should be merged with partial fraction decomposition over the reals.
isn't this POV ?
The fact that people mix up both subjects is not enough. I think it is a good idea to distinguish
* the algebraic view of partial fractions (associated to a given denominator polynomial), and problems of decomposition of an element of R(X) into those (for different types of rings R)
* the analysis point of view, over the reals (and, not to forget, over the complex numbers, useful also for the real case), with methods for this, including limits, derivatives, changes of variables,... 
There is enough to be said about both subject to justify two separate pages, of course with tight links especially in the first sections of both pages. — MFH: Talk 18:28, 26 May 2005 (UTC)
thread_title:  shouldn't be merged 

message: Does a partial fraction expansion have to have a power of an irreducible polynomial in the denominator? If I had started this article, I would have said that "partial fraction expansion" is the act of changing a fraction A(x)/B(x) into the sum of multiple fractions. Is this general definition not correct? Fresheneesz 21:01, 21 April 2006 (UTC)
replies: text: No, the partial fraction decomposition is more than that. The fractions in the sum have a special form. Namely, each denominator is a power of a polynomial that doesn't factor into polynomials of positive degree (irreducible) and the numerators have degree strictly smaller than that irreducible in the denominator. Cactus0192837465 (talk) 03:41, 6 January 2019 (UTC)
thread_title:  must have irreducible polynom in denom??? 

message: Don't you think that the newly-added "simple example" is kinda dumb? I mean, you can simply turn (x) to (x+a-a), and then (x+a)/(x+a) -a/(x+a) = 1-a/(x+1). There's no reason to mess around with partial fraction decomposition here....
:I think the simple examples used here (and elsewhere) are valuable. They may be obvious to you and I, but for someone meeting partial fractions for the first time it will be a valuable first step before they meet the more complicated examples. DavidMcKenzie 10:46, 26 February 2007 (UTC)
thread_title:  the new "simple example" 

message: :See also the merge proposal in the next section
This merge was proposed before but opposed with the argument that it is a good idea to distinguish the algebraic view of partial fractions and the analysis point of view. While that may be true:
#the articles do a good job of making that distinction;
#the distinction, to the extent it is needed, can well be made within a single article;
#there is no relationship between the distinction and the names of the articles – or, in fact, it is, rather, the reverse, as the article entitled "Partial fraction" talks a whole lot more about decomposition, even going into algorithms for computing the decomposition (most of which can be adapted to any Euclidean ring);
#other articles referring to the process of partial fraction decomposition understandably have links to Partial fraction decomposition, while links to Partial fraction would usually be more appropriate in the context (see, e.g., Rational function); 
#what is said in the "P.f. decomposition" article should really also be said in the other article;
#the "P.f. decomposition" article is a stub; to flesh it out to a full-fledged article would result in a lot of duplication.
 --LambiamTalk 07:40, 18 June 2007 (UTC)
thread_title:  Merge with [[Partial fraction decomposition]] 

message: :See also the merge proposal in the preceding section
This merge was proposed before but opposed with the argument that it is a good idea to distinguish the algebraic view of partial fractions and the analysis point of view. While that may be true:
#the articles don't do a good job of making that distinction;
#the distinction, to the extent it is needed, can well be made within a single article;
#the article named Partial fraction decomposition over the reals would be an orphan if it was not for a "See also" link here;
#most of what is said in the other article is also said here; there is a lot of overlap and duplication.
 --LambiamTalk 08:00, 18 June 2007 (UTC)
replies: text: In fact the merge has already happened. Charles Matthews (talk) 23:07, 1 January 2009 (UTC)
thread_title:  Merge with [[Partial fraction decomposition over the reals]] 

message: A major issue was lacking in this very long article: the useful formula for the coefficients of a PFD via Taylor polynomial expansion. I have therefore added a small section on the subject with a sketch of the proof. It seems to me that this article is not bad locally, but still contains things repeated several times in different places; sooner or later it should be rectified--PMajer (talk) 20:30, 31 December 2008 (UTC)
thread_title: Where's Taylor?

message: Moved from Talk:Partial fraction/Comments, but it'll have a better chance of being seen here. Astronaut (talk) 08:14, 2 April 2009 (UTC)
thread_title:  External link error 

message: Connection with the rational canonical form of a matrix, and therefore with finitely generated modules over PIDs, is missing. Also missing in the rational canonical form page.  franklin  14:38, 16 December 2009 (UTC)
thread_title:  Connection with matrix canonical form. Rational canonical form 

message: The characterization at the beginning of the article:
:In symbols, one can use partial fraction expansion to change a rational function in the form
:: \frac{f(x)}{g(x)} 
:where ƒ and g are polynomials, into a function of the form
:: \sum \frac{a_n}{h_n(x)} 
:where hn are polynomials that are factors of g(x), and are in general of lower degree.
is not correct.  In addition, I don't know what "are in general of lower degree" means.  (A good rule for clear writing: When making a comparison, make sure that what is being compared to what is clear.)
I think this is what was intended:
:In symbols, one can use the partial fraction expansion technique to change a rational function in the form
:: \frac{f(x)}{g(x)} 
:where ƒ and g are polynomials, into a function of the form
:: \sum \frac{a_n(x)}{h_n(x)} 
:where an and hn are polynomials, the hns are factors of g, and each an is of lower degree than the corresponding hn.
SDCHS (talk) 00:19, 28 May 2010 (UTC)
thread_title:  Symbolic Characterization Has Errors 

message: I think this article should be completely rewritten. Its arrangement is just a mess—what article first gives you examples then explain what it is? And it seems to me the examples are too many.--Netheril96 (talk) 10:39, 25 October 2010 (UTC)
replies: text: The rewrite should probably take place here rather than in user space.  This way it is more inclusive and transparent.  Moreover, changes can more easily be discussed on the talk page.  Also, I see that major changes have taken place here after the rewrite started.  I am going to remove the template, since we should not have people simultaneously editing two versions of the article.  Sławomir Biały (talk) 11:28, 5 November 2010 (UTC)
replies: text:  If you think pages in userspace are inappropriate, I can move it to somewhere like Partial fraction/draft. This article need not just revisions but complete rewrite instead. To do that, we need to start from almost scratch because this article contains too much redundant and unencylopediac information and before we finish the main article should remain intact.--Netheril96 (talk) 13:28, 5 November 2010 (UTC)

text: Major edits are already underway here, though.  I think the article should be left where it is while this happens. Sławomir Biały (talk) 14:20, 5 November 2010 (UTC)
replies: text: Fine. I'll wait until you guys finish the trunk of the new article.--Netheril96 (talk) 14:57, 5 November 2010 (UTC)
replies: text: I think it's down to a basic trunk.  I don't think the article requires a complete rewrite anymore, although there is certainly still plenty of room for improvement.  Sławomir Biały (talk) 12:37, 7 November 2010 (UTC)
thread_title:  Complete rewrite 

message: The recent edit  highlights an issue with the notation in this article. Specifically, the divisor polynomial is denoted by Q(x) as in rational function, whereas in polynomial long division it is denoted by D(x), with Q(x) denoting the quotient polynomial. Can the notation be improved so that we get closer to P(x) = D(x)Q(x) + R(x)? Isheden (talk) 10:37, 15 February 2012 (UTC)
thread_title:  Issue with notation 

message: Hello everyone, when comparing Example 4 with the results given by Wolfram Alpha, i noticed that the result is quite different. Since i'm absolutely no expert on this subject, could someone review this and correct the example? 134.28.202.206 (talk) 15:07, 10 March 2012 (UTC)
replies: text: Wolfram Alpha gives the following partial fraction expansion: \frac{i}{z-i} - \frac{i}{z+i} - \frac{3}{2(z+1)} + \frac{3}{2(z-1)} Isheden (talk) 20:00, 10 March 2012 (UTC)

text: Starting with the expression \frac{z^2+5}{(z^2-1)(z^2+1)}, the partial fraction expansion is -\frac{2}{z^2+1}-\frac{3}{2(z+1)}+\frac{3}{2(z-1)}. Isheden (talk) 20:38, 10 March 2012 (UTC)

text: It turned out that the numerator should have been z^2-5 instead of z^2+5. I've fixed it and expanded the example to clarify what idea it is based on. Isheden (talk) 21:49, 10 March 2012 (UTC)
thread_title:  Example 4 might be wrong 

message: The first example of incorrect reasoning is the following:
: Therefore:
: \lim_{x \to 1} \frac{1}{(x - 1)(x^2 + x + 1)} =  \lim_{x \to 1}\frac{A}{x - 1},
: and thus:
: A = \lim_{x \to 1}{\frac{1}{x^2 + x + 1}} = \frac{1}{3}
This simply does not follow. For example,
\lim_{x \to 0}{\frac{1}{x}} = \lim_{x \to 0}{\frac{2}{x}}. But  1 \neq 2. From the "equality" of two limits which are equal by virtue of both being equal to infinity, we can conclude absolutely nothing!
The second case of bad reasoning in Example 5 is the following:
: Therefore:
:   \lim_{x \to \infty} \frac{A + B}{x} = \lim_{x \to \infty} \frac{1}{(x - 1)(x^2 + x + 1)} =0,
: which implies A + B = 0 and so
From
  \lim_{x \to \infty} \frac{A + B}{x} =0,
one cannot conclude that    A + B = 0  .
For example:
  \lim_{x \to \infty} \frac{2}{x} =0,
But  2 \neq 0.
I believe I have a line of reasoning that corrects both of these problems. However, there are two problems with it:
# It is more complex and difficult to follow.
# It is my original work, and therefore not eligible to be put in Wikipedia.
Basically, the idea is instead of considering and comparing two separate expressions which are assumed to be equal, one considers the ratios of those expressions which should be 1. Someone please help!
Pelliott11 (talk) 00:51, 22 May 2017 (UTC)
replies: text: I agree that the reasoning was wrong. I have fixed it. D.Lazard (talk) 07:41, 22 May 2017 (UTC)
thread_title: Example 5 has bad reasoning leading to correct result.

message: This is also the most efficient way to compute this, the ugly formula involving derivatives mentioned in the section about the residue method is not the most practical way to compute series expansions.  Also note that if the degree of the numerator is larger than that of the numerator then the point at infinity is a pole, the principal part of the Laurent expansion around infinity is then precisely the polynomial you would get if you perform a division to reduce the degree of the numerator to below that of the denominator. Count Iblis (talk) 23:48, 5 April 2013 (UTC)
replies: text: This could be said somewhere in the article, but this article should be understandable to people that know only polynomials and rational fractions and ignore the theory of analytic functions (poles, Laurent expansion, ....). Nevertheless, this article is badly written, and several sections, in particular that on the residue method, need to be completely rewritten. About your assertion on the ugly formula, I suppose that it is the formula for aij. I agree with you, that the general formula is not useful. But in case of a simple root of the denominator, the formula a_{i1}=P(\alpha)/Q'(\alpha) is the simplest way to compute the "logarithmic part". D.Lazard (talk) 09:22, 6 April 2013 (UTC)
replies: text: Yes, but then because we're dealing with rational functions only, you can formulate and prove this using only elementary means (a sketch of the proof is that if you subtract from a rational function its complete singular behavior for each singularity, the result doesn't have any singular behavior left anymore and must therefore be a polynomial. Then if the original rational function tends to zero at infinity, that polynomial must be identical to zero). So, one has the same result that all you need to do is expand around each singularity and that's something every first year student ought to be capable of. Count Iblis (talk) 16:15, 6 April 2013 (UTC)
thread_title:  Why not simply say that it is the sum of all the principal parts of the Laurent expansions around each pole? 

message: Substituted at 02:27, 5 May 2016 (UTC)
thread_title: Assessment comment

message: I think the article is lacking proper explanation and examples on how to perform partial fraction decomposition over the Complex Numbers. In particular, Example 2 is not at all clear on how the values of D and E were derived from the initial decomposition.  — Preceding unsigned comment added by Frikdt (talk • contribs) 08:49, 23 December 2016 (UTC)
replies: text: There is nothing special with complex numbers, except that every denominator of the decomposition is a linear polynomial or a power of a linear polynomial. Nevertheless, I have edited example 2, for clarification. Note also that if the denominator of the input fraction is a product of distinct linear polynomials, the residue method is almost alway the simplest method for computing the decomposition. D.Lazard (talk) 09:38, 23 December 2016 (UTC)
thread_title:  Decomposition over the Complex Numbers 

message: Why is deg(Cf)<deg(P) and deg(Cf)<deg(Q)? See http://marasingha.org/mathspages/partialfrac/html/node2.html for the right way to do this. Sraianu (talk) 06:53, 19 July 2020 (UTC)
replies: text: The section was not wrong; it was incomplete. I have completed it. By the way, the link that you provided is not reliable, as it is titled "The partial fraction algorithm" when several algorithms are commonly used. D.Lazard (talk) 16:36, 19 July 2020 (UTC)
thread_title:  The beginning of the section Basic principles