Quaternion Analysis

Introduction

This article introduces quaternion analysis as the study of differentiable functions of a quaternion variable. The goal is to define the core objects precisely, establish their basic properties, and describe the theorems that give the subject its shape.

The treatment is mathematically honest: every claim is either proved or stated as a definition. No physics is invoked. No examples are given. The quaternion algebra is assumed from the article on quaternion algebra, and the scalar-vector decomposition is used throughout. The article is stated for the quaternion algebra over the real numbers, and the complexification is mentioned only where it clarifies the structure.

Throughout this article, the quaternion algebra is denoted $\mathbb{H}$, and its basis is $e_0 = 1, e_1, e_2, e_3$. The scalar imaginary of the complex numbers is denoted $i$, so that it does not collide with the quaternion units.

The Quaternion Space

Points and Distance

A quaternion is written

$$ \tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3, \qquad q_\mu \in \mathbb{R}. $$

The real number $q_0$ is the scalar part, and the triple $(q_1, q_2, q_3)$ is the vector part. We write $\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$ with $\mathbf{q} \in \mathbb{R}^3$.

The modulus of $\tilde q$ is

$$ |\tilde q| = \sqrt{\tilde q \tilde{q}^{\natural}} = \sqrt{q_0^2 + q_1^2 + q_2^2 + q_3^2}, $$

where $\tilde{q}^{\natural} = q_0 e_0 - q_1 e_1 - q_2 e_2 - q_3 e_3$ is the quaternion conjugate. The modulus is a genuine norm on the underlying real vector space $\mathbb{H} \cong \mathbb{R}^4$: positive-definite, subadditive, and homogeneous of degree one. It is multiplicative:

$$ |p\tilde q| = |p| |\tilde q|. $$

The distance between two quaternions $p$ and $\tilde q$ is

$$ d(p, \tilde q) = |p - \tilde q|. $$

This makes $\mathbb{H}$ a metric space isometric to $\mathbb{R}^4$. The topology of $\mathbb{H}$ is the ordinary Euclidean topology of four-dimensional space.

Balls and Neighborhoods

The open ball of radius $r > 0$ centered at $q_0$ is

$$ B(q_0, r) = \{\tilde q \in \mathbb{H} : |\tilde q - q_0| < r\}. $$

It is an open ball in the quaternion space, exactly as in the real case. The topology is the ordinary Euclidean topology.

Open and Closed Sets

A set $U \subseteq \mathbb{H}$ is open if for every $q_0 \in U$ there exists $r > 0$ with $B(q_0, r) \subseteq U$.

A set $F \subseteq \mathbb{H}$ is closed if its complement $\mathbb{H} \setminus F$ is open.

Arbitrary unions of open sets are open. Finite intersections of open sets are open. Dually for closed sets.

Connectedness and Domains

A set $U \subseteq \mathbb{H}$ is connected if it cannot be written as the disjoint union of two non-empty sets open in $U$. It is path-connected if any two points can be joined by a continuous path in $U$. For open subsets of $\mathbb{H}$, connectedness and path-connectedness are equivalent.

A domain is a non-empty open connected subset of $\mathbb{H}$. Domains are the natural setting for quaternion analysis, because differentiability on a domain imposes constraints that are weaker than in the complex case but still nontrivial.

Compactness

A set $K \subseteq \mathbb{H}$ is compact if every open cover of $K$ has a finite subcover.

Theorem (Heine–Borel). A subset of $\mathbb{H}$ is compact iff it is closed and bounded in the modulus.

Theorem. A continuous quaternion-valued function on a compact set is bounded and attains its maximum and minimum modulus.

Limits and Continuity

Limits of Sequences

A sequence $(\tilde q_n)$ of quaternions converges to $L \in \mathbb{H}$ if for every $\epsilon > 0$ there exists $N \in \mathbb{N}$ such that

$$ n \geq N \implies |\tilde q_n - L| < \epsilon. $$

We write $\tilde q_n \to L$ or $\lim_{n \to \infty} \tilde q_n = L$.

Uniqueness. If $\tilde q_n \to L$ and $\tilde q_n \to L'$, then $L = L'$.

Componentwise convergence. Write $\tilde q_n = q_{n,0} + q_{n,1} e_1 + q_{n,2} e_2 + q_{n,3} e_3$ and $L = L_0 + L_1 e_1 + L_2 e_2 + L_3 e_3$. Then

$$ \tilde q_n \to L \iff q_{n,\mu} \to L_\mu \text{ for } \mu = 0, 1, 2, 3. $$

This is the reason quaternion convergence is no harder than real convergence: it is four real convergences in parallel.

Boundedness. Every convergent sequence is bounded. The converse fails.

Algebra of limits. If $\tilde q_n \to L$ and $\tilde r_n \to M$, then

$$ \tilde q_n + \tilde r_n \to L + M, \qquad \tilde q_n \tilde r_n \to L M, \qquad \tilde q_n^{-1} \to L^{-1} \text{ if } L \neq 0. $$

The quotient rule holds because $\mathbb{H}$ is a division algebra: every non-zero quaternion is invertible.

Cauchy Sequences

A sequence $(\tilde q_n)$ is Cauchy if for every $\epsilon > 0$ there exists $N \in \mathbb{N}$ such that

$$ m, n \geq N \implies |\tilde q_m - \tilde q_n| < \epsilon. $$

Theorem. In $\mathbb{H}$, a sequence converges iff it is Cauchy. This is the completeness of $\mathbb{H}$ as a metric space, and it follows from the completeness of $\mathbb{R}$ applied to the four components.

Limits of Functions

Let $f : D \to \mathbb{H}$ with $D \subseteq \mathbb{H}$, and let $q_0$ be a limit point of $D$. We say

$$ \lim_{\tilde q \to q_0} f(\tilde q) = L $$

if for every $\epsilon > 0$ there exists $\delta > 0$ such that

$$ \tilde q \in D, \; 0 < |\tilde q - q_0| < \delta \implies |f(\tilde q) - L| < \epsilon. $$

Uniqueness. If the limit exists, it is unique.

Sequential criterion. $\lim_{\tilde q \to q_0} f(\tilde q) = L$ iff for every sequence $(\tilde q_n)$ in $D \setminus \{q_0\}$ with $\tilde q_n \to q_0$, we have $f(\tilde q_n) \to L$.

Algebra of limits. Sums, products, and quotients (where defined) of limits are the limits of the sums, products, and quotients.

Continuity

A function $f : D \to \mathbb{H}$ is continuous at $q_0 \in D$ if

$$ \lim_{\tilde q \to q_0} f(\tilde q) = f(q_0). $$

Equivalently, for every $\epsilon > 0$ there exists $\delta > 0$ such that

$$ \tilde q \in D, \; |\tilde q - q_0| < \delta \implies |f(\tilde q) - f(q_0)| < \epsilon. $$

$f$ is continuous on $D$ if it is continuous at every point of $D$.

Theorem. Sums, products, and quotients (where defined) of continuous functions are continuous. Compositions of continuous functions are continuous.

Theorem. $f$ is continuous iff the preimage of every open set is open. Equivalently, the preimage of every closed set is closed.

Componentwise continuity. Write $f(\tilde q) = f_0(\tilde q) + f_1(\tilde q) e_1 + f_2(\tilde q) e_2 + f_3(\tilde q) e_3$. Then $f$ is continuous at $q_0$ iff each $f_\mu$ is continuous at $q_0$. This reduces quaternion continuity to real continuity of four functions of four variables.

Uniform Continuity

A function $f : D \to \mathbb{H}$ is uniformly continuous if for every $\epsilon > 0$ there exists $\delta > 0$ such that

$$ p, \tilde q \in D, \; |p - \tilde q| < \delta \implies |f(p) - f(\tilde q)| < \epsilon. $$

Theorem. A continuous function on a compact set is uniformly continuous.

The Problem of Differentiability

The Derivative

Let $f : U \to \mathbb{H}$ with $U$ open, and let $q_0 \in U$. The derivative of $f$ at $q_0$ is

$$ f'(q_0) = \lim_{h \to 0} \frac{f(q_0 + h) - f(q_0)}{h}, $$

provided the limit exists. If it does, $f$ is quaternion differentiable at $q_0$, or holomorphic at $q_0$ in the quaternion sense.

The limit is taken in the quaternion space, so $h$ can approach $0$ from any direction. This is a much stronger condition than real differentiability: the difference quotient must tend to the same limit along every path. The condition is so strong that it forces the function to be affine.

Theorem. Quaternion differentiability implies continuity. The converse fails.

The Cauchy–Riemann Equations

Write $f(\tilde q) = u(\tilde q) + v_1(\tilde q) e_1 + v_2(\tilde q) e_2 + v_3(\tilde q) e_3$, where $\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$ and $u, v_k : U \to \mathbb{R}$.

Theorem. $f$ is quaternion differentiable at $q_0$ iff the components $u, v_1, v_2, v_3$ are real differentiable at $q_0$ and satisfy the quaternion Cauchy–Riemann equations

$$ \frac{\partial u}{\partial q_0} = \frac{\partial v_1}{\partial q_1} = \frac{\partial v_2}{\partial q_2} = \frac{\partial v_3}{\partial q_3}, $$

$$ \frac{\partial u}{\partial q_k} = -\frac{\partial v_k}{\partial q_0}, \qquad k = 1, 2, 3, $$

and the remaining equations obtained by cyclically permuting the indices $1, 2, 3$.

Proof. Write $h = h_0 + h_1 e_1 + h_2 e_2 + h_3 e_3$. The difference quotient is

$$ \frac{f(q_0 + h) - f(q_0)}{h} = \frac{\sum_\mu h_\mu \partial_\mu f}{h} + o(1). $$

For the limit to exist independently of the direction of $h$, the numerator must be a left multiple of $h$, and the multiplication must be consistent with the quaternion relations. This forces the equations.

The Liouville Theorem

Theorem (Liouville). Every bounded quaternion differentiable function on all of $\mathbb{H}$ is constant.

Proof. The quaternion Cauchy–Riemann equations force the components to be harmonic and to satisfy strong constraints. The only bounded solutions on all of $\mathbb{H}$ are constants.

This is the quaternion analogue of the Liouville theorem in complex analysis, and it shows that the class of quaternion differentiable functions is very rigid.

The Class of Quaternion Differentiable Functions

The quaternion Cauchy–Riemann equations are so restrictive that the quaternion differentiable functions are exactly the functions of the form

$$ f(\tilde q) = a \tilde q + b, \qquad a, b \in \mathbb{H}. $$

That is, the only quaternion differentiable functions are the affine functions. This is the fundamental difference from complex analysis, where the class of holomorphic functions is rich.

Proof. The Cauchy–Riemann equations force all second derivatives to vanish, so the function is affine. The details are a computation using the quaternion relations.

So the naive notion of quaternion differentiability is too restrictive to be useful. This is the reason quaternion analysis is not the direct analogue of complex analysis.

Regular Functions on $\mathbb{H}$

Definition

A function $f : U \to \mathbb{H}$ is monogenic (or regular) if it satisfies the Cauchy–Riemann–Fueter equation

$$ \frac{\partial f}{\partial q_0} + e_1 \frac{\partial f}{\partial q_1} + e_2 \frac{\partial f}{\partial q_2} + e_3 \frac{\partial f}{\partial q_3} = 0. $$

Equivalently, if $D = \partial_{q_0} + e_1 \partial_{q_1} + e_2 \partial_{q_2} + e_3 \partial_{q_3}$ is the Cauchy–Riemann operator, then $f$ is monogenic iff $D f = 0$.

The Cauchy–Riemann operator and its conjugate $\bar{D} = \partial_{q_0} - e_1 \partial_{q_1} - e_2 \partial_{q_2} - e_3 \partial_{q_3}$ satisfy

$$ D \bar{D} = \bar{D} D = \partial_{q_0}^2 + \partial_{q_1}^2 + \partial_{q_2}^2 + \partial_{q_3}^2 = \Delta, $$

where $\Delta$ is the Laplacian on $\mathbb{R}^4$. So monogenic functions are harmonic:

$$ \Delta f = 0. $$

Properties

Monogenic functions satisfy:

Closure under left and right multiplication by constants. If $f$ is monogenic and $a, b \in \mathbb{H}$, then $a f$ and $f b$ are monogenic.

The Cauchy integral formula. If $f$ is monogenic on a domain containing a ball $B(q_0, r)$, then

$$ f(\tilde q) = \frac{1}{2\pi^2} \int_{\partial B(q_0, r)} \frac{(w - \tilde q)^{-1}}{|w - \tilde q|^2} n(w) f(w) \, dS(w), $$

where $n(w)$ is the outward unit normal and $dS$ is the surface measure on the sphere. This is the quaternion analogue of the Cauchy integral formula.

The maximum principle. If $f$ is monogenic on a domain and $|f|$ attains its maximum at an interior point, then $f$ is constant.

The Liouville theorem. Every bounded monogenic function on all of $\mathbb{H}$ is constant.

The identity theorem. If two monogenic functions agree on a set with an accumulation point in a domain, they agree on the whole domain.

So monogenic functions are the correct analogue of holomorphic functions in quaternion analysis. They form a rich class, and they satisfy the standard theorems of complex analysis.

The Relation to the Cauchy–Riemann Operator

The Cauchy–Riemann operator $D$ is the quaternion analogue of $\bar{\partial}$, the Cauchy–Riemann operator of one complex variable. It factors the Laplacian:

$$ \bar{D} D = \Delta, $$

and the monogenic functions are the kernel of $D$. This is the starting point of Clifford analysis, which generalizes the theory to $\mathbb{R}^n$ with Clifford algebra coefficients.

The Cauchy–Riemann–Fueter Equation

The Cauchy–Riemann–Fueter equation $D f = 0$ is a first-order system of four real equations for the four components of $f$. It is elliptic, and its solutions are harmonic. The equation is the quaternion analogue of the Cauchy–Riemann equations, and it is the correct notion of differentiability in quaternion analysis.

Integration

Contour Integrals

Let $\gamma : [a, b] \to \mathbb{H}$ be a piecewise continuously differentiable path, and let $f$ be continuous on the image of $\gamma$. The contour integral of $f$ along $\gamma$ is

$$ \int_\gamma f(\tilde q) \, dq = \int_a^b f(\gamma(t)) \gamma'(t) \, dt. $$

Linearity. $\int_\gamma (af + bg) = a \int_\gamma f + b \int_\gamma g$ for constant $a, b \in \mathbb{H}$.

Reversal. $\int_{-\gamma} f = -\int_\gamma f$.

Additivity. If $\gamma$ is the concatenation of $\gamma_1$ and $\gamma_2$, then $\int_\gamma f = \int_{\gamma_1} f + \int_{\gamma_2} f$.

Estimation. If $|f(\tilde q)| \leq M$ on $\gamma$ and $L$ is the length of $\gamma$, then

$$ \left| \int_\gamma f(\tilde q) \, dq \right| \leq M L. $$

The Cauchy–Goursat Theorem

Theorem (Cauchy–Goursat, quaternion version). If $f$ is monogenic on a simply connected domain $U$ and $\gamma$ is a closed contour in $U$, then

$$ \oint_\gamma f(\tilde q) \, dq = 0. $$

Proof. The integral of a monogenic function over a closed surface in $\mathbb{R}^4$ vanishes by the divergence theorem, because the Cauchy–Riemann operator annihilates $f$. The contour case follows by approximation.

Corollary. On a simply connected domain, the integral of a monogenic function is path-independent. The function

$$ F(\tilde q) = \int_{q_0}^q f(w) \, dw $$

is well-defined, monogenic, and satisfies $D F = f$ in the appropriate sense.

The Cauchy Integral Formula

Theorem (Cauchy Integral Formula, quaternion version). Let $f$ be monogenic on a domain containing the closed ball $\overline{B}(q_0, r)$. Then for every $\tilde q$ in the open ball,

$$ f(\tilde q) = \frac{1}{2\pi^2} \int_{\partial B(q_0, r)} \frac{(w - \tilde q)^{-1}}{|w - \tilde q|^2} n(w) f(w) \, dS(w), $$

where $n(w)$ is the outward unit normal and $dS$ is the surface measure on the sphere.

Corollary (derivatives). Under the same hypotheses, $f$ is infinitely differentiable, and the derivatives are given by differentiating the kernel.

Liouville's Theorem

Theorem (Liouville). Every bounded monogenic function on all of $\mathbb{H}$ is constant.

Proof. Apply the Cauchy estimates to the derivatives of $f$ on a ball of radius $r$ around $q_0$. Since $f$ is bounded by $M$, the derivatives are bounded by $M/r^n$. Let $r \to \infty$ to get that all derivatives vanish. So $f$ is constant.

Morera's Theorem

Theorem (Morera). If $f$ is continuous on a domain $U$ and $\oint_\gamma f = 0$ for every closed contour $\gamma$ in $U$, then $f$ is monogenic on $U$.

Proof. Define $F(\tilde q) = \int_{q_0}^q f(w) \, dw$. The hypothesis makes $F$ well-defined, and $D F = f$. Since $F$ is monogenic, $F$ is infinitely differentiable, so $f$ is monogenic.

Power Series

Definition

A power series centered at $q_0$ is

$$ \sum_{n=0}^\infty c_n (\tilde q - q_0)^n, \qquad c_n \in \mathbb{H}. $$

The radius of convergence is

$$ R = \frac{1}{\limsup_{n \to \infty} |c_n|^{1/n}}, $$

with the conventions $R = 0$ if the limsup is $\infty$ and $R = \infty$ if the limsup is $0$.

Theorem. The series converges absolutely for $|\tilde q - q_0| < R$ and diverges for $|\tilde q - q_0| > R$. On $|\tilde q - q_0| < R$ it converges uniformly on compact subsets.

Theorem. A power series is monogenic on $|\tilde q - q_0| < R$, and its derivative is obtained by term-by-term differentiation:

$$ D \sum_{n=0}^\infty c_n (\tilde q - q_0)^n = \sum_{n=1}^\infty n c_n (\tilde q - q_0)^{n-1}. $$

The differentiated series has the same radius of convergence.

Taylor Series

Theorem (Taylor, quaternion version). If $f$ is monogenic on a domain containing the closed ball $\overline{B}(q_0, r)$, then $f$ has a power series expansion

$$ f(\tilde q) = \sum_{n=0}^\infty \frac{f^{(n)}(q_0)}{n!} (\tilde q - q_0)^n $$

valid for $|\tilde q - q_0| < r$.

Corollary. A monogenic function is analytic: it equals its Taylor series in a neighborhood of every point.

Corollary (identity theorem). If two monogenic functions on a domain $U$ agree on a set with an accumulation point in $U$, they agree on all of $U$.

Laurent Series

Theorem (Laurent). If $f$ is monogenic on an annulus $r < |\tilde q - q_0| < R$, then $f$ has a unique expansion

$$ f(\tilde q) = \sum_{n=-\infty}^\infty c_n (\tilde q - q_0)^n $$

valid on the annulus, with

$$ c_n = \frac{1}{2\pi^2} \int_{\partial B(q_0, \rho)} \frac{(w - q_0)^{-n-1}}{|w - q_0|^2} n(w) f(w) \, dS(w), \qquad r < \rho < R. $$

Singularities

Classification

Let $f$ be monogenic on a punctured ball $0 < |\tilde q - q_0| < R$.

Removable singularity. $q_0$ is removable if $f$ extends to a monogenic function on $|\tilde q - q_0| < R$. Equivalently, the Laurent expansion has $c_n = 0$ for $n < 0$.

Pole. $q_0$ is a pole of order $m \geq 1$ if the Laurent expansion has $c_{-m} \neq 0$ and $c_n = 0$ for $n < -m$.

Essential singularity. $q_0$ is an essential singularity if the Laurent expansion has infinitely many non-zero $c_n$ with $n < 0$.

Theorem (Riemann). $q_0$ is removable iff $f$ is bounded near $q_0$.

Theorem (Casorati–Weierstrass). If $q_0$ is an essential singularity, then $f$ takes values arbitrarily close to every quaternion in every neighborhood of $q_0$.

Residues

The residue of $f$ at an isolated singularity $q_0$ is the coefficient $c_{-1}$ in the Laurent expansion:

$$ \operatorname{Res}(f, q_0) = c_{-1}. $$

Theorem (Residue Theorem). Let $f$ be monogenic on a simply connected domain except for isolated singularities $q_1, \dots, q_n$. Let $\gamma$ be a closed contour in the domain that does not pass through any $q_k$ and winds once around each. Then

$$ \oint_\gamma f(\tilde q) \, dq = 2\pi^2 \sum_{k=1}^n \operatorname{Res}(f, q_k). $$

The factor $2\pi^2$ is the surface area of the unit sphere in $\mathbb{R}^4$, and it is the quaternion analogue of the factor $2\pi i$ in complex analysis.

The Differential Operators

Definition

The Cauchy–Riemann operator is

$$ D = \partial_{q_0} + e_1 \partial_{q_1} + e_2 \partial_{q_2} + e_3 \partial_{q_3}. $$

It acts on functions $f : \mathbb{H} \to \mathbb{H}$ by

$$ D f = \partial_{q_0} f + e_1 \partial_{q_1} f + e_2 \partial_{q_2} f + e_3 \partial_{q_3} f. $$

Properties

Factorization of the Laplacian. $\bar{D} D = \Delta$.

Ellipticity. The symbol of $D$ is $\sigma_D(\xi) = \xi_0 + e_1 \xi_1 + e_2 \xi_2 + e_3 \xi_3$, which is invertible for every non-zero $\xi \in \mathbb{R}^4$. So $D$ is elliptic.

Fundamental solution. The fundamental solution of $D$ is

$$ E(\tilde q) = \frac{\tilde q^{-1}}{|\tilde q|^2} = \frac{\tilde{q}^{\natural}}{|\tilde q|^4}, $$

which satisfies $D E = 2\pi^2 \delta_0$ in the sense of distributions.

The Relation to the Cauchy–Riemann Operator

In complex analysis, the Cauchy–Riemann operator is

$$ \bar{\partial} = \frac{1}{2}(\partial_a + i \partial_{a'}), $$

and the holomorphic functions are the kernel of $\bar{\partial}$. In quaternion analysis, the Cauchy–Riemann operator is the analogue of $\bar{\partial}$, and the monogenic functions are the kernel of $D$. The Cauchy–Riemann operator is the correct generalization of the Cauchy–Riemann operator of one complex variable to higher dimensions.

Applications

Harmonic Analysis

Monogenic functions are harmonic, so quaternion analysis is closely related to harmonic analysis on $\mathbb{R}^4$. The Cauchy integral formula gives a representation of monogenic functions in terms of their boundary values, and the boundary values form a Hardy space.

Clifford Analysis

Quaternion analysis is the case $n = 4$ of Clifford analysis, which generalizes the theory to $\mathbb{R}^n$ with Clifford algebra coefficients. The Cauchy–Riemann operator is defined for any $n$, and the monogenic functions are the kernel of the Cauchy–Riemann operator. The theory is used in:

• The theory of harmonic forms and Hodge theory.
• The theory of boundary value problems for elliptic systems.
• The theory of Hardy spaces and singular integrals.

The last two are the subject of Riemann Boundary Value Problems and Singular Integral Equations, which develops the linear boundary value problem with a jump and the singular integral equation of Cauchy type for the quaternion case among the other systems.

The Radon Transform

The quaternion Radon transform is the analogue of the Radon transform in complex analysis. It is defined by integrating a function over spheres in $\mathbb{R}^4$, and it is inverted by a formula involving the Cauchy–Riemann operator. It is used in:

• The theory of integral geometry in four dimensions.
• The inversion of the Radon transform for the Cauchy–Riemann operator.
• The theory of the X-ray transform in higher dimensions.

Comparison with Complex Analysis

The differences between quaternion analysis and complex analysis are consequences of the non-commutativity of $\mathbb{H}$ and the higher dimension of the space.

The complex case is the case $n = 2$ of the general theory, and the quaternion case is the case $n = 4$. The general theory is Clifford analysis, and the pattern is the same: the biquaternionic Cauchy–Riemann operator replaces the classical Cauchy–Riemann operator, the monogenic functions replace the holomorphic functions, and the Cauchy integral formula holds with the appropriate kernel.

Summary

Quaternion analysis is the study of differentiable functions of a quaternion variable $\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$. The space carries its quaternion norm, and with it the convergent sequences and the continuous functions on which the subject is built.

The naive derivative $f'(q_0) = \lim_{h \to 0} (f(q_0 + h) - f(q_0))/h$ exists only for very special functions, because $\mathbb{H}$ is non-commutative and the limit must be independent of the direction of $h$. The correct notion replaces it: a function is monogenic, or regular, when it satisfies the Cauchy–Riemann–Fueter equation, and monogenic functions are the analogues of the holomorphic functions of one complex variable.

The article develops what that notion supports: contour integrals along paths, power series and their radius of convergence, and the classification of the isolated singularities as removable, a pole, or essential. The Cauchy–Riemann operator is studied in its own right, and the applications record the harmonicity of monogenic functions and the representation given by the Cauchy integral formula. The final section compares the subject with complex analysis and traces each difference to the non-commutativity of $\mathbb{H}$ and to the higher dimension of the space.

Summary of Notation

Further Reading

• William Rowan Hamilton, Lectures on Quaternions (1853), for the original formulation.
• Rudolf Fueter, "Die Funktionentheorie der Differentialgleichungen $\Delta u = 0$ und $\Delta \Delta u = 0$ mit vier reellen Variablen" (1935), for the origin of monogenic functions.
• F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis (Pitman, 1982), for the general Clifford theory.
• John Ryan, Clifford Algebras in Analysis and Related Topics (CRC Press, 1996), for the analytic theory.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge, 2001), for the connection to Clifford algebras.
• H. Blaine Lawson and Marie-Louise Michelsohn, Spin Geometry (Princeton, 1989), for the Clifford-algebra and spinor background of the operator.