Re: Analytic Functions in 3D?-

Jeff wrote:

Hello

I've been studying the remarkable properties of analytic functions in the complex plane and their relationship to conformal mapping.

I was wondering if there was any extension of this to 3 dimensions - though I suspect not.
And what about higher dimensions? Will the quaternions do for 4D what complex numbers do for 2?

Any pointers or keywords for searching would be appreciated.

Many Thanks
Jeff.


I am afraid you are right to guess that there is on extension to 3D -

The ingredients for differentiability are the four principal arithmetical operations and the limit concept,
in other words: one needs a topological field, preferably commutative, the topology preferably derived from a metric, the field being complete as a metric space.

Well, there is no 3D field over the reals.

As regards quaternions: the norm in the quaternion skew field yields the well-known Euclidean metric with its accompanying topology. Multiplication is not commutative, so one needs to distinguish between left-division
( p^(-1).q ) and right-division ( q.p^(-1) ), and consequently, between left- and right-difference and differential quotients.
I am not aware of a fully developed quaternion differential calculus. Google and Wikipedia might be helpful.

Concerning conformal mappings of 4D regions:
First look at the 2D case. In the plane we have at every point a combined translation, rotation and magnification, plus something infinitesimal.

Something similar happens in 4D: 4D translation, 4D rotation, magnification, plus something infinitesimal.

The snag is in the 4D rotations. They come in several different kinds. Read more about 4D rotations in Wikipedia > SO(4) and in my arXiv articles at http://arxiv.org/abs/math.GM/0501249 and http://arxiv.org/abs/math.GM/0701759 ,
all of this also reachable from my web page on 4D rotations at http://www.xs4all.nl/~jemebius/4drot.htm .

Also read about hyper-Kähler mainfolds, a class of 4D Riemann manifolds with special, quaternion-based porperties.

Cheers: Johan E. Mebius
.

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