Does Quaternion Calculus Extend Classical Complex Analysis Theorems?

[mhill]

let be a quaternion $$a+ib+cj+dk$$ and a,b,c,d are functions of (x,y,z,u)

my questions are.

- is there an anlogue of Cauchy integral theorem ?? , if an analytic function of a quaternion z , defined by f(z) , has a pole at the point 1+i+2j-3k How could you calculate its residue ??

- If a function of a quaternion is ANALYTIC does it satisfy $$\Box f =0$$

this would be a consequence that if Q is a quaternion (a,b,c,d) then

$$QQ^{*} = a^{2}-b^{2}-c^{2}-d^{2}$$ * = conjugate , so QQ* is a real number.

i would be interested to find solution to integrals on 4 dimensions or to construct Laurent series for functions f(x,y,z,t)

 

[fresh_42]

These are very specific questions. The algebra nature of quaternions is often the main subject which is addressed, calculus not so much. But I have found a book https://math.dartmouth.edu/~jvoight/quat-book.pdf which mentions an analogon to Cauchy's integral theorem (p. 188).