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▲ srean 5 days ago

Questions for mathematicians out here.

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.

▲Koshkin 5 days ago | parent | next [-]

I am not familiar with this area, but see, e.g. https://link.springer.com/book/10.1007/978-3-0348-0622-0

There are also these notes: https://www.geometrictools.com/Documentation/Quaternions.pdf

▲srean 5 days ago | parent [-]

Thanks a bunch

▲xeonmc 4 days ago | parent | prev [-]

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 4 days ago | parent [-]

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