Talk:Quaternionic analysis

Latest comment: 1 month ago by 2601:643:8D00:5A40:5C45:8F0D:4CB3:E24F in topic calculus with a quaternionic variable

Question

edit

What is the "R" under the limit symbol in the definition of the Gateaux derivative? — Preceding unsigned comment added by 133.86.80.122 (talk) 02:10, 4 June 2012 (UTC)Reply

The R means that t is restricted to the real line. As the text says, "h shows the direction" in which the derivative is taken; for emphasis the R was included by an editor to limit the range of t.Rgdboer (talk) 23:40, 4 June 2012 (UTC)Reply

Renaming the article to "Quaternionic analysis"

edit

Sometime soon, I hope to rename the article to "Quaternionic analysis". I had considered "quaternion analysis," however google scholar seems to indicate that the former is more highly used. This is an improvement from "quaternion variable" according to the WP naming guidlines for the following reasons:

  • Recognizability/Naturalness: "quaternion variable" is just a hacked-off version of "function of a quaternion variable," and is not likely to be the phrase people use to search. It does not turn up relevant hits like "quaternionic analysis" does.
  • Consistency: "complex variable" redirects to complex analysis, and "real variable" redirects to a stub which is basically a redirect to real analysis, so to follow this pattern it would be more sensical to use "quaternionic analysis"

Redirects for "quaternion variable" and "quaternion analysis" would definitely be part of the plan. Feedback welcome. Rschwieb (talk) 15:06, 4 January 2013 (UTC)Reply

The current title can be used in phrases like "differentiation with respect to a quaternion variable", and "quaternion variable domain", as in domain (mathematical analysis). The title you propose is something of a conversation stopper. Reviewing the article mathematical analysis there is the section called Subdivisions. There functional analysis is put on a par with geometric analysis. But textbooks today frequently use Complex Variable in preference to Complex Analysis.
As for the adjective Quaternionic, note that we have quaternion group though references often use ionic for the designation of that group. The longer adjective sounds like ionic bond. The current title refers to variable which connotes the extent of quaternions. This comment may serve to explain why the title was "Quaternion variable", should a Move be made.Rgdboer (talk) 01:19, 7 January 2013 (UTC)Reply
I've got no problem with the phrase "a quaternion variable," and it can continue to be used in articles, but I just think it's not a good title for this article.
Judging from googlebook hits, it is very difficult to agree with you that "complex variable" is preferred. It rather looks like a large majority of hits use the phrase "function of a complex variable." I don't dispute that "function(s) of a quaternion variable," is a fine title, but I am pointing out that "function of a complex variable" and "function of a real variable" are both (basically) redirects to complex analysis and real analysis. There is a definite pattern suggested. Rschwieb (talk) 20:31, 7 January 2013 (UTC)Reply

It looks like there isn't any major disagreement (right?). Sometime soon I'll make the move, making sure to make appropriate redirects to preserve the former use of "quaternion variable." I might also expand the lead to make the analogy to functions of real and complex variables a little clearer. I hope this satisfies all parties. Thanks! Rschwieb (talk) 15:54, 11 January 2013 (UTC)Reply

Recommend that this article be deleted.

edit

The article is apparently not written by a mathematician, since many statements are not mathematically rigorous and suggest a harmful kind of "original research". For example, the part about extending a function to the quaternions if its real part is even and if its pure quaternion part is odd, does not even define what "extending" the function means. Without any special definition, all such functions can be extended to the quaternions. Shortly after that, an expression involves a variable t, and although the other variables are specified, it is left unspecified whether t is an arbitrary quaternion or is restricted to having real values, or something else. Et cetera.Daqu (talk) 21:27, 7 October 2015 (UTC)Reply

There is no t in the argument, perhaps it was f that you read as t. Check it out, the conditions are necessary for a function of a quaternion variable to be an extension of a complex variable function. — Rgdboer (talk) 22:27, 8 October 2015 (UTC)Reply

Thank you for the correction. Apparently I misread or mistyped something; I'm not sure what. Nevertheless, the article is a train wreck, with no explanation of what conditions "extending" a function from ℂ to ℍ is required to satisfy, what the concept of "linear" means in the quaternions (and why "affine" appears to have a decidedly different definition with only left multiplication allowed before the translation), etc., etc., etc. The article appears to have been written by a very intelligent person who does not have a lot of experience in writing about mathematics or writing encyclopedia articles.Daqu (talk) 15:04, 15 October 2015 (UTC)Reply

Thank you for your interest in this article. As to the extension of functions, the proposition is concerned with extending a complex function to the quaternions strictly as a function of two real variables derived from a quaternion: the scalar part and the norm of the vector part. Naturally polynomials and power series with real coefficients can be formally extended without exception. The proposition concerns quaternionic analysis as it might naturally arise from considerations commonly made with complex functions as functions of two real variables. — Rgdboer (talk) 21:29, 15 October 2015 (UTC)Reply
edit

Hello fellow Wikipedians,

I have just modified one external link on Quaternionic analysis. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

When you have finished reviewing my changes, please set the checked parameter below to true or failed to let others know (documentation at {{Sourcecheck}}).

This message was posted before February 2018. After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{source check}} (last update: 5 June 2024).

  • If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
  • If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot (Report bug) 16:44, 16 July 2016 (UTC)Reply

edit

Hello fellow Wikipedians,

I have just modified 2 external links on Quaternionic analysis. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

When you have finished reviewing my changes, please set the checked parameter below to true or failed to let others know (documentation at {{Sourcecheck}}).

This message was posted before February 2018. After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{source check}} (last update: 5 June 2024).

  • If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
  • If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot (Report bug) 18:20, 20 July 2016 (UTC)Reply

Analytic conjugate ?

edit

The following comment was removed:

An immediate corollary of which is that the quaternion conjugate is analytic everywhere in   Compare this to the seemingly identical complex conjugate,   for   and   which is not analytic in  .

The arithmetic expression of conjugation of quaternions is asserted to be an analytic function, but defining an analytic quaternion function is problematic. — Rgdboer (talk) 23:35, 8 September 2018 (UTC)Reply

Notation "U" needs clarifying in the homographies section

edit

I suspect it indicates that the row vector following it is a homogeneous vector. But that's not clear to a reader Svennik (talk) 21:33, 9 November 2019 (UTC)Reply

Click on "homography" in the first line. It brings you to Homography#Over a ring where the notation is described. — Rgdboer (talk) 00:38, 10 November 2019 (UTC)Reply
I've never seen that notation prior to this. I would personally recommend dropping the   and simply clarifying that the row vector is homogeneous. Other than that, it's a nice derivation of the action of the dual quaternions on 3D space. --Svennik (talk) 13:07, 11 November 2019 (UTC)Reply
Walter Benz used the notation with   denoting the group of units of the ring, then   denotes a point in the "Projektive Gerade uber einem Ringe", page 84, in his book Vorlesungen uber Geometrie der Algebren, available at 84 at Google Books. — Rgdboer (talk) 02:09, 2 December 2019 (UTC)Reply
I still think it's more confusing than helpful. The   could be misinterpreted as a matrix. --Svennik (talk) 09:49, 3 December 2019 (UTC)Reply
If U were a matrix, then it would be found on the other side of the row vector ! Recall matrix multiplication. — Rgdboer (talk) 02:42, 4 December 2019 (UTC)Reply
I think that's asking too much from the reader. You should always define your notation. See my edit. Svennik (talk) 10:23, 14 February 2020 (UTC)Reply

for a left homomorphism

edit

This is in reference to section "Linear maps". The mistake is with the  . Svennik (talk) 11:11, 14 February 2020 (UTC)Reply

Additionally, the term "linear map" might not be appropriate. What's actually discussed is a homomorphism between modules. Svennik (talk) 11:15, 14 February 2020 (UTC)Reply
It turns out it's just a regular linear map, where the scalar field is the real numbers. Svennik (talk) 15:52, 23 February 2020 (UTC)Reply

Tensor section

edit

Since quaternions express some linear algebra, and tensors are frequently used, someone inserted this section, now removed as unreferenced and disputed:

Linear map

edit

The map   of quaternion algebra is called linear, if following equalities hold

 
 
 

where   is real field. Since   is linear map of quaternion algebra, then, for any  , the map

 

is linear map. If   is identity map ( ), then, for any  , we identify tensor product   and the map

 

For any linear map   there exists a tensor  ,  , such that

 

So we can identify the linear map   and the tensor  .

Comments

edit

Discussion can proceed here if necessary. — Rgdboer (talk) 02:29, 18 February 2020 (UTC)Reply

I was confused by the notation. The letter   was used to represent both a quaternion and a real number. I changed one of those to a  , which now denotes a real number. Svennik (talk) 15:55, 23 February 2020 (UTC)Reply

More specification

edit

In the derivative section, what is   and  ? Wilson868 (talk) 04:53, 18 February 2021 (UTC)Reply

calculus with a quaternionic variable

edit

See the work of C. Schwartz, J. Math. Phys. 50, 013523 (2009) He studies the differential, not the derivative, of a function of quaternionic variable. Detailed results are very interesting. 2601:643:8D00:5A40:5C45:8F0D:4CB3:E24F (talk) 14:57, 10 September 2024 (UTC)Reply