From: "G. A. Edgar"
Subject: Re: Quaternion Differenciation
Date: Thu, 25 Mar 1999 10:15:31 -0500
Newsgroups: sci.math
Keywords: Definitions of analyticity do not generalize well to quaternions.
In article , Uday Patil wrote:
> How does one define Quaternion Derivative of
> a Quaternion valued function of a Quaternion variable?
>
> In other words, what is the Quaternionic equivalent of a
> "complex-analytic function"?
>
There is no definition that will do everything the complex
case does. Try this one:
limit (h -> 0) (f(z+h)-f(z)) h^{-1}
where the limit is as quaternion h goes to 0 (in the sense of
the quaternion distance |h| -> 0 ). With this definition, very
few functions are differentiable. Even f(z) = z^2 is non-differentiable.
Should a "differentiable" function be conformal (as in the cases
of R and C)? There are very few conformal maps of R^4 to itself.
--
Gerald A. Edgar edgar@math.ohio-state.edu
Department of Mathematics telephone: 614-292-0395 (Office)
The Ohio State University 614-292-4975 (Math. Dept.)
Columbus, OH 43210 614-292-1479 (Dept. Fax)
==============================================================================
From: Pertti Lounesto
Subject: Re: Quaternion Differenciation
Date: Thu, 25 Mar 1999 14:43:43 +0200
Newsgroups: sci.math
Uday Patil wrote:
> How does one define Quaternion Derivative of
> a Quaternion valued function of a Quaternion variable?
>
> In other words, what is the Quaternionic equivalent of a
> "complex-analytic function"?
There are five (5) different generalizations of "complex
analytic" to "quaternion analytic" depending on which
feature of analyticity you want to emphasize (quaternion
differentiable, quaternion power series with quaternion
coefficients, quaternion power series with real coefficients,
conformal mappings in R^4, or generalization of Cauchy-
Riemann conditions). Only one of these alternative
generalizations leads to a fruitful class of functions.
For further information, you may consult my book
"Clifford algebras and spinors", CUP, 1997/98, URL
http://www.cup.org/Titles/59/0521599164.html, Chapter
5 "Quaternions", section 5.8 "Function theory of
quaternion variables", pages 74-76.
Pertti Lounesto
==============================================================================
From: Pertti Lounesto
Subject: Re: Quarternion Analysis?
Date: Sat, 31 Jul 1999 14:28:04 +0200
Newsgroups: sci.math
"G. A. Edgar" wrote:
> In article <37A0A51A.C89925C3@cwcom.net>, wrote:
>
> > Is there such a thing as quarternion analysis, analogous to complex
> > analysis?
> > If so can anyone supply references to books and survey, or introductory,
> >
> > papers?
>
> The analytic function of complex analysis has many nice properties,
> each of which may be used as definition.
> However, when generalized to quaternions these properties are
> no longer equivalent. So we get no theory analogous to
> the complex theory. FOR EXAMPLE, the quaternion function
> z^2 is not differentiable... There are very few conformal maps
> in 4-dimensional real space...
The crux of G.A. Edgar's comment is that quaternion analysis cannot
be as elegant, as wide connecting, or as succint as complex analysis.
There are different generalizations of function theory in 2D to 4D:
1. Derivative of a function f at x is a linear mapping f'(x):R^4->R^4,
h->f'(x)h which can be represented by quaternion multiplication
f'(x)h = qh for some quaternions q,h in H = R^4. The only functions
in this category are the affine right linear functions q -> aq+b.
2. Function f:R^4->R^4, x->f(x) is a power series of x, with coefficients in
a) quaternions; this results in real analytic functions of 4 real
variables
b) reals; this results in axial symmetric functions, obtained by
"rotating"
the graph of an analytic function C->C "around" the real axis in R^4.
3. Conformal mappings in R^4 are just restrions of Mo:bius transformations
of R^4, which in turn are compositions of 4 mappings: rotations,
translations,
dilatations, and transversion (a term introduced by Ahlfors).
4. Generalizations of Cauchy-Riemann equaitons do result in an interesting
class of functions. _But_, these functions are not quaternions
differentiable,
neither conformal, nor representable by power series of the arguments.
==============================================================================
From: nospam@127.0.0.1
Subject: Re: Quarternion Analysis?
Date: Sun, 08 Aug 1999 16:39:16 GMT
Newsgroups: sci.math
On Thu, 29 Jul 1999 20:01:46 +0100, m04fuj00@cwcom.net wrote:
>Is there such a thing as quarternion analysis, analogous to complex
>analysis?
>If so can anyone supply references to books and survey, or introductory,
>
>papers?
Tony Sudbery's papers are probably the best available on the net in
English.
ftp://ftp.netcom.com/pub/hb/hbaker/quaternion/sudbery/Quaternionic-analysis-memo.ps.gz
ftp://ftp.netcom.com/pub/hb/hbaker/quaternion/sudbery/Quaternionic-analysis-memo.dvi.gz
ftp://ftp.netcom.com/pub/hb/hbaker/quaternion/sudbery/Quaternionic-analysis.ps.gz
ftp://ftp.netcom.com/pub/hb/hbaker/quaternion/sudbery/Quaternionic-analysis.dvi.gz