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The quaternions are associative, so nonnegative integer powers of quaternions are well-defined, and one can consider analytic functions on $\mathbb{H}$ (functions that are given locally by power series). The octonions are not associative, but from what I have read, I think they are "associative enough" that nonnegative integer powers of octonions are defined unambiguously (please correct me if I'm wrong), and one can define analytic functions of octonions.

My question is, how much of this have people actually done? Has a lot of theory been developed about analytic functions on $\mathbb{H}$ and $\mathbb{O}$? I haven't heard of very much. If not, is it because their properties are very similar to those of analytic functions on $\mathbb{C}$?

I know that a lot of people in engineering, kinematics, and computer graphics use quaternions (perhaps more than mathematicians do) so it is not the case that no one is interested in quaternions. I have heard that some physicists are interested in octonions, but I don't know much about that.

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    $\begingroup$ Things are not that great. You can define any power series, say with real coefficients, certainly for the quaternions, but you don't get $e^{p+q}$ coming out well because $pq \neq qp$ usually; here we might as well be using the standard representation of quaternions as certain 4 by 4 matrices. $\endgroup$ – Will Jagy Dec 31 '13 at 3:52
  • $\begingroup$ Addition of quaternions is commutative though. So $ e^{p}*e^{q} = e^{p+q} = e^{q+p} = e^{q}*e^{p}$. $\endgroup$ – Mr X Nov 24 '18 at 20:24
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Here's some information on quaternion analysis. The list of references is probably the most useful part.

http://www.zipcon.net/~swhite/docs/math/quaternions/analysis.html

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    $\begingroup$ @stefanSmith Quaternion analysis sometimes falls under the heading "hypercomplex analysis" and I would suspect that if anyone tried it for octonions too, it might appear under that name also. $\endgroup$ – rschwieb Jan 2 '14 at 14:01
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I am familiar with the theory of holomorphicc functions on H totally similar to the theory of holomorphicc functions on C. See http://vixra.org/abs/1609.0006 Bilberry

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