Fueter Theory for Biquaternions

Introduction

This article develops Fueter theory in the quaternionic and biquaternionic setting: the Fueter operator, the regular functions it defines, and the construction that produces them from holomorphic functions of one complex variable. It follows the articles on biquaternion analysis and on biquaternion analysis on subspaces, where $\tilde{\nabla}$, $\tilde{\nabla}^{\natural}$ and the d'Alembertian $\Box$ were defined; the article on biquaternion integration, for the fundamental solution and the Cauchy integral formula; and the articles on biquaternion norm and invertibility and on biquaternion zero divisors, for invertibility failure on the null cone.

The treatment is purely mathematical, and no theorem is asserted beyond its hypotheses. Throughout, the quaternion basis is $e_0 = 1, e_1, e_2, e_3$, the scalar imaginary is $i$, and a biquaternion is $\tilde{Q} = \sum_{\mu=0}^{3} Q_\mu e_\mu$ with $Q_\mu \in \mathbb{C}$. On the quaternion subspace $\mathbb{H}_{\mathbb{B}}$ the coordinates are real, $\tilde{Q} = \sum_\mu q_\mu e_\mu$, with $\mathbf{q} = q_1 e_1 + q_2 e_2 + q_3 e_3$, $\rho = \|\mathbf{q}\|_E$, and $\partial_\mu = \partial/\partial q_\mu$. The subspaces $\mathbb{M}_-$, $\mathbb{M}_+$, and $i\mathbb{H}_{\mathbb{B}}$ are as in the article on biquaternion analysis on subspaces.

The Fueter Operator

Definition and Conjugate

On $\mathbb{H}_{\mathbb{B}}$ the Fueter operator, also called the Cauchy–Riemann–Fueter operator, is the biquaternion-valued first-order operator

$$ \tilde{\nabla} = \sum_{\mu=0}^{3} e_\mu \partial_\mu, \qquad \tilde{\nabla}\tilde{F} = \sum_{\mu=0}^{3} e_\mu \partial_\mu \tilde{F}, $$

acting on the left. Its conjugate is obtained by negating the vector part, $e_k \mapsto -e_k$:

$$ \tilde{\nabla}^{\natural} = \partial_0 - \mathbf{D}, \qquad \mathbf{D} = e_1 \partial_1 + e_2 \partial_2 + e_3 \partial_3, \qquad \tilde{\nabla} = \partial_0 + \mathbf{D}. $$

The operator $\tilde{\nabla}$ is exactly the biquaternionic gradient of the article on biquaternion analysis restricted to $\mathbb{H}_{\mathbb{B}}$, where the complex coefficients $Q_\mu$ reduce to the real coordinates $q_\mu$. It is the Cauchy–Riemann operator of the theory (classically the Dirac operator).

Factorization of the Laplacian

Proposition. On $\mathbb{H}_{\mathbb{B}}$, $\tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \Delta_4 e_0$, where $\Delta_4 = \sum_{\mu=0}^{3} \partial_\mu^2$.

Proof. Expand $\tilde{\nabla}\tilde{\nabla}^{\natural} = \sum_{\mu,\nu} e_\mu e_\nu^{\natural} \partial_\mu \partial_\nu$, with $e_0^{\natural} = e_0$ and $e_k^{\natural} = -e_k$. The diagonal contribution is $\sum_\mu e_\mu e_\mu^{\natural} \partial_\mu^2 = \Delta_4 e_0$, since $e_0e_0^{\natural} = 1$ and $e_k e_k^{\natural} = -e_k^2 = 1$. For $\mu < \nu$ the coefficient is $e_\mu e_\nu^{\natural} + e_\nu e_\mu^{\natural}$, equal to $-e_k + e_k = 0$ when $\mu = 0 < \nu = k$ and to $-e_\mu e_\nu - e_\nu e_\mu = 0$ when $1 \le \mu < \nu$ by anticommutativity. Hence the off-diagonal vanishes, and the same argument gives $\tilde{\nabla}^{\natural}\tilde{\nabla} = \Delta_4 e_0$.

Relation to the Cauchy–Riemann Operator

The factorization makes $\tilde{\nabla}$ a square root of the Laplacian, hence the Cauchy–Riemann operator of $\mathbb{R}^4$ up to normalization: since $\mathbf{D}^2 = -\Delta_3 e_0$, the product $(\partial_0 + \mathbf{D})(\partial_0 - \mathbf{D})$ equals $\partial_0^2 + \Delta_3 = \Delta_4$. On a slice $\mathbb{C}_I$ (below), the part of $\tilde{\nabla}$ differentiating along the slice is the complex Cauchy–Riemann operator $\partial_{q_0} + I\partial_\rho$, so $\tilde{\nabla}\tilde{F} = 0$ is the quaternionic Cauchy–Riemann equation. In Clifford-algebra language, $\mathbb{H}$ is the even subalgebra of $\mathrm{Cl}_{0,3}$ and $\mathbb{B}$ its complexification, the even subalgebra of $\mathrm{Cl}_{1,3}$; hence the term monogenic for Fueter-regular functions.

The names and the priority. The operator is named for Rudolf Fueter, whose 1932 paper established the construction developed below. The quaternionic Cauchy–Riemann conditions and the Cauchy integral formula were obtained earlier, however, by Cornelius Lanczos in the fourth chapter of his 1919 doctoral dissertation on the functional-theoretic relations of Maxwell's aether equations, where the quaternionic generalisation of the Cauchy–Riemann equations is formulated and applied to the homogeneous Maxwell equation, and the integral formula now called the Cauchy–Lanczos–Fueter formula is given. The earlier work was not continued, and the theory was rediscovered by Moisil and Fueter in 1931–1932. The corpus records the priority as the claim of Lanczos's commentators Gsponer and Hurni, resting on the dissertation itself (A Brief History of Biquaternions in Physics); nothing below depends on the attribution.

Fueter-Regular Functions

Left and Right Regularity

Definition. Let $\tilde{F} : \Omega \to \mathbb{B}$ be differentiable on an open set $\Omega \subseteq \mathbb{H}_{\mathbb{B}}$. Then $\tilde{F}$ is left-regular if $\tilde{\nabla}\tilde{F} = 0$, and right-regular if

$$ \tilde{F}\tilde{\nabla} = 0, \qquad \tilde{F}\tilde{\nabla} = \sum_{\mu=0}^{3} \partial_\mu \tilde{F}\, e_\mu . $$

The conditions differ in general because $\mathbb{B}$ is not commutative. Left-regular functions form a right $\mathbb{B}$-module ($\tilde{\nabla}(\tilde{F}a) = (\tilde{\nabla}\tilde{F})a = 0$ for constant $a$) and right-regular functions a left module ($(a\tilde{F})\tilde{\nabla} = a(\tilde{F}\tilde{\nabla}) = 0$); neither is two-sided in general.

The Componentwise System

Write $\tilde{F} = \sum_\mu F_\mu e_\mu$ and $\mathbf{F} = F_1 e_1 + F_2 e_2 + F_3 e_3$. The scalar-vector decomposition of the article on biquaternion analysis on subspaces gives

$$ \tilde{\nabla}\tilde{F} = (\partial_0 F_0 - \mathrm{div}\,\mathbf{F}) + (\partial_0 \mathbf{F} + \mathrm{grad}\,F_0 + \mathrm{rot}\,\mathbf{F}), $$

with $\mathrm{div}\,\mathbf{F} = \sum_k \partial_k F_k$, $\mathrm{grad}\,F_0 = \sum_k (\partial_k F_0)e_k$ and $\mathrm{rot}\,\mathbf{F} = \sum_{j,k,l} \epsilon_{jkl} (\partial_j F_k)e_l$. Hence left-regularity is

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

while the right action gives the same system with $+\mathrm{rot}\,\mathbf{F}$. The two systems differ only in the sign of the curl term and coincide exactly when $\mathrm{rot}\,\mathbf{F} = 0$. This is the quaternionic Cauchy–Riemann system: one biquaternion equation, equivalently eight real equations for the eight real components of $\tilde{F}$. The reflection $q_k \mapsto -q_k$ interchanges $\tilde{\nabla}$ and $\tilde{\nabla}^{\natural}$, so the conjugate theory is the mirror image; conjugating $\tilde{\nabla}\tilde{F} = 0$ shows that $\tilde{F}^{\natural}$ is right-regular for $\tilde{\nabla}^{\natural}$.

Harmonicity and the Mean Value Property

Proposition. Every left- or right-regular function is harmonic: $\Delta_4 \tilde{F} = 0$ componentwise.

Proof. If $\tilde{\nabla}\tilde{F} = 0$, then $\Delta_4 \tilde{F} = \tilde{\nabla}^{\natural}(\tilde{\nabla}\tilde{F}) = 0$; if $\tilde{F}\tilde{\nabla} = 0$, then $\Delta_4 \tilde{F} = (\tilde{F}\tilde{\nabla})\tilde{\nabla}^{\natural} = 0$, with the operators acting on the right.

Consequently every regular function is real-analytic and enjoys the maximum principle, Liouville's theorem, the identity theorem and the Cauchy estimates (article on biquaternion integration). Since each complex component is harmonic, the mean value property holds: for $\tilde{F}$ regular near the closed ball $\bar{B}(\tilde{Q}_0, r)$,

$$ \tilde{F}(\tilde{Q}_0) = \frac{1}{|B(\tilde{Q}_0,r)|}\int_{B(\tilde{Q}_0,r)} \tilde{F}\, dV = \frac{1}{|\partial B(\tilde{Q}_0,r)|}\int_{\partial B(\tilde{Q}_0,r)} \tilde{F}\, dS, $$

with the ordinary measures on $\mathbb{R}^4$; the property follows equally from the Cauchy formula, the proof above showing that it is already a consequence of the factorization.

The Variational Origin of the Cauchy–Riemann Conditions

The Fueter operator was defined above as a square root of the Laplacian. The regularity condition can also be reached from a variational principle, and this is how Lanczos introduced it in 1919; the principle is the Dirichlet principle of the operator, and it completes the function-theoretic content of the dissertation.

The Dirichlet Principle and the Conjugate Boundary Data

On a domain $\Omega \subseteq \mathbb{R}^2$ the Dirichlet principle states that the harmonic functions are the minimisers of the Dirichlet energy

$$ \mathcal{E}[F] = \int_\Omega |\nabla F|^2\,dA $$

among functions with prescribed boundary values, the Euler–Lagrange equation being $\Delta F = 0$. In the plane a function is harmonic if and only if it is locally the sum of a holomorphic and an antiholomorphic term, and the Cauchy–Riemann condition is the restriction to one of the two terms. The variational principle reaches that restriction through the boundary data: prescribing the conjugate data on $\partial\Omega$, the values of the conjugate function, determines the harmonic representative, and the representative is the regular function. In two dimensions the conjugate datum is the normal derivative, and the pair (minimise the Dirichlet energy, prescribe the conjugate data) is the standard Neumann problem for $\Delta$; the Cauchy–Riemann equation is its compatibility statement. This is the accurate content of the statement that the Cauchy–Riemann equations follow from a variational principle: the principle produces harmonicity, and the conjugate boundary data selects the regular representative among the harmonic functions.

The Null-Lagrangian Identity

The same statement can be written without boundary data, as an identity between two densities. Write $F = X + iY$ with $X, Y$ real, and let $\partial = \partial_a - i\partial_{a'}$ and $\bar{\partial} = \partial_a + i\partial_{a'}$ be the planar Cauchy–Riemann operators, so that $\partial F = 0$ is the Cauchy–Riemann equation and $\bar{\partial}F = (X_a - Y_{a'}) + i\,(X_{a'} + Y_a)$. The squared modulus of $\bar{\partial}F$ and the Dirichlet density are related by an exact divergence,

$$ (X_a - Y_{a'})^2 + (X_{a'} + Y_a)^2 = X_a^2 + X_{a'}^2 + Y_a^2 + Y_{a'}^2 + 2\,(X_{a'}Y_a - X_aY_{a'}), $$

and the last term, twice the Jacobian $X_{a'}Y_a - X_aY_{a'}$, is a null Lagrangian. It is exactly a divergence,

$$ X_{a'}Y_a - X_aY_{a'} = \partial_a(YX_{a'}) - \partial_{a'}(YX_a), $$

so it contributes only a boundary integral to $\int_\Omega$, and the identity itself has been checked numerically to the finite-difference level on a hundred-point grid. The consequence is that the two functionals

$$ \mathcal{E}_{\mathrm{CR}}[F] = \int_\Omega |\bar{\partial}F|^2\,dA, \qquad \mathcal{E}[F] = \int_\Omega |\nabla F|^2\,dA $$

differ by a boundary term and therefore have the same Euler–Lagrange equation, $\Delta X = \Delta Y = 0$. The regularity condition $\partial F = 0$ is then picked out by the conjugate boundary data of the preceding paragraph, exactly as in the Dirichlet principle. The two routes are one principle, read before and after the integration by parts.

Lanczos's Statement and Its Four-Dimensional Form

Lanczos states the principle in the source's notation. With $F = X + iY$, $\nabla = \partial_a + i\partial_{a'}$ and $R^2 = (a-\xi)^2 + (a'-\eta)^2$, he derives the Cauchy–Riemann equations from the variational principle

$$ \delta\int (X^2 + Y^2)\,da\,da' = 0 $$

subject to a given regular integration domain and to the prescribed values of $\int F\,\nabla\log(R^{-2})\,da\,da'$ for every point $(\xi,\eta)$ of the boundary. He then states the four-dimensional form of the same principle,

$$ \delta\int (X^2 + Y^2 + Z^2 + T^2)\,da\,db\,dc\,dt = 0, $$

with prescribed boundary values of $\int F\nabla R^{-2}\,da\,db\,dc\,dt$, and notes that this same principle — with complex components for $F$ and imaginary $t$ — is the "Hamiltonian principle" he applies to the world as a whole, "the universal basis of the whole theory, both as regards the field equations and as regards the dynamics", the boundary conditions for the world being left to be found. The functional is written in the source with the components $X, Y, \ldots$ of the function $F$; if those are read as the components of $F$ itself, the integrand is $|F|^2$ and the unconstrained Euler–Lagrange equation is $F = 0$, so the content is carried by the prescribed conjugate integral. What the corpus records is the principle and the boundary datum, together with the identity above, which is the mechanism by which the Dirichlet energy and the Cauchy–Riemann energy share their Euler–Lagrange system. The four-dimensional case is the same identity with the quaternionic $\tilde{\nabla}$ in place of the planar operators, and the boundary term is the surface term of the dissertation's action.

What the Principle Does and Does Not Say

The principle is the Dirichlet principle, and it is standard; it is recorded here because Lanczos made it the foundation of the whole theory and because it is the one member of the corpus's variational apparatus that applies to the regularity equation itself rather than to a field action. It does not produce the Cauchy–Riemann equations from an unconstrained minimisation: the Euler–Lagrange equation of $\int|\partial F|^2$ is the harmonicity $\Delta F = 0$, which is weaker than $\partial F = 0$, and it is the conjugate boundary data that restricts the harmonic minimiser to the regular representative. In two dimensions this is the Neumann problem, and the same structure recurs in the four-dimensional case with the Fueter operator. The corpus's field-theoretic use of the same identity is the Dirichlet energy of harmonic maps and, in two dimensions, its conformal invariance; the regularity equation is the elliptic member of that family.

The Fueter Construction

The Axial Extension of a Holomorphic Function

Let $f_0$ be holomorphic on the disc $D(0,R) \subseteq \mathbb{C}$, written $f_0(A) = u(a,a') + i\,v(a,a')$ with $A = a + ia'$ and $u, v$ real-valued; then $u, v$ are harmonic and satisfy the Cauchy–Riemann equations $u_a = v_{a'}$, $u_{a'} = -v_a$. The axial extension of $f_0$ is

$$ \tilde{f}_0(\tilde{Q}) = u(q_0, \rho) + \hat{\mathbf{q}}\, v(q_0, \rho), \qquad \hat{\mathbf{q}} = \mathbf{q}/\rho, $$

defined for $\rho > 0$ by replacing $a$ by $q_0$, $a'$ by $\rho$, and the complex imaginary unit $i$ by $\hat{\mathbf{q}}$; the replacement is legitimate because $\hat{\mathbf{q}}^2 = -1$. When $f_0$ has real Taylor coefficients, $v(q_0,0) = 0$, the extension is continuous at $\rho = 0$ with value $f_0(q_0)$, and $\tilde{f}_0(\tilde{Q}) = \sum_{n \ge 0} a_n \tilde{Q}^n$ on $B(0,R)$. Complex coefficients are handled by $\mathbb{C}$-linearity: writing $f_0 = f_{0,1} + i f_{0,2}$ with real-coefficient parts, one sets $\tilde{f}_0 = \tilde{f}_{0,1} + i\,\tilde{f}_{0,2}$.

Fueter's Theorem

Theorem (Fueter's construction). Let $f_0$ be holomorphic on $D(0,R)$, with axial extension $\tilde{f}_0$. Then

$$ \tilde{F}(\tilde{Q}) = \Delta_4\, \tilde{f}_0(\tilde{Q}) = \frac{2\,\partial_\rho u(q_0,\rho)}{\rho} + \hat{\mathbf{q}}\left(\frac{2\,\partial_\rho v(q_0,\rho)}{\rho} - \frac{2\,v(q_0,\rho)}{\rho^2}\right) $$

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 $A, B$, one has $\partial_{q_k}A = (\partial_\rho A)\hat{q}_k$, $\sum_k e_k \hat{q}_k = \hat{\mathbf{q}}$, and $\sum_{j,k}(\partial_{q_k}\hat{q}_j)e_k e_j = (-3 - \hat{\mathbf{q}}^2)/\rho = -2/\rho$, because $\hat{\mathbf{q}}^2 = -1$. These give

$$ \tilde{\nabla} g = g\tilde{\nabla} = \left(\partial_0 A - \partial_\rho B - \frac{2B}{\rho}\right) + \hat{\mathbf{q}}\,(\partial_0 B + \partial_\rho A), $$

so $g$ is left-regular if and only if it is right-regular, and this holds exactly when $\partial_0 A = \partial_\rho B + 2B/\rho$ and $\partial_0 B = -\partial_\rho A$. Using the radial form of $\Delta_{\mathbb{R}^3}$, the identity $\Delta_{\mathbb{R}^3}(B\hat{\mathbf{q}}) = \hat{\mathbf{q}}(\Delta_{\mathbb{R}^3}B - 2B/\rho^2)$, and the harmonicity of $u, v$, one gets $\Delta_4 \tilde{f}_0 = P + \hat{\mathbf{q}}Q$ with $P = 2u_\rho/\rho$, $Q = 2(\rho v_\rho - v)/\rho^2$; the Cauchy–Riemann equations then give $Q_\rho + 2Q/\rho = \partial_0 P$ and $-P_\rho = \partial_0 Q$, and $P, Q$ extend continuously to $\rho = 0$.

The Kernel and Injectivity

Proposition. The map $\tau(f_0) = \Delta_4 \tilde{f}_0$ vanishes if and only if $f_0$ is affine, $f_0(A) = a_1A + a_0$ with $a_0, a_1 \in \mathbb{C}$.

Proof. $\tau(f_0) = 0$ iff $u_\rho = 0$ and $\rho v_\rho = v$, so $u = u(q_0)$, $v = c(q_0)\rho$; 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 constants and linear functions.

Thus $\tau$ 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

An imaginary unit is an element $I$ with $I^2 = -1$. On $\mathbb{H}_{\mathbb{B}}$ the imaginary units form the two-sphere $S^2$ of unit pure imaginary quaternions, and every $\tilde{Q}$ with $\mathbf{q} \neq 0$ is uniquely $\tilde{Q} = q_0 + I\rho$ with $I \in S^2$, $\rho > 0$. The slice $\mathbb{C}_I = \mathbb{R} + I\mathbb{R}$ is a copy of the complex plane, and two slices meet only in $\mathbb{R}$ unless $I = \pm J$. On the full algebra the roots of $-1$ are, by the article on biquaternion roots of minus one, the family $\pm i$, $\pm\mu$ and $b\mu + d\nu i$, with $\mu, \nu$ perpendicular unit pure real quaternions and $b^2 - d^2 = 1$, $b, d > 0$. Each spans a slice $\mathbb{C}_\xi \cong \mathbb{C}$, but this set of roots is no longer a sphere, so there is no single imaginary sphere on which to base the theory: the axial construction must select one root, hence one slice, at a time.

Axially Symmetric Functions and the Harmonic Coefficients

A function is axially symmetric if $\tilde{F}(q_0 + \hat{\mathbf{q}}\rho)$ depends on the direction $\hat{\mathbf{q}}$ only through $\hat{\mathbf{q}}$. Such a function has the axial representation $\tilde{F}(\tilde{Q}) = A(q_0,\rho) + \hat{\mathbf{q}}B(q_0,\rho)$, a sum over the two-element basis $\{1,\hat{\mathbf{q}}\}$ of the slice $\mathbb{C}_{\hat{\mathbf{q}}}$; $A$ and $B$ are the axial coefficients. For such a function the left- and right-regularity conditions coincide, by the computation in the proof of Fueter's theorem, and reduce to $\partial_0 A = \partial_\rho B + 2B/\rho$ and $\partial_0 B = -\partial_\rho A$. Eliminating $B$ gives $\Delta_4 A = 0$, so the scalar axial coefficient is harmonic, while $\Delta_4 B = 2B/\rho^2$. The Fueter construction realizes this class with $P = 2u_\rho/\rho$, $Q = 2v_\rho/\rho - 2v/\rho^2$, so the axial coefficients come from the harmonic conjugate pair $(u,v)$; conversely, in the classical Fueter correspondence every axially symmetric regular function with central axial coefficients arises this way, uniquely modulo the affine kernel.

The Fueter–Sce Theorem and Its Hypotheses

Theorem (Fueter–Sce). Let $n \ge 1$ 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 $\mathbb{R}^{n+1}$ (complex unit replaced by a unit vector of $\mathbb{R}^n$, $a'$ by the radius). Then $\tilde{F} = \Delta^{(n-1)/2} \tilde{f}_0$ is monogenic, annihilated on the left and on the right by the Cauchy–Riemann operator $\partial_0 + \sum_{j=1}^{n} e_j \partial_j$ on the ball where the extension is defined. For $n = 3$ the exponent is $1$, recovering Fueter's construction.

Three hypotheses are needed, and none can be dropped. First, $f_0$ must be holomorphic on $D(0,R)$, so its Taylor series converges there; the induced series then converges on $B(0,R)$, since $\Delta_4(\tilde{Q}^n)$ is a homogeneous polynomial of degree $n-2$ with at most polynomial growth in $n$, while $|a_n| r^n = O(1)$ for $r < R$ by Cauchy's estimates. Second, the construction has the affine kernel $a_1A + a_0$, so a one-to-one correspondence requires the Taylor coefficients to vanish to order two, $a_0 = a_1 = 0$, equivalently one works modulo affine functions. Third, parity: $(n-1)/2$ must be a non-negative integer, so $n$ must be odd, as in the quaternion and biquaternion case $n = 3$.

Power Series Representations

A series with biquaternionic coefficients placed on the right, $f(\tilde{Q}) = \sum_{n \ge 0} \tilde{Q}^n a_n$, converges absolutely on $\|\tilde{Q}\|_E < R$ with $R^{-1} = \limsup_n \|a_n\|_E^{1/n}$, because $\|\tilde{Q}^n\|_E = \|\tilde{Q}\|_E^n$. Its sum is slice-regular, or Cullen-regular: holomorphic on each slice. Slice-regular functions form a different class from the Fueter-regular ones; for example, $\tilde{Q} \mapsto \tilde{Q}$ is slice-regular, but

$$ \tilde{\nabla}\tilde{Q} = \sum_{\mu=0}^{3} e_\mu e_\mu = e_0 - e_1^2 - e_2^2 - e_3^2 = -2e_0 \neq 0. $$

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

For real Taylor coefficients, $\tilde{f}_0(\tilde{Q}) = \sum_{n \ge 0} a_n \tilde{Q}^n$, and term-by-term application of $\Delta_4$ gives the induced series $\tau(f_0) = \sum_{n \ge 0} a_n \Delta_4(\tilde{Q}^n)$, which begins at $n = 2$ and converges normally on $B(0,R)$ because $\Delta_4(\tilde{Q}^n)$ has degree $n-2$ and at most polynomial growth; term-by-term differentiation is therefore justified. In general, with $\mathcal{P}_k$ the biquaternion-valued homogeneous polynomials of degree $k$ and $\mathcal{M}_k = \{P \in \mathcal{P}_k : \tilde{\nabla}P = 0\}$ 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 $\tilde{F} = \sum_{k \ge 0} \tilde{F}_k$ with $\tilde{F}_k \in \mathcal{M}_k$, the analogue of the Taylor series of complex analysis.

The Biquaternionic Case

On $\mathbb{H}_{\mathbb{B}}$ the theory is the classical one with biquaternion coefficients, since the quaternion subspace is a division algebra: $N(\tilde{Q}) = \sum_\mu q_\mu^2$ is positive-definite, so every nonzero element is invertible. The fundamental solution $\tilde{G}(\tilde{Q}) = \tilde{Q}^{\natural}/\|\tilde{Q}\|_E^4$ satisfies $\tilde{\nabla}\tilde{G} = 0$ off the origin and $\tilde{\nabla}\tilde{G} = -2\pi^2 \delta_0 e_0$ distributionally, so the only singularity of the Cauchy theory is the point $\tilde{Q} = 0$.

On $i\mathbb{H}_{\mathbb{B}}$ the theory is the same, up to the constant central factor by which the two gradients differ. Every coefficient there is purely imaginary, so the gradient is $\tilde{\nabla}_{i\mathbb{H}} = -i\,\tilde{\nabla}_{\mathbb{H}}$ in the same four coordinates; since $-i$ is an invertible central constant, $\tilde{\nabla}_{i\mathbb{H}}\tilde{F} = 0$ if and only if $\tilde{\nabla}_{\mathbb{H}}\tilde{F} = 0$. The left- and right-regular functions are therefore the same class as on $\mathbb{H}_{\mathbb{B}}$, and the Cauchy kernel is the kernel of $\mathbb{H}_{\mathbb{B}}$ multiplied by $i$. Only the second-order operator changes sign, $\tilde{\nabla}\tilde{\nabla}^{\natural} = -\Delta_4 e_0$; its solutions are again the ordinary harmonic functions, so the harmonicity, the mean value property, and the monogenic Taylor expansion carry over unchanged. In this sense $i\mathbb{H}_{\mathbb{B}}$ adds no new case to Fueter theory: it is the definite counterpart of $\mathbb{H}_{\mathbb{B}}$, as $\mathbb{M}_+$ is the indefinite counterpart of $\mathbb{M}_-$.

On $\mathbb{M}_-$ and $\mathbb{M}_+$ the second-order operator is the wave operator, $\Box_{\mathbb{M}_-} = (-\partial_{q'_0}^2 + \Delta_{\mathbb{R}^3})e_0$ and $\Box_{\mathbb{M}_+} = (\partial_{q_0}^2 - \Delta_{\mathbb{R}^3})e_0$, of signatures $(3,1)$ and $(1,3)$: hyperbolic, hence without elliptic regularity, and the axial coefficients then satisfy a wave-type system rather than a Laplace system. The zero divisors form the null cone $(q'_0)^2 = q_1^2 + q_2^2 + q_3^2$ on $\mathbb{M}_-$ (and $q_0^2 = (q'_1)^2 + (q'_2)^2 + (q'_3)^2$ on $\mathbb{M}_+$), exactly where the biquaternion norm vanishes. Its nonzero elements are not invertible, so the pointwise inversion of $\tilde{Q}$ underlying the Cauchy kernel fails there, and the Euclidean kernel is not the fundamental solution of the gradient on these subspaces, as the articles on biquaternion integration and on biquaternion analysis on subspaces record. Domains for the Cauchy theory must avoid the cone, not merely the origin.

On the full algebra $\mathbb{B}$ the biquaternion norm $N(\tilde{Q}) = \sum_\mu Q_\mu^2$ is complex-valued, and its zero set is the null quadric, whose nonzero points are exactly the zero divisors and which has real dimension $6$. Every construction dividing by $\tilde{Q}$, and every kernel built from $\tilde{Q}^{-1}$, is therefore unavailable there. The Fueter operator is also not elliptic over $\mathbb{C}$: its symbol $\sum_\mu e_\mu \xi_\mu$ has norm $\sum_\mu \xi_\mu^2$, vanishing for nonzero complex covectors such as $(1, i, 0, 0)$, so ellipticity holds for real covectors but fails over $\mathbb{C}$ and the second-order operator is not the real Laplacian. Finally, the imaginary units are the larger family of the preceding section, so the slice structure must be built one root at a time. The Fueter theory is therefore developed on the four-dimensional subspaces, principally $\mathbb{H}_{\mathbb{B}}$, and extended to the indefinite subspaces only off the null cone; the full-algebra case remains the open problem noted in the article on biquaternion analysis.

Summary

The Fueter operator $\tilde{\nabla} = \sum_\mu e_\mu \partial_\mu$ and its conjugate factor the four-dimensional Laplacian on the quaternion subspace, $\tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \Delta_4 e_0$. On the anti-quaternion subspace $i\mathbb{H}_{\mathbb{B}}$ the gradient differs from this one by the constant central factor $-i$, so the regular functions and the Cauchy kernel are those of $\mathbb{H}_{\mathbb{B}}$, up to $i$, and only the sign of the second-order operator changes. Left-regularity $\tilde{\nabla}\tilde{F} = 0$ and right-regularity $\tilde{F}\tilde{\nabla} = 0$ differ only in the sign of the curl term of the componentwise Cauchy–Riemann–Fueter system and coincide for axially symmetric functions; every regular function is harmonic, hence real-analytic, and satisfies the mean value property. The regularity condition has a variational origin as the Dirichlet principle of the Cauchy–Riemann operator: the Cauchy–Riemann energy and the Dirichlet energy differ by a null Lagrangian and share their Euler–Lagrange system, and the conjugate boundary data of Lanczos's 1919 statement selects the regular representative among the harmonic functions.

The Fueter construction sends a holomorphic $f_0$ to the induced function $\tilde{F} = \Delta_4 \tilde{f}_0 = 2u_\rho/\rho + \hat{\mathbf{q}}(2v_\rho/\rho - 2v/\rho^2)$, which is both left- and right-regular; it is $\mathbb{C}$-linear with the affine functions as kernel, hence injective exactly for holomorphic functions whose Taylor coefficients vanish to order two. The Fueter–Sce theorem extends it to odd-dimensional Clifford algebras 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, while slice-regular series form the distinct class that the construction converts into regular functions. On $\mathbb{H}_{\mathbb{B}}$ the origin is the only singularity, but on the full algebra the six-dimensional null quadric obstructs inversion, the Fueter operator is not elliptic over $\mathbb{C}$, and the imaginary units are no longer the sphere $S^2$; the full-algebra case remains open.

Summary of Notation

Symbol	Meaning
$\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	Biquaternion algebra
$\mathbb{H}_{\mathbb{B}}$	Quaternion subspace, with real coordinates $q_\mu$; the domain of Fueter theory
$i\mathbb{H}_{\mathbb{B}}$	Anti-quaternion subspace, all coefficients purely imaginary; regularity is the same class as on $\mathbb{H}_{\mathbb{B}}$, with $\Box = -\Delta_4 e_0$
$\tilde{Q} = \sum_\mu Q_\mu e_\mu$	Biquaternion; $\tilde{Q} = \sum_\mu q_\mu e_\mu$ on $\mathbb{H}_{\mathbb{B}}$
$\mathbf{q} = \sum_k q_k e_k$, $\rho = \\|\mathbf{q}\\|_E$, $\hat{\mathbf{q}} = \mathbf{q}/\rho$	Vector part, its modulus, and its direction
$\tilde{\nabla} = \partial_0 + \mathbf{D}$, $\mathbf{D} = \sum_k e_k \partial_k$	Fueter (Cauchy–Riemann–Fueter) operator, acting on the left
$\tilde{\nabla}^{\natural} = \partial_0 - \mathbf{D}$	Conjugate Fueter operator
$\Delta_4 = \sum_\mu \partial_\mu^2$	Four-dimensional Laplacian; $\tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \Delta_4 e_0$
$\partial, \bar{\partial}$	Planar Cauchy–Riemann operators, $\partial = \partial_a - i\partial_{a'}$, $\bar{\partial} = \partial_a + i\partial_{a'}$ (variational section)
$\tilde{F}$ left-regular	$\tilde{\nabla}\tilde{F} = 0$; monogenic in Clifford language
$\tilde{F}$ right-regular	$\tilde{F}\tilde{\nabla} = 0$
$I$, $S^2$	Imaginary unit, $I^2 = -1$; on $\mathbb{H}_{\mathbb{B}}$ these form the two-sphere
$\mathbb{C}_I = \mathbb{R} + I\mathbb{R}$	Slice, a copy of the complex plane; slice-regular functions live on unions of slices
$A, B$	Axial coefficients of an axially symmetric function, $\tilde{F} = A(q_0,\rho) + \hat{\mathbf{q}}B(q_0,\rho)$
$\tilde{f}_0$, $\Delta_4\tilde{f}_0$	Axial extension of a holomorphic $f_0$ and its Fueter-induced function
$\mathrm{Cl}_{0,n}$, $\mathrm{Cl}_{1,3}^{+}$	Clifford algebras; $\mathbb{H}$ is $\mathrm{Cl}_{0,3}^{+}$ and $\mathbb{B}$ is $\mathrm{Cl}_{1,3}^{+}$