Talk:Lagrange inversion theorem

[Untitled]
What about this:



\left. g(z) = a + \sum_{n=1}^{\infty} \frac{d^{n-1}}{(dw)^{n-1}} \left( \frac{(w-a)^n}{(f(w) - b)^n} \right) \right| _{w = a} {\frac{(z - b)^n}{n!}} $$

&infin;   dn-1  /  (w - a)n    \ |      (z - b)n g(z) = a +  &sum;  -- | ---  | | n=1 (dw)n-1 \ (f(w) - b)n / |         n! | w=a

--Edmund 02:23 Feb 22, 2003 (UTC)

Yup, the TeX is correct now. AxelBoldt 20:45 Mar 2, 2003 (UTC)

Examples
I think the use of the "group" in binary tree example unnecessary complicates things. If I didn't know what this is all about I wouldn't have guessed what is meant here. Can someone rewrite this in plain language, like "removing the root splits a binary tree into two subtrees, so accounting for the missing root vertex we have..."

Also, while I am here - the example with enumeration of labeled trees giving Cayley's formula is much more interesting, I say. Mhym 23:14, 22 July 2006 (UTC)

Generalizations? How?
The article states that the theorem is readily generalized to functions of several variables, but I don't see how, not the least of which is because the dimensions of the domain need not be the same as the dimensions of the range (for example, a vector-valued function of a scalar would have what kind of series, or vice versa?).--Jasper Deng (talk) 06:03, 20 January 2014 (UTC)

Proof of the theorem is not given
The proof of the theorem is not to be found anywhere. Not even in Lagrange's original paper. — Preceding unsigned comment added by 2601:9:8180:1029:C5A7:2EC4:2CEB:1843 (talk) 18:19, 30 April 2014 (UTC)
 * You may check the link to the wiki article on formal power series, where a few lines elementary proof is given. --pm a 14:07, 3 May 2014 (UTC)

An addition that needs discussion (and a source)
A couple of weeks ago, made the following addition to the article, at the end of the section Lagrange_inversion_theorem: In fact, integration by parts doesn't work for:



n=0\rightarrow \frac{1}{2\pi i} \oint_{C} \frac{ \omega f'(\omega)}{ f(\omega) - f(a) } d\omega = \frac{1}{2\pi i} \oint_{f(C)} \frac{ f^{-1}(u)}{ u - f(a) } du= f^{-1}(f(a))=a $$ Fortunately, writing the integral as a residue in the last line fixes the problem. I've reverted this because of the odd formating and the apparent editorialization, without making any attempt to determine the validity. The whole derivation is (sadly) unsourced; does anyone have access to a handy reference that would allow a sourced derivation, rather than the apparent WP:OR we have here? --JBL (talk) 20:51, 16 June 2024 (UTC)


 * Is the source really necessary? I think that as a mathematician you understand what I presented. Every tool I used is already on Wikipedia. Mathematical.Analysis (talk) 22:44, 25 June 2024 (UTC)
 * Yes, sources are really necessary for everything on Wikipedia, including mathematical proofs. This is for several fundamental reasons, including the policies WP:OR and WP:DUE.  --JBL (talk) 22:57, 25 June 2024 (UTC)
 * If I send a link from Instagram, will it be enough? (I'm afraid not?) Mathematical.Analysis (talk) 22:59, 25 June 2024 (UTC)
 * I usually also try to find reliable sources (e.g. my correction to the article about Norlund Rice's integral), but here I only generalized something that a certain individual shared on Instagram. The problem is that this person also didn't provide any sources there, so I don't know if referring to the Instagram post would be ok. Mathematical.Analysis (talk) 23:05, 25 June 2024 (UTC)
 * No, of course "something someone shared on Instagram" is not a reliable source. Since you are quite explicit that the section is your own work, I've removed it in accordance with our policy WP:OR.  There are many high-quality sources in existence that discuss Lagrange inversion; you are more than welcome to add a section on its derivation based on one of them.  --JBL (talk) 23:29, 25 June 2024 (UTC)
 * Exactly: "To demonstrate that you are not adding original research, you must be able to cite reliable, published sources that are directly related to the topic of the article and directly support the material being presented." I think that the derivation is based on elementary techniques and it is nothing new, which is consistent with what I quoted. Mathematical.Analysis (talk) 23:11, 25 June 2024 (UTC)
 * Cauchy integral formula, geometric series, integration by parts, residue. This is what the derivation is based on after all. Mathematical.Analysis (talk) 23:13, 25 June 2024 (UTC)
 * "I think this is nothing new" is not a citation of a reliable, published source. (I mean, really.) --JBL (talk) 23:30, 25 June 2024 (UTC)
 * Okay, I'll keep this knowledge to myself. I wanted to be nice by sharing my knowledge, but it's impossible because of selfish people like you. Continue deleting as you wish, you sad pseudo-mathematician. Mathematical.Analysis (talk) 23:32, 25 June 2024 (UTC)
 * Sure, bot. It's best to ignore the beauty of mathematics and be an internet troll rather than have a normal conversation. Mathematical.Analysis (talk) 23:29, 25 June 2024 (UTC)

IMHO
The whole article should be rewritten, starting from eliminating awkward formulas like

f^{-1}(z) = \sum_{n=0}^\infty \frac{({z - f(a) })^{n}}{n!} \lim_{ \omega \to a} \frac{d^{n-1}}{d\omega ^{n-1}} \left(\frac{\omega  - a }{f(\omega) - f(a)}\right)^{n}, $$ that no mathematician would write. One should avoid keeping throughout these proofs, that are algebraic in nature, an analytic/topologic language (limits, path integration,..) here useless and misleading. The point is that, dealing with analytic functions (like, in fact, we do in all mathematics), whenever possible computations are made at a formal level, relying on suitable general representation theorems that allows to do so (here enters the topology, once and for all). Here we can map germs of meromorphic functions at 0, to their formal Laurent series, and perform all computations in that algebraic setting. pm a 15:42, 21 June 2024 (UTC)


 * The formula you've quoted here is in the (uncited) section Lagrange_inversion_theorem, which was added about six months ago by and is also the subject of the discussion section just above this one.  Are the points you're concerned about largely limited to that section?  If so, perhaps it should be removed.  --JBL (talk) 19:59, 21 June 2024 (UTC)
 * First of all, what does it mean that "no mathematician would write such a formula?" I think there's been a big misunderstanding. Secondly, I am not misleading anyone. The derivation uses elementary (almost trivial) techniques. Additionally, what I did is more useful than some draft proof that also contained gaps in the previous version of the article. Mathematical.Analysis (talk) 22:56, 25 June 2024 (UTC)

I agree with to the extent that I think the section should be rewritten without contour integration. As pma says, the theorem holds in a purely algebraic setting (in the ring of formal power series, where it is more general). Tito Omburo (talk) 18:05, 26 June 2024 (UTC)