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is it possible do somehow define regular function(in the way that some analogy to Cauchy integral formula would hold) over quaternions that function of form $$f(q)= \sum_{n=0}^\inf a_n q^n b_n$$ $a,b,c,q \in \mathbb{H}$, would be regular?

With classic definition of regular function linear functions $(f(q)= qa +b )$ are not regular.

In Graziano Gentili, Daniele C. Struppa they define different type of regular function. So functions $ \sum q^n a_n$ are regular. But I would like to have functions of form $\sum a_n q^n b_n$ to be regular and have for them analogy of Cauchy integral formula. Is it possible?

(please note that I have no real knowledge in this topic. I just got interested in barycentric coordinates and its nice connection to complex numbers in 2d. So I thought if it could be extended to 3d via quaternions. But I ran in trouble reproducing rotations. So I thought I need functions of form $R(q) = rqr^*$ to be regular )

Thank you for answers, Tom!

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up vote 1 down vote accepted

Check out and references therein.

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