What is all right(left) differential function $f :Q \to Q $, where $Q$ is the quaternions ring ?
I was thinking about the notion of analytic function on functions defined on quaternions. If $f(x)=ax+b$ then $\lim_{h\to 0}[f(x+h)-f(x)]h^{-1}=a$(right derivative). If $f(x)=xa+b$ then $\lim_{h\to 0}h^{-1}[f(x+h)-f(x)]=a$(left derivative). $f$ is call right(left)analytic if it has right(left)derivative at any point of $Q$. I think all right analytic function $f$ is in the form $f(x)=ax+b$ and all left differentiable(analytic) function $f$ is in the form $f(x)=xa+b$ . If it is right what is the proof?
I think there would be a technique like the Cauchy-Riemann equation to prove it.