Hello,


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](https://neo.math.unifi.it/users/gentili/lavoripdf/AdvancesCP.pdf) 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!



regular quaternion functions of form  f(q) =c + a*q*b + ...
Check out <http://www.zipcon.net/~swhite/docs/math/quaternions/analysis.html> and references therein.


