Fueter Theory for Quaternions

Introduction

Fueter theory is the four-variable function theory attached to the quaternion algebra: a first-order operator, the class of functions it annihilates, and a construction that produces those functions from holomorphic functions of one complex variable. This article develops that theory for $\mathbb{H}$, whose definite norm makes it the classical and unobstructed case. It is the quaternion member of the family's Fueter-theory pair; its counterpart is the biquaternion case, where the indefinite form introduces the null cone and the imaginary units are no longer a sphere. Here the imaginary units form the sphere $S^2$, the only singularity of the theory is the origin, and the Fueter construction holds in the form Fueter gave it.

The article uses Quaternion Regular Functions for the Cauchy–Riemann operator and its conjugate, the regularity system, harmonicity and the Cauchy integral formula; Quaternion Roots of Minus One for the sphere of imaginary units; Quaternion Special Functions for the exponential; Quaternion Integration for the fundamental solution and the mean value property; and Clifford Algebras in Finite Dimensions for the Clifford identifications. It does not re-derive the Cauchy formula, which belongs to Quaternion Regular Functions. The Clifford analysis referred to below is the function theory of the Cauchy–Riemann operator on the even Clifford algebra; the biquaternion comparison is Fueter Theory for Biquaternions.

The corpus's default base is a commutative ring with identity; the analysis requires the real numbers, so domains are taken in $\mathbb{R}^4$ and functions are quaternion-valued.

Throughout, $\mathbb{H}$ is the quaternion algebra over $\mathbb{R}$ with basis $e_0 = 1, e_1, e_2, e_3$, $e_k^2 = -e_0$; the variable is $\tilde q = q_0+\mathbf{q}$, $\mathbf{q} = q_1e_1+q_2e_2+q_3e_3$, with $\rho = |\mathbf{q}|$ and, for $\rho>0$, the direction $\hat{\mathbf{q}} = \mathbf{q}/\rho$. The partials are $\partial_\mu = \partial/\partial q_\mu$, and $\Delta_4 = \sum_\mu\partial_\mu^2$ is the four-dimensional Laplacian. Vector operations on $\mathbf{F} = F_1e_1+F_2e_2+F_3e_3$ are as in Quaternion Regular Functions.

The Fueter Operator

Definition and Conjugate

Definition. The Fueter operator, also called the Cauchy–Riemann–Fueter operator, is the quaternion-valued first-order operator

$$ D = \partial_0+\mathbf{D} = \sum_{\mu=0}^{3}e_\mu\partial_\mu, \qquad D F = \sum_{\mu=0}^{3}e_\mu\partial_\mu F, $$

acting on the left, with $\mathbf{D} = e_1\partial_1+e_2\partial_2+e_3\partial_3$. Its conjugate is $\bar D = \partial_0-\mathbf{D} = \sum_\mu e_\mu^{\natural}\partial_\mu$, obtained by negating the vector part, $e_0^{\natural} = e_0$, $e_k^{\natural} = -e_k$.

Remark. The Fueter operator is the Cauchy–Riemann operator of Quaternion Regular Functions, under the name used in the Fueter tradition; it is the same operator and this article does not restate its definition independently.

Factorization of the Laplacian

Theorem. On $\mathbb{H}$ the operator and its conjugate factor the Laplacian,

$$ D\bar D = \bar DD = \Delta_4 , $$

so $D$ is a square root of $\Delta_4$.

Proof. Expand $D\bar D = \sum_{\mu,\nu}e_\mu e_\nu^{\natural}\partial_\mu\partial_\nu$. The diagonal term is $\sum_\mu e_\mu e_\mu^{\natural}\partial_\mu^2 = \sum_\mu\partial_\mu^2$, since $e_0e_0^{\natural} = 1$ and $e_k e_k^{\natural} = -e_k^2 = 1$; for $\mu\neq\nu$ the coefficient is $e_\mu e_\nu^{\natural}+e_\nu e_\mu^{\natural} = 0$ by the Clifford relation of Quaternion Regular Functions. Hence all mixed terms cancel, leaving $\Delta_4$; the same argument gives $\bar DD = \Delta_4$.

Relation to the Cauchy–Riemann Operator

Since $\mathbf{D}^2 = -\Delta_3$ on the vector part, the product $(\partial_0+\mathbf{D})(\partial_0-\mathbf{D})$ equals $\partial_0^2+\Delta_3 = \Delta_4$, so the Fueter operator is the four-dimensional Cauchy–Riemann operator up to normalization. On a slice $\mathbb{C}_I$ (below) the part differentiating along the slice is the classical operator $\partial_{q_0}+I\partial_\rho$, so $DF = 0$ is the quaternionic Cauchy–Riemann equation. In the Clifford language of Clifford Algebras in Finite Dimensions, the quaternion algebra is the even subalgebra of $\mathrm{Cl}_{0,3}$,

$$ \mathbb{H}\cong\mathrm{Cl}^0_{0,3}\cong\mathrm{Cl}_{0,2}, $$

and the Fueter operator is the restriction of the first-order Clifford operator of that even algebra; for this reason Fueter-regular functions are called monogenic.

Fueter-Regular Functions

Left and Right Regularity

Definition. Let $F : \Omega\to\mathbb{H}$ be differentiable on an open $\Omega\subseteq\mathbb{H}$. Then $F$ is left-regular, or Fueter-regular, if $DF = 0$, and right-regular if $F D = \sum_\mu\partial_\mu F\,e_\mu = 0$.

Proposition. Left-regular functions form a right $\mathbb{H}$-module, $D(Fa) = (DF)a = 0$ for constant $a$; right-regular functions form a left module, $(aF)D = a(FD) = 0$. Neither class is two-sided in general, and conjugation exchanges the two: $DF = 0\iff\bar F\bar D = 0$.

Proof. Both regularities are linear and are preserved by multiplication by a constant on the appropriate side; the exchange by conjugation is Quaternion Regular Functions.

The Componentwise System

Theorem. Writing $F = F_0+\mathbf{F}$, the operator decomposes as

$$ DF = \bigl(\partial_0F_0-\mathrm{div}\,\mathbf{F}\bigr) + \bigl(\partial_0\mathbf{F}+\mathrm{grad}\,F_0+\mathrm{rot}\,\mathbf{F}\bigr), $$

so left-regularity is the quaternionic Cauchy–Riemann system

$$ \partial_0F_0 = \mathrm{div}\,\mathbf{F}, \qquad \partial_0\mathbf{F} = -\mathrm{grad}\,F_0-\mathrm{rot}\,\mathbf{F}, $$

one quaternion equation, equivalently four real equations for the four coefficients. The right action gives the same system with $+\mathrm{rot}\,\mathbf{F}$ in the second equation, so the two systems differ only in the sign of the curl and coincide when $\mathrm{rot}\,\mathbf{F} = 0$.

Proof. Multiplication out of $e_\mu\partial_\mu(F_0+\mathbf{F})$ with $\mathbf{a}\mathbf{b} = -\langle\mathbf{a},\mathbf{b}\rangle+\mathbf{a}\times\mathbf{b}$; the right action reverses the order of $e_\mu$ and the differentiated component.

Corollary. The system is elliptic, with principal symbol $s(\xi) = \sum_\mu\xi_\mu e_\mu$ invertible for every real $\xi\neq0$, since $s(\xi)s^{\natural}(\xi) = |\xi|^2e_0$; hence Fueter-regular functions are real-analytic.

The Debye-Type Reformulation of the System

The system can also be written, away from the origin, as identities that trade each first-order equation for a radial derivative of one component and angular derivatives of the others. The rewriting is classical in one complex variable, and it is the source of the name.

The complex case. Let $f_0(A) = u(a,a')+iv(a,a')$ be holomorphic on a disc, put $\mathbf{a} = (a,a')$ with $\rho = |\mathbf{a}|>0$, and write $[\mathbf{a}\times\nabla g]_3 = a\,\partial_{a'} g-a'\,\partial_a g$ for the out-of-plane component of the two-dimensional cross product. The Cauchy–Riemann relations are equivalent to the two identities

$$ u = (\hat{\mathbf{a}}\cdot\nabla)(\rho u) - [\mathbf{a}\times\nabla v]_3 , \qquad v = (\hat{\mathbf{a}}\cdot\nabla)(\rho v) + [\mathbf{a}\times\nabla u]_3 . $$

Since $(\hat{\mathbf{a}}\cdot\nabla)(\rho g) = g+a\partial_ag+a'\partial_{a'}g$, the first identity states $0 = a\partial_au+a'\partial_{a'}u-[\mathbf{a}\times\nabla v]_3$ and the second states $0 = a\partial_av+a'\partial_{a'}v+[\mathbf{a}\times\nabla u]_3$; the Cauchy–Riemann relations imply both, and conversely the two rows $(a,a',a',-a)$ and $(-a',a,a,a')$ in the partials $(\partial_au,\partial_{a'}u,\partial_av,\partial_{a'}v)$ are orthogonal and their span contains $(1,0,0,-1)$ and $(0,1,1,0)$, so away from $\rho = 0$ the pair is equivalent to $\partial_au = \partial_{a'}v$, $\partial_{a'}u = -\partial_av$. The rewriting is called Debye because the two real scalars $\psi_E,\psi_M$ that generate the source-free Maxwell solutions and satisfy the wave equation are the Debye potentials: the identity is the sense in which an analytic function serves as its own Debye potential, the cross term playing the role of the companion component.

The quaternion case. The same contraction applies to the quaternion system. For $F = F_0+\mathbf{F}$ regular with $\mathbf{F} = F_1e_1+F_2e_2+F_3e_3$, and $\mathbf{q} = q_1e_1+q_2e_2+q_3e_3$, $\rho = |\mathbf{q}|>0$, the component $F_0$ satisfies

$$ F_0 = (\hat{\mathbf{q}}\cdot\nabla)(\rho F_0) + \partial_0(\mathbf{q}\cdot\mathbf{F}) + (\mathbf{q}\times\nabla F_1)_1+(\mathbf{q}\times\nabla F_2)_2+(\mathbf{q}\times\nabla F_3)_3 , $$

and the three companions are obtained by applying the same statement to $-Fe_1$, $-Fe_2$, $-Fe_3$, each again regular, since $D(Fc) = (DF)c$ for a constant $c$ and the scalar component of $-Fe_j$ is $F_j$. The companion for $F_1$ reads

$$ F_1 = (\hat{\mathbf{q}}\cdot\nabla)(\rho F_1) + \partial_0(-q_1F_0-q_2F_3+q_3F_2) - (\mathbf{q}\times\nabla F_0)_1-(\mathbf{q}\times\nabla F_3)_2+(\mathbf{q}\times\nabla F_2)_3 , $$

and the remaining two follow the same pattern; in each case one component is expressed through its own radial derivative, the $q_0$-derivative of the inner product of the radius with the other three, and the components of $\mathbf{q}\times\nabla$ of those three.

Structural content. The identity for $F_0$ collapses. Because $(\hat{\mathbf{q}}\cdot\nabla)(\rho F_0) = F_0+\mathbf{q}\cdot\nabla F_0$ and $\sum_k(\mathbf{q}\times\nabla F_k)_k = \mathbf{q}\cdot\mathrm{rot}\,\mathbf{F}$, the term $F_0$ cancels on the two sides and the relation states

$$ 0 = \mathbf{q}\cdot\bigl(\partial_0\mathbf{F}+\mathrm{grad}\,F_0+\mathrm{rot}\,\mathbf{F}\bigr) , $$

which is the vector part of $DF$ contracted with the radius. Each of the four relations is therefore the radius contraction of one component equation of the system, and every regular $F$ satisfies it; the vector part vanishes in each radial direction precisely because it vanishes outright. The four relations together are a reformulation of the system in the source's sense — each component is recovered from radial and angular derivatives of the four — and the source notes that they are not explicitly found in the literature; they are an identity satisfied by regular functions rather than a construction of them, the construction remaining Fueter's.

Harmonicity and the Mean Value Property

Theorem. Every left- or right-regular function is harmonic,

$$ DF = 0\ \text{or}\ FD = 0 \implies \Delta_4 F = 0 $$

componentwise, and the converse fails.

Proof. $DF = 0$ gives $\Delta_4F = \bar D(DF) = 0$, and $FD = 0$ gives $\Delta_4F = (FD)\bar D = 0$. The coordinate $q_0$ is harmonic with $Dq_0 = e_0\neq0$.

Theorem (mean value property). A regular function $F$ near a closed ball $\bar B(q_0,r)$ satisfies

$$ F(q_0) = \frac{1}{|B(q_0,r)|}\int_{B(q_0,r)}F\,dV = \frac{1}{|\partial B(q_0,r)|}\int_{\partial B(q_0,r)}F\,dS, $$

with the ordinary Lebesgue measures on $\mathbb{R}^4$.

Proof. By the preceding theorem each component of $F$ is harmonic, and the mean value property for harmonic functions gives the two equalities; equivalently, the property follows from the Cauchy integral formula of Quaternion Regular Functions by shrinking the boundary of the ball.

Corollary. Regular functions satisfy the maximum principle, Liouville's theorem, the identity theorem and the Cauchy estimates on $\mathbb{H}$; all of these are consequences of harmonicity together with ellipticity, and none of them requires an exceptional set because the quaternion norm is definite.

The Fueter Construction

The Axial Extension of a Holomorphic Function

Definition. Let $f_0$ be holomorphic on the disc $D(0,R)\subseteq\mathbb{C}$, written $f_0(A) = u(a,a')+iv(a,a')$ with $A = a+ia'$ and $u,v$ real-valued. The axial extension of $f_0$ is

$$ \tilde f_0(\tilde q) = u(q_0,\rho)+\hat{\mathbf{q}}\,v(q_0,\rho), $$

defined for $\rho>0$ by replacing $a$ by $q_0$, $a'$ by $\rho$ and the complex unit $i$ by $\hat{\mathbf{q}}$, which is legitimate because $\hat{\mathbf{q}}^2 = -1$. When $f_0$ has real Taylor coefficients, $v(q_0,0) = 0$, the extension extends continuously to $\rho = 0$ with value $f_0(q_0)$, and $\tilde f_0(\tilde q) = \sum_{n\geq0}a_n\tilde q^n$ on $B(0,R)$.

Fueter's Theorem

Theorem (Fueter's construction). Let $f_0$ be holomorphic on $D(0,R)$ with axial extension $\tilde f_0$. Then

$$ F(\tilde q) = \Delta_4\tilde f_0(\tilde q) = \frac{2\,\partial_\rho u(q_0,\rho)}{\rho}+\hat{\mathbf{q}}\left(\frac{2\,\partial_\rho v(q_0,\rho)}{\rho}-\frac{2\,v(q_0,\rho)}{\rho^2}\right) $$

is defined and real-analytic on $B(0,R)$, by continuity at $\rho = 0$, and is both left- and right-Fueter-regular there.

Proof. For $g = A(q_0,\rho)+\hat{\mathbf{q}}B(q_0,\rho)$ with central coefficients $A,B$, one has $\partial_{q_k}A = (\partial_\rho A)\hat q_k$, $\sum_ke_k\hat q_k = \hat{\mathbf{q}}$ and $\sum_{j,k}(\partial_{q_k}\hat q_j)e_ke_j = (-3-\hat{\mathbf{q}}^2)/\rho = -2/\rho$, because $\hat{\mathbf{q}}^2 = -1$. Hence $Dg = gD = (\partial_0A-\partial_\rho B-2B/\rho)+\hat{\mathbf{q}}(\partial_0B+\partial_\rho A)$, so $g$ is left-regular if and only if it is right-regular, and this holds exactly when $\partial_0A = \partial_\rho B+2B/\rho$ and $\partial_0B = -\partial_\rho A$. Applying the radial form of the three-dimensional Laplacian and the identity $\Delta_{\mathbb{R}^3}(B\hat{\mathbf{q}}) = \hat{\mathbf{q}}(\Delta_{\mathbb{R}^3}B-2B/\rho^2)$ to the harmonic pair $(u,v)$ gives $\Delta_4\tilde f_0(\tilde q) = P+\hat{\mathbf{q}}Q$ with $P = 2u_\rho/\rho$, $Q = 2(\rho v_\rho-v)/\rho^2$, and the Cauchy–Riemann equations give $Q_\rho+2Q/\rho = \partial_0P$, $-\partial_0Q = P_\rho$; thus $P,Q$ satisfy the regularity system and extend continuously to $\rho = 0$.

The Kernel and Injectivity

Proposition. The Fueter map $\tau(f_0) = \Delta_4\tilde f_0$ vanishes if and only if $f_0$ is affine, $f_0(A) = CA+D$.

Proof. $\tau(f_0) = 0$ forces $u_\rho = 0$ and $\rho v_\rho = v$, so $u = u(q_0)$ and $v = c(q_0)\rho$; then $c' = 0$ and $u = cq_0+d$ with $c,d\in\mathbb{R}$, and complex linearity gives all affine functions. Conversely, $\Delta_4$ annihilates the constants and the linear monomials.

Corollary. The Fueter map is injective exactly on the holomorphic functions whose Taylor coefficients vanish to order two, $a_0 = a_1 = 0$, its kernel being the affine functions.

The Axial and Slice Approach

Imaginary Units and Slices

Definition. An imaginary unit is an element $I$ with $I^2 = -1$. The slice through $I$ is $\mathbb{C}_I = \mathbb{R}+I\mathbb{R}\cong\mathbb{C}$.

Theorem. In $\mathbb{H}$ the imaginary units form the two-sphere

$$ S^2 = \{I\in\operatorname{Im}\mathbb{H} : N(I) = 1\}, $$

and every non-real quaternion has a unique representation $\tilde q = q_0+I\rho$ with $I\in S^2$ and $\rho>0$. Two slices meet only in $\mathbb{R}$ unless $I = \pm J$, in which case they coincide.

Proof. The roots of $-1$ are the pure unit quaternions, which form the unit sphere of the three-dimensional space $\operatorname{Im}\mathbb{H}$, by Quaternion Roots of Minus One; the direction $\hat{\mathbf{q}}$ is pure and unit and $\hat{\mathbf{q}}^2 = -1$, giving the representation, and the uniqueness is the uniqueness of the polar form of the vector part. Two slices $\mathbb{C}_I,\mathbb{C}_J$ intersect in the real span of $I,J$, which is larger than $\mathbb{R}$ exactly when $I = \pm J$.

Axially Symmetric Functions and the Harmonic Coefficients

Definition. A function $F$ on $\mathbb{H}$ is axially symmetric if it depends on the direction $\hat{\mathbf{q}}$ only through $\hat{\mathbf{q}}$ itself, that is, if it has the axial representation

$$ F(\tilde q) = A(q_0,\rho)+\hat{\mathbf{q}}B(q_0,\rho), $$

with $A,B$ the axial coefficients, functions of the two variables $q_0,\rho$.

Proposition. For an axially symmetric $F$ the left- and right-regularity conditions coincide, and they reduce to

$$ \partial_0A = \partial_\rho B+\frac{2B}{\rho}, \qquad \partial_0B = -\partial_\rho A . $$

Eliminating $B$ gives $\Delta_4A = 0$, so the scalar axial coefficient is harmonic, while the vector coefficient satisfies $\Delta_4B = 2B/\rho^2$.

Proof. The first display is the computation in the proof of Fueter's theorem, valid for any central-coefficient axial function and the same on the left and on the right; applying the radial Laplacian and eliminating gives $\Delta_4A = 0$ from the two equations, and substituting back gives $\Delta_4B = 2B/\rho^2$.

Theorem. The Fueter construction realizes every axially symmetric regular function with central axial coefficients: such an $F$ arises as $\Delta_4\tilde f_0$ for a holomorphic $f_0$, uniquely modulo the affine kernel.

Proof. The axial coefficients $P = 2u_\rho/\rho$, $Q = 2(\rho v_\rho-v)/\rho^2$ are produced from the harmonic conjugate pair $(u,v)$ by Fueter's theorem, and conversely the regularity system solved for $A,B$ recovers a holomorphic $f_0$ up to the affine functions, by the injectivity of the Fueter map.

The Fueter–Sce Theorem and Its Hypotheses

Theorem (Fueter–Sce). Let $n\geq1$ be odd, let $\mathrm{Cl}_{0,n}$ have generators $e_1,\dots,e_n$, and let $f_0$ be holomorphic on a disc, with axial extension $\tilde f_0$ to $\mathbb{R}^{n+1}$. Then $\tilde F = \Delta^{(n-1)/2}\tilde f_0$ is monogenic, annihilated on the left and on the right by the operator $\partial_{q_0}+\sum_{j=1}^{n}e_j\partial_{q_j}$, on the ball where the extension is defined. For $n = 3$ the exponent is one, recovering Fueter's construction.

Proof. The power $(n-1)/2$ is a non-negative integer because $n$ is odd, so the operator is ordinary iteration of the Laplacian; the argument of Fueter's theorem generalizes to each odd $n$, the radial computation with $-3$ replaced by $-n$.

Remark. Three hypotheses are needed. The function $f_0$ must be holomorphic on $D(0,R)$, so that $\Delta^{(n-1)/2}(\tilde q^n)$ is a homogeneous polynomial of degree $n-1-(n-1)/2$ with at most polynomial growth in $n$ and the induced series converges on $B(0,R)$; the parity exponent $(n-1)/2$ must be a non-negative integer, so $n$ is odd; and the affine kernel $az+b$ remains, so a one-to-one correspondence requires the Taylor coefficients to vanish to order two. None of the three can be dropped.

Power Series Representations

A series with quaternionic coefficients on the right, $F(\tilde q) = \sum_{n\geq0}\tilde q^na_n$, converges absolutely on $|\tilde q| < R$ with $R^{-1} = \limsup_n|a_n|^{1/n}$, because $|\tilde q^n| = |\tilde q|^n$. Its sum is slice-regular, or Cullen-regular: holomorphic on each slice. Slice-regular functions form a class distinct from the Fueter-regular ones; for instance $\tilde q\mapsto\tilde q$ is slice-regular but

$$ Dx = \sum_{\mu=0}^{3}e_\mu e_\mu = e_0^2+e_1^2+e_2^2+e_3^2 = e_0-e_0-e_0-e_0 = -2e_0\neq0 . $$

The Fueter construction is exactly the operation converting slice-regular, or holomorphic, data into Fueter-regular functions.

For real Taylor coefficients, $\tilde f_0(\tilde q) = \sum_{n\geq0}a_n\tilde q^n$ and termwise application of $\Delta_4$ gives the induced series $\tau(f_0) = \sum_{n\geq0}a_n\Delta_4(\tilde q^n)$, beginning at $n = 2$ and converging normally on $B(0,R)$, since $\Delta_4(\tilde q^n)$ has degree $n-2$ and at most polynomial growth; termwise differentiation is therefore justified. In general, with $\mathcal{P}_k$ the homogeneous quaternion-valued polynomials of degree $k$ and $\mathcal{M}_k = \{P\in\mathcal{P}_k : DP = 0\}$ the monogenic homogeneous polynomials, the Fischer decomposition

$$ \mathcal{P}_k = \bigoplus_{j=0}^{k}\tilde q^j\mathcal{M}_{k-j} $$

gives every Fueter-regular function on $B(0,R)$ a normally convergent expansion $F = \sum_{k\geq0}F_k$ with $F_k\in\mathcal{M}_k$, the four-variable analogue of the Taylor series of one complex variable.

Relation to the Biquaternion Theory and Clifford Analysis

The biquaternion algebra $\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ carries Fueter theory on its quaternion subspace $\mathbb{H}_{\mathbb{B}}$, and when the values are taken in $\mathbb{H}$ rather than in $\mathbb{B}$ that theory is exactly the theory of this article: same operator, same regularity, same Fueter construction. The quaternion case is therefore the real-coefficient case of the biquaternion theory, and it is the case in which no obstruction appears.

Feature	Quaternion case $\mathbb{H}$	Biquaternion case $\mathbb{B}$
Norm	definite	indefinite
Imaginary units	the sphere $S^2$ of pure unit quaternions	the larger family of Biquaternion Square Roots of Minus One, Zero and Plus One, not a sphere
Slice structure	a single sphere of slices	one slice selected at a time from a non-spherical root set
Singularity of the Cauchy theory	the origin only	the null quadric, real dimension $6$
Ellipticity of the Fueter operator	everywhere for real covectors	fails over $\mathbb{C}$
Fueter construction	Fueter's theorem, exponent one	same on $\mathbb{H}_{\mathbb{B}}$; open on the full algebra

The relation to Clifford analysis is the identification $\mathbb{H}\cong\mathrm{Cl}^0_{0,3}$ of Clifford Algebras in Finite Dimensions: Fueter-regular functions are the monogenic functions of the even Clifford algebra in three dimensions, and the biquaternion theory is the monogenic theory of the complexified even Clifford algebra $\mathrm{Cl}^0_{1,3}$. The Fueter–Sce theorem is the general statement of that correspondence for odd $n$, of which the quaternion case $n = 3$ is the first non-trivial instance. Because the quaternion norm is definite, the Clifford analysis here is elliptic throughout, and the only singularity of the Cauchy theory is the origin.

Summary

The Fueter operator $D = \sum_\mu e_\mu\partial_\mu$ and its conjugate $\bar D = \partial_0-\mathbf{D}$ factor the four-dimensional Laplacian, $D\bar D = \bar DD = \Delta_4$, and on a slice $\mathbb{C}_I$ reduce to the classical operator $\partial_{q_0}+I\partial_\rho$. Fueter-regular functions, the solutions of $DF = 0$, satisfy the quaternionic Cauchy–Riemann system $\partial_0F_0 = \mathrm{div}\,\mathbf{F}$, $\partial_0\mathbf{F} = -\mathrm{grad}\,F_0-\mathrm{rot}\,\mathbf{F}$; left- and right-regularity differ only in the sign of the curl and coincide for axially symmetric functions. The system is elliptic, and every regular function is harmonic, hence real-analytic and subject to the mean value property, the maximum principle, Liouville's theorem and the Cauchy estimates, with no exceptional set. The system also admits a Debye-type reformulation, in which each component is expressed through its own radial derivative, the $q_0$-derivative of the radius contracted with the others, and the components of $\mathbf{q}\times\nabla$ of the others; the four relations are the radius contractions of the four component equations, so they hold identically for a regular function, and in one complex variable the same pair of identities is equivalent to the Cauchy–Riemann relations.

The Fueter construction sends a holomorphic $f_0$ to $F = \Delta_4\tilde f_0 = 2u_\rho/\rho+\hat{\mathbf{q}}(2v_\rho/\rho-2v/\rho^2)$, which is both left- and right-regular; it is injective exactly on holomorphic functions whose Taylor coefficients vanish to order two, with the affine functions as kernel. The imaginary units of $\mathbb{H}$ form the two-sphere $S^2$, so that every non-real quaternion has a unique slice representation $\tilde q = q_0+I\rho$, and the axial representation $F = A(q_0,\rho)+\hat{\mathbf{q}}B(q_0,\rho)$ reduces regularity to $\partial_0A = \partial_\rho B+2B/\rho$, $\partial_0B = -\partial_\rho A$, with $\Delta_4A = 0$ and $\Delta_4B = 2B/\rho^2$. The Fueter–Sce theorem extends the construction to odd $n$ with the power $(n-1)/2$ under the hypotheses of holomorphic convergence, order-two vanishing and odd dimension.

Regular functions have the induced series and the monogenic Taylor expansion of the Fischer decomposition, while slice-regular series form the distinct class the construction converts into regular functions. The quaternion case is the real-coefficient case of the biquaternion Fueter theory; the biquaternion case differs through the indefiniteness of the norm, the non-spherical root set, the null quadric obstruction, and the failure of ellipticity over $\mathbb{C}$. In Clifford terms, $\mathbb{H}\cong\mathrm{Cl}^0_{0,3}$, and this article treats the elliptic classical instance of the Fueter–Sce correspondence.

Summary of Notation

Symbol	Meaning
$\mathbb{H}$	The quaternion algebra, a division algebra
$\tilde q = q_0+\mathbf{q}$, $\rho = \lvert\mathbf{q}\rvert$, $\hat{\mathbf{q}} = \mathbf{q}/\rho$	Variable, modulus of the vector part, direction
$D = \partial_0+\mathbf{D}$, $\mathbf{D} = \sum_k e_k\partial_k$	Fueter (Cauchy–Riemann–Fueter) operator, acting on the left
$\bar D = \partial_0-\mathbf{D}$	Conjugate Fueter operator
$\Delta_4 = \sum_\mu\partial_\mu^2$	Four-dimensional Laplacian; $D\bar D = \bar DD = \Delta_4$
$DF = 0$	Left-regularity; monogenic in Clifford language
$FD = 0$	Right-regularity
$\mathrm{div}, \mathrm{grad}, \mathrm{rot}$	Vector operators in the componentwise system
$\partial_0F_0 = \mathrm{div}\,\mathbf{F}$, $\partial_0\mathbf{F} = -\mathrm{grad}\,F_0-\mathrm{rot}\,\mathbf{F}$	Quaternionic Cauchy–Riemann system
$(\mathbf{q}\times\nabla g)_k$	Components of the radius–gradient cross product in the Debye-type identities
$\psi_E$, $\psi_M$	Debye potentials, the two real scalar generators of source-free Maxwell solutions
$I$, $S^2$	Imaginary unit, $I^2 = -1$; the imaginary units of $\mathbb{H}$ form the two-sphere
$\mathbb{C}_I = \mathbb{R}+I\mathbb{R}$	Slice through $I$, a copy of the complex plane
$A, B$	Axial coefficients, $F = A(q_0,\rho)+\hat{\mathbf{q}}B(q_0,\rho)$
$\tilde f_0$, $\tau(f_0) = \Delta_4\tilde f_0$	Axial extension of a holomorphic $f_0$ and its Fueter-induced regular function
$\mathcal{P}_k$, $\mathcal{M}_k$	Homogeneous polynomials of degree $k$; monogenic ones; Fischer decomposition
$\mathrm{Cl}^0_{0,3}\cong\mathrm{Cl}_{0,2}\cong\mathbb{H}$	Clifford identifications of the quaternion algebra
$\mathbb{B}$, $\mathbb{H}_{\mathbb{B}}$	Biquaternion algebra and its quaternion subspace, the complexified case