Is there such a thing as quaternion analysis -- calculus of functions from quaternions to quaternions.
What would be their key theorems ? What would be the analogue of conformal mappings, if any ?
Any book recommendations would be gratefully appreciated.
jacobolus 159 days ago [-]
Conformal mappings are not nearly as rich in >2 dimensions. There is a much stronger rigidity constraint and you end up limited to just Möbius transformations. The 2 dimensional case is special.
Yes of course, but I am curious about any interesting structures that functions from quaternion to quaternion may possess. I used conformal mapping as an example of an interesting structure. I could have used Cauchy Riemann as another example.
jacobolus 159 days ago [-]
You're probably looking for something like Sudbery 1977,
A quaternion encodes uniform scaling + rotation. The logarithm of a quaternion is its axis-angle-nepers form, and vice versa.
quat = sqrt( exp( nepers + radians * <axis> ) )
So I think with this exponential map, the rest of its calculus can be extended from that.
srean 166 days ago [-]
Heard the word 'nepers' after many decades. Are you by any chance an Electrical major ?
Thanks for your comment. To be fair, I had not done due diligence before asking. There's a Wikipedia pages on quaternion calculus.
Complex analysis (calculus on functions from 2D rotations to 2D rotations) is beautiful -- Once differentiability guarantees infinite differentiability. Wondering what would the analogue of that be for quaternions
JKCalhoun 167 days ago [-]
Very cool.
On MacOS: no audio on Safari, worked in Chrome though.
I was odd to me to see that Ben Eater was involved. His talents are broad.
calebm 167 days ago [-]
Wow, this is very cool.
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Is there such a thing as quaternion analysis -- calculus of functions from quaternions to quaternions.
What would be their key theorems ? What would be the analogue of conformal mappings, if any ?
Any book recommendations would be gratefully appreciated.
See: https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal...
https://dougsweetser.github.io/Q/Stuff/pdfs/Quaternionic-ana...
(published 1979, doi: 10.1017/S0305004100055638)
There are also these notes: https://www.geometrictools.com/Documentation/Quaternions.pdf
Thanks for your comment. To be fair, I had not done due diligence before asking. There's a Wikipedia pages on quaternion calculus.
Complex analysis (calculus on functions from 2D rotations to 2D rotations) is beautiful -- Once differentiability guarantees infinite differentiability. Wondering what would the analogue of that be for quaternions
On MacOS: no audio on Safari, worked in Chrome though.
I was odd to me to see that Ben Eater was involved. His talents are broad.