I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true division algebra in $\mathbb{R}^3$. Likewise, Liouville's theorem requires all three-dimensional conformal maps to be a composition of Mobius transforms. If I sacrificed commutativity, I could work with the Quaternions, which are a true division algebra; however, bicomplex numbers in $\mathbb{C}_2$ preserve commutativity at the expense of allowing zero divisors. I found a helpful diagram comparing the algebras at the following post.

A comparison of algebras

I am particularly interested in producing a biholomorphic or holomorphic mapping from a subset in one of these algebras to a subset of $\mathbb{C}$ using contour integration. I feel that the following paper would be more suited for defining the Cauchy-Riemann equations in the quaternions as opposed to the multicomplex numbers. I assume that helpful properties such as infinite differentiability as a result of the generalized Cauchy integral formula will not hold, especially in the case of multicomplex numbers when the projection of the contour onto one complex plane yields intervals of overlap or self-intersection.

Could the bicomplex numbers be trivially restricted to represent polar coordinates ($r$, $\theta$, $\varphi$) in $\mathbb{R}^3$. I only ask about this seemingly contradictory result as the Mandelbulb was defined using a three-dimensional analog of complex multiplication.

Furthermore, if the notion of contour integration were to exist in either of the quaternions or bicomplex numbers, would Stoke's theorem for 1-manifolds be applicable? Could a contour in this algebra even be shown to be diffeomorphic to an interval residing in $\mathbb{R}$?

I intend to avoid cross-posting, and for those interested, I will direct you all to my similar question posed on StackExchange mathematics.

Thank you all.

  • 1
    $\begingroup$ Is there a particular reason why you wouldn't accept any of the answers below? We can expand them if necessary. $\endgroup$
    – M.G.
    Apr 9, 2021 at 17:57

4 Answers 4


I will address the commutative case.

For every finite-dimensional commutative associative unital $\mathbb{C}$-algebra $\mathcal{A}$ there exists a complete analogue of the theory of Complex Analysis of a Single Variable in the following precise sense. If you consider functions $f:U \to \mathcal{A}$, $U \subseteq \mathcal{A}$ open, with $\mathcal{A}$-linear $Df$, then one can establish the following 4 pillars of complex function theory about them:

  1. Generalized Cauchy-Riemann equations;
  2. Generalized Cauchy and Morera Integral Theorems;
  3. Generalized Cauchy Integral Formula;
  4. Analyticity in the sense of $\mathcal{A}\{Z\}$;

The key starting points really are that $U(\mathcal{A})$ is (path-)connected and $\omega := f(Z) \mathrm{d}Z$ is a closed $\mathcal{A}$-valued differential form. In particular, you get things like $$ f(W) = \frac{1}{2\pi i} \oint_\gamma \frac{f(Z)}{Z-W} \mathrm{d}Z $$ whenever well-defined.

One could say, this is one-variable theory of (certain) functions of several complex variables, so they behave in some regards like functions of a single complex variable.

Here is a painfully detailed reference for the local theory: https://arxiv.org/abs/1812.11661 (beware of typos!) It's in a dire need of an update, but unfortunately I haven't had the time to do so. It was a part of my Msc. thesis (apologies for the shameless self-advertisement). The paper actually considers morphisms of such algebras instead of the objects themselves, so just replace a general morphism by the identity map.

  • $\begingroup$ Marin, are you still interested in seeing a copy of the 1952 Lincei paper of Rizza? $\endgroup$ Mar 3, 2021 at 19:46
  • $\begingroup$ @DanieleTampieri: yes, that would be great for completeness sake. Thank you! $\endgroup$
    – M.G.
    Mar 3, 2021 at 20:03
  • $\begingroup$ Could I use the e-mails in your Arxiv paper? $\endgroup$ Mar 3, 2021 at 20:42
  • $\begingroup$ @DanieleTampieri: yes, please send it to the gmail, the other one is probably no longer working. $\endgroup$
    – M.G.
    Mar 3, 2021 at 20:44

This is really not my expertise: however, some friends and knowledgeable researchers in these fields have shared with me some insight, so I can add my two cents by sharing with you what they shared with me.
The differential and integral calculus for algebras of functions over bicomplex numbers was comprehensively dealt by Griffith Baley Price in his (perhaps classical) monograph [2] on the subject: there he deals also with general multicomplex function algebras and extends the classical Cauchy formulas to those settings.
A more modern exposition, dealing with Cauchy formulas but also with recent progress in the field and also with Stokes and Pompeiu formulas in the context of (only) bicomplex variables is [1] (see in this regard chapter 11, pp. 211-217): it is also a beautiful, smooth, clean and clear, read.
Dealing with differential and integral calculus over this kind of function algebras, both these monographs deal with the concept of differentiability for such functions, and thus give a definition of the (generalized) Cauchy-Riemann equations in these settings.

An historical remark

For general complex function algebras, commutative or not, the classical Cauchy integral theorem was first proved by Giovanni Battista Rizza in 1950: the classical Cauchy integral formula was later extended by him to functions on commutative real normed algebras isomorphic to a complex algebra two years later, in 1952. Many of the details of the researches of Rizza and his school are described in the survey [3].
An addendum. Marin Genov, in his comment below, points out that, earlier than G. B. Rizza, Edgar Raymond Lorch was able to prove a Cauchy formula for infinite dimensional commutative Banach $\Bbb C$-algebras in reference [A1]: this seems the earlier result of this kind, thus I thank Dr. Genov for his reporting.


[1] M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac, Bicomplex holomorphic functions. The algebra, geometry and analysis of bicomplex numbers. (English) Frontiers in Mathematics. Cham: Birkhäuser/Springer, ISBN 978-3-319-24866-0, pp. viii+231 (2015), MR3410909, Zbl 1345.30002.

[2] G. Baley Price, An introduction to multicomplex spaces and functions, (English) Pure and Applied Mathematics, 140. New York etc: Marcel Dekker, Inc. pp. xiii+402 (1991),  ISBN: 0-8247-8345-X, MR1094818, Zbl 0729.30040.

[3] Giovanni Battista Rizza, "Contributi recenti alla teoria delle funzioni nelle algebre", Rendiconti del Seminario Matematico e Fisico di Milano (in Italian), 43 (1): 45–54, doi:10.1007/BF02924838, MR0350025, S2CID 123219540, Zbl 0325.30040.

Addendum reference

[A1] Edgar R. Lorch, "The theory of analytic functions in normed Abelian vector rings", (English), Transaction of the American Mathematical Society 54, pp. 414-425 (1943), DOI:10.2307/1990255, MR0009090, Zbl 0060.27202.

  • 2
    $\begingroup$ To your references I would also add Lorch's “The Theory of Analytic Functions in Normed Abelian Vector Rings” (1943), where he shows Cauchy's Integral Theorem and Integral Formula (as well as some other related results) directly in the setting of (infinite-dimensional) commutative Banach $\mathbb{C}$-algebras by functional-analytic methods. $\endgroup$
    – M.G.
    Mar 3, 2021 at 20:29

To address the question you are particularly interested in, the easiest way to consider holomorphic functions from subsets of the quaternions $\mathbb{Q}$ to $\mathbb{C}$ is to restrict the domain to a plane $\mathbb{P}$ inside $\mathbb{Q}$ and consider functions defined on the plane with complex values that are $\mathbb{P}$-differentiable in the sense that the limit of the derivative exists if $h \to 0$ while $h$ is always a vector on $\mathbb{P}$. Then one would get the following version of Cauchy's integral theorem for a closed curve $C$ enclosing a region $R$:

$$\int_C f(z)dz=\left(\iint_R f'(z)dA\right) \eta_{\mathbb{P}}$$

where $\eta_{\mathbb{P}}=vw-wv$ for two normal unit vectors $v, w$ in $\mathbb{P}$ (it is easy to check that this coefficient does not depend on the choice of such vectors). In particular if one chooses a plane for which the vectors commute one gets the value $0$, while on the plane spanned by, say, $i, j$, the value of the integral is non-zero in general.

Of course this formula is not surprising taking into account that it is essentially Green theorem: the differentiability conditions on $f=u+iv+js+kt$ impose precisely the restrictions on $u, v, s, t$ that makes Green theorem appear.

One could also consider $3$-dimensional subspaces $\mathbb{S}$ of $\mathbb{Q}$ and express the contour integral as a surface integral using the derivative, but there the result is less interesting since there are really few $\mathbb{S}$-differentiable functions (and in fact, for the four-dimensional $\mathbb{Q}$ the only $\mathbb{Q}$-differentiable functions are the linear ones).

Regarding the bicomplex contour integration, one could define the limit of the derivative if $h \to 0$ while $h$ is not a zero divisor and obtain formulas much similar to Cauchy's integral formula, but in the end it would be essentially replicating this latter twice in each of the two planes formed by the zero-divisors, so it would not really count as a genuine generalization.

  • $\begingroup$ The bicomplex numbers are indeed boring in this regard, because as an algebra they are just two copies of $\mathbb{C}$. Also, IIRC, functions that are "plane"-holomorphic when restricted to a $\mathbb{C}$-subplane of $\mathbb{H}$ are called slice holomorphic. Google returns a lot of relevant results with this search term. $\endgroup$
    – M.G.
    Mar 4, 2021 at 5:51
  • $\begingroup$ @M.G. well, not exactly two copies of $\mathbb{C}$. One can consider them as a combination of complex numbers and hyperbolic numbers as well. So, they are not that trivial. Yes, they have two copies of complex numbers, but also a copy of hyperbolic numbers. $\endgroup$
    – Anixx
    Mar 23, 2021 at 16:37
  • $\begingroup$ @Anixx: as a $\mathbb{C}$-algebra, the bicomplex numbers is indistinguishable from the product algebra $\mathbb{C} \times \mathbb{C}$ because they are isomorphic as $\mathbb{C}$-algebras. Maybe you mean these algebras plus some additional structure on them that makes them distinct? $\endgroup$
    – M.G.
    Mar 23, 2021 at 17:07
  • $\begingroup$ @M.G. bicomplex numbers and tessarines are 4-dimensional. One dimension is real, two are complex and the fourth dimension is hyperbolic (split-complex). So, besides complex numbers, aloso split-complex numbers are included. $\endgroup$
    – Anixx
    Mar 23, 2021 at 17:11
  • $\begingroup$ @Anixx: if I understand you correctly, this is just a special choice of a (nice) $\mathbb{R}$-basis for the algebra. $\endgroup$
    – M.G.
    Mar 23, 2021 at 17:17

For what it worth, in tessarines ($j^2=1$) the following rule holds:

$$f((a+i b)+j (c+i d))=\frac{1}{2} (f(a+i b+c+i d)+f(a+i b-c-i d))+\frac{1}{2} j (f(a+i b+c+i d)-f(a+i b-c-i d))$$

So, I think the theorems from complex numbers are directly applicable.

  • $\begingroup$ What is $f$ here? Surely you need some assumptions on it, it doesn't hold for all functions. $\endgroup$
    – Wojowu
    Mar 23, 2021 at 17:12
  • $\begingroup$ @Wojowu well, this is what you get from the matrix representation. So, I think it is general enough. I think, the function should be analytic. $\endgroup$
    – Anixx
    Mar 23, 2021 at 17:14

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