Talk:Quaternionic analysis

Question

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)

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)

Renaming the article to "Quaternionic analysis"

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)

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)

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)

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)

Recommend that this article be deleted.

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)

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)

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)

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)

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Analytic conjugate ?

The following comment was removed:

An immediate corollary of which is that the quaternion conjugate is analytic everywhere in $\mathbb {H} .$ Compare this to the seemingly identical complex conjugate, $(x+iy)^{*}=x-iy,$ for $x,y\in \mathbb {R} ,$ and $i^{2}=-1,$ which is not analytic in $\mathbb {C}$.

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)

Notation "U" needs clarifying in the homographies section

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)

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)

I've never seen that notation prior to this. I would personally recommend dropping the $U$ 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)

Walter Benz used the notation with ${\mathcal {R}}$ denoting the group of units of the ring, then ${\mathcal {R}}(x_{1},\ x_{2})$ 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)

I still think it's more confusing than helpful. The $U$ could be misinterpreted as a matrix. --Svennik (talk) 09:49, 3 December 2019 (UTC)

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)

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)

$(afb)(x)\neq af(x)b$ for $f$ a left $\mathbb {H}$ homomorphism

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

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)

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)

Tensor section

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

Linear map

The map $f:\mathbb {H} \rightarrow \mathbb {H}$ of quaternion algebra is called linear, if following equalities hold

$f(x+y)=f(x)+f(y)$

$f(\lambda x)=\lambda f(x)$

$x,y\in \mathbb {H} ,\lambda \in \mathbb {R}$

where $\mathbb {R}$ is real field. Since $f$ is linear map of quaternion algebra, then, for any $a,b\in \mathbb {H}$, the map

$(afb)(x)=af(x)b$

is linear map. If $f$ is identity map ($f(x)=x$), then, for any $a,b\in \mathbb {H}$, we identify tensor product $a\otimes b$ and the map

$(a\otimes b)\circ x=axb$

For any linear map $f:\mathbb {H} \rightarrow \mathbb {H}$ there exists a tensor $a\in \mathbb {H} \otimes \mathbb {H}$, $a=\sum _{s}a_{s0}\otimes a_{s1}$, such that

$f(x)=a\circ x=(\sum _{s}a_{s0}\otimes a_{s1})\circ x=\sum _{s}a_{s0}xa_{s1}$

So we can identify the linear map $f$ and the tensor $a$.

Comments

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

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

More specification

In the derivative section, what is $d_{s0}$ and $d_{s1}$? Wilson868 (talk) 04:53, 18 February 2021 (UTC)

calculus with a quaternionic variable

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)