# Generalizing Contour Integration to Quaternions

I have recently entertained the possibility of defining complex contour integration for the quaternions. I am somewhat aware that the Frobenius theorem dictates that no division algebra can exist in $$\mathbb{R}^3$$; however, Cayley-Dickinson constructions can generalize these division algebras to spaces of the form $$\mathbb{R}^{2^n}$$ at the expense of commutativity and later associativity. Even if the holomorphicity and analyticity are not equivalent (although they are for $$\mathbb{C}$$) generalizations of the Cauchy-Riemann equations have been made to higher dimensions with some success. [1] Additionally, a field analyzing differential quaternions exists in computer graphics, though I believe it is more concerned with aptly representing rotations in $$\mathbb{R}^3$$. [2] Would it be reasonable to believe that contour integration may be generalized to one of these division algebras by designing a hypercomplex function on a 1-manifold map (i.e., one that is diffeomorphic to a subset of $$\mathbb{R}$$.)

Although Liouville's theorem requires that the amount of holomorphic maps significantly decreases as the dimensionality increases and those that do exist must be representable as a composition of Mobius transforms, would a reasonable definition of quaternion complex integration be the sum of standard contour integration along each component of the one manifold for a given basis? (I also assume that the result would have to be invariant of the choice of basis in order to be well-defined.) In particular, I'm wondering if a generalization of contour integration could be used to define a biholomorphic map (if not a holomorphic map) in the quaternions or complex-like numbers spaces above? I'm very new to the topic and haven't the experience of many members on this website. As a semi-related question, why does the existence of quaternions not provide for the existence of division algebras in $$\mathbb{R}^3$$? I assume that taking the subset of all quaternions where only the same three components are nonzero would not suffice as the existence of a multiplicative inverse or closure under division would not hold.

As a final soft question, does anyone suggest textbooks for hypercomplex numbers? I had considered purchasing Hypercomplex Numbers by Isaiah Kantor. Are there any papers that expound on the notion of integration on Hypercomplex Manifolds described on Wikipedia? [3]

Thank you all.

• Your question doesn't apparently consider the several complex variables "approach" (the "several" beginning at $2$). Feb 28, 2021 at 5:59
• Thank you for your prompt reply. Is this theory a result of Kyoshi Oka's research? Would the usage of several complex numbers work in $\mathbb{R}^3$? Feb 28, 2021 at 6:46
• I understand now that your issue deals with functions $\mathbb{C} \to \mathbb{H}$. Feb 28, 2021 at 6:50
• Do you think it would be reasonable to perform a generalization of a contour integral on a hypercomplex manifold by using a notion similar to $\alpha^{*}: \Omega(\mathcal{T}_p^k(\mathcal{M})) \to \Omega(\mathcal{T}_x^k(U))$ where $p \in \mathcal{M}$, $x \in U$, and $\alpha:U \to V$ with $V \cap \mathcal{M} \neq \varnothing$ is a coordinate patch? The nontrivial task would then be defining $D \alpha$. I am about to sleep, but I will return tomorrow. Thank you for all of your help. Feb 28, 2021 at 6:59
• integration of functions $f: M \to \Bbb R^m$ for any manifold $M$ and dimension $m$ is already defined. That you are building additional structures on $\Bbb R^m$ does not change this, so there is nothing related to integration that needs defining. How your additional structures interact with that integration is something to be explored, and as with the case of integration along curves in $\Bbb C$, it may prove to have a rich and interesting theory. But the definition itself has already been done. Feb 28, 2021 at 15:15