# Split-Complex Analysis: Foundations & Cauchy Theory

Mannucci Mirco A.

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# Split-Complex Analysis: Foundations & Cauchy Theory

Mirco A. Mannucci

Holomathics, LLC

mirco@holomathics.com

September 1, 2025

## Abstract

We investigate the development of a holomorphic function theory over the split-complex algebra $D = \text{Cl}(1, 0)$, revealing a profound connection between algebraic structure and wave physics. While complex holomorphic functions encode elliptic phenomena through the Laplace equation, D-holomorphic functions are fundamentally hyperbolic—they are the mathematical embodiment of wave propagation. The zero divisors that appear as “obstacles” to classical function theory are actually the characteristic curves where wave information propagates, and the apparent “failures” of complex analysis in the D-setting encode the essential physics of hyperbolic systems. Building on recent breakthrough results by Luna-Elizarraras et al., we demonstrate that D-holomorphic functions naturally factorize the wave operator and that their singular behavior on zero divisor lines captures shock wave formation, characteristic data propagation, and branching phenomena observed in supersonic flow, electromagnetic wave fronts, and relativistic spacetimes. Rather than overcoming the mathematical difficulties, we embrace them as the emergence of wave physics, proposing hyperfunction methods, stratified analysis, and sheaf-theoretic frameworks that systematically exploit this wave structure. Our central insight is that D-holomorphic function theory is not complex analysis with complications—it is the native mathematical language of hyperbolic wave phenomena.

**Main Contributions.** (1) A two-annulus Cauchy reconstruction theorem for D-holomorphic functions with independent characteristic radii and double Laurent expansions; (2) A characteristic residue calculus on product domains with line-residues along $u = \text{constant}$ and $v = \text{constant}$; (3) A corrected retarded Green's function with proper causal support and D-holomorphic decomposition; (4) An idempotent interface transfer analysis demonstrating energy balance in wave scattering; (5) A comprehensive framework connecting classical D-holomorphic theory to systematic wave physics applications; (6) Rigorous jump laws connecting residues to primitive discontinuities across characteristics.
# Contents

<table><tr><td>1</td><td>Introduction</td><td>3</td></tr><tr><td>1.1</td><td>Historical Context and Recent Breakthroughs</td><td>5</td></tr><tr><td>2</td><td>The D-Holomorphic Function Problem</td><td>6</td></tr><tr><td>2.1</td><td>Foundations and Modern Developments</td><td>6</td></tr><tr><td>2.2</td><td>Idempotent Structure and Basic Theory</td><td>6</td></tr><tr><td>2.3</td><td>Basic D-Function Toolbox</td><td>7</td></tr><tr><td>2.4</td><td>Limited Identity Theorem</td><td>8</td></tr><tr><td>3</td><td>The Wave Physics Essence of D-Holomorphic Theory</td><td>9</td></tr><tr><td>3.1</td><td>Zero Divisors as Wave Characteristic Curves</td><td>10</td></tr><tr><td>4</td><td>Zero Divisors as Wave Information Carriers</td><td>10</td></tr><tr><td>4.1</td><td>Wave Characteristic Structure</td><td>11</td></tr><tr><td>5</td><td>Two-Annulus Cauchy Formula and Characteristic Line Residues</td><td>11</td></tr><tr><td>6</td><td>Causal Green's Function in D-Form</td><td>14</td></tr><tr><td>7</td><td>Characteristic Residues, Interface Transfer, and Jump Laws</td><td>15</td></tr><tr><td>7.1</td><td>Formal Residue-Jump Law</td><td>15</td></tr><tr><td>7.2</td><td>Interface Transfer in Idempotent Variables and Energy Balance</td><td>17</td></tr><tr><td>7.3</td><td>Computed Example: Gaussian Pulse Scattering at an Interface</td><td>18</td></tr><tr><td>7.4</td><td>Validation of the Wave Physics Framework</td><td>19</td></tr><tr><td>8</td><td>Proposed Approaches</td><td>19</td></tr><tr><td>8.1</td><td>Stratified Regular Analysis</td><td>19</td></tr><tr><td>8.2</td><td>Hyperfunction Framework</td><td>19</td></tr><tr><td>8.3</td><td>Sheaf-Theoretic Approach</td><td>20</td></tr><tr><td>9</td><td>Applications and Physical Relevance</td><td>20</td></tr><tr><td>9.1</td><td>Wave Physics and Hyperbolic PDEs</td><td>20</td></tr><tr><td>9.2</td><td>Mathematical Physics Applications</td><td>21</td></tr><tr><td>9.3</td><td>Computational and Numerical Methods</td><td>21</td></tr><tr><td>10</td><td>Conclusion and Future Directions</td><td>21</td></tr><tr><td>A</td><td>Rosetta Table: Complex vs. D-Analysis</td><td>23</td></tr></table>
<table>
  <tbody>
    <tr>
      <td><b>B Computational Exploration Library</b></td>
      <td><b>24</b></td>
    </tr>
    <tr>
      <td>B.1 Library Design Philosophy</td>
      <td>24</td>
    </tr>
    <tr>
      <td>B.2 Core Implementation</td>
      <td>24</td>
    </tr>
    <tr>
      <td>B.2.1 SplitComplex Class</td>
      <td>25</td>
    </tr>
    <tr>
      <td>B.2.2 DHolomorphic Class</td>
      <td>25</td>
    </tr>
    <tr>
      <td>B.2.3 Visualization Tools</td>
      <td>25</td>
    </tr>
    <tr>
      <td>B.3 Key Explorations Enabled</td>
      <td>26</td>
    </tr>
    <tr>
      <td>B.4 Educational Impact</td>
      <td>26</td>
    </tr>
    <tr>
      <td>B.5 Library Design Philosophy</td>
      <td>26</td>
    </tr>
    <tr>
      <td>B.6 Core Implementation</td>
      <td>26</td>
    </tr>
    <tr>
      <td>B.6.1 SplitComplex Class</td>
      <td>26</td>
    </tr>
    <tr>
      <td>B.6.2 DHolomorphic Class</td>
      <td>27</td>
    </tr>
    <tr>
      <td>B.6.3 Visualization API</td>
      <td>27</td>
    </tr>
    <tr>
      <td>B.7 Key Explorations Enabled</td>
      <td>27</td>
    </tr>
    <tr>
      <td>B.8 Educational Impact</td>
      <td>28</td>
    </tr>
  </tbody>
</table>

# 1 Introduction

**Scope/novelty.** The split-complex CR system and the decomposition $f = F(u)e_+ + G(v)e_-$ are standard (e.g., Sobczyk; Kisil), and hyperbolic Cauchy-type formulas exist (e.g., Libine). We formulate a Cauchy/annulus framework aligned with the two characteristics (“two-annulus” with independent radii), *derive* residue-jump correspondences tailored to characteristic interfaces, and emphasize physical interpretations (left/right movers, characteristic flux) and interface transfer. We do not claim global algebraic equivalences beyond the stated hypotheses. The remarkable success of complex analysis stems from the intimate connection between holomorphic functions and elliptic partial differential equations. A function $f(z) = u(x, y) + iv(x, y)$ holomorphic in $\mathbb{C}$ satisfies the Cauchy-Riemann equations

$$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad (1) $$

$$ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}, \qquad (2) $$

which immediately implies that both $u$ and $v$ are harmonic: $\Delta u = \Delta v = 0$.

**Classic Example:** The function $f(z) = z^2 = (x + iy)^2 = (x^2 - y^2) + i(2xy)$ gives us $u(x, y) = x^2 - y^2$ and $v(x, y) = 2xy$. We can verify: $\frac{\partial u}{\partial x} = 2x = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -2y = -\frac{\partial v}{\partial x}$. Both components are harmonic: $\Delta u = 2 - 2 = 0$ and $\Delta v = 0 + 0 = 0$.

This connection motivates investigating analogous function theories over other Clifford algebras.
Of particular interest is the split-complex algebra $D = \text{Cl}(1,0)$, spanned by $\{1, j\}$ where $j^2 = +1$. Elements of $D$ take the form $z = x + jy$ with $x, y \in \mathbb{R}$, and D-holomorphic functions satisfy modified Cauchy-Riemann equations leading to the wave equation rather than Laplace's equation.

**Clarification of Novelty:** While the basic connection between D-holomorphic functions and the wave equation is known in the literature [4, 7], its profound physical implications have been underexplored. This work moves beyond the basic observation to develop a complete wave-physics-based function theory, including characteristic residue theorems, energy-conserving interface transfer laws, and retarded Green's functions derived directly from the algebraic structure. Our central contribution is demonstrating that D-holomorphic theory is not complex analysis adapted to hyperbolic settings—it is the native mathematical language that nature uses for wave phenomena.

**D-Holomorphic Example:** Consider $F(z) = z^2 = (x + jy)^2 = x^2 + y^2 + j(2xy)$. Here $u(x, y) = x^2 + y^2$ and $v(x, y) = 2xy$. The D-Cauchy-Riemann equations become: $\frac{\partial u}{\partial x} = 2x = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = 2y = \frac{\partial v}{\partial x}$. This gives us the wave equation: $\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 2^2 - 2^2 = 0$ and $\frac{\partial^2 v}{\partial x^2} - \frac{\partial^2 v}{\partial y^2} = 0 - 0 = 0$.

Notice the striking difference: where complex $z^2$ gives harmonic functions (like the real part $x^2 - y^2$), D-holomorphic $z^2$ gives wave-type functions (real part $x^2 + y^2$).

<figure><img src="image_2.png" /><figcaption>Figure 1: Comparison between complex and D-holomorphic behavior for $z^2$. Left: Complex holomorphic $z^2$ yields harmonic functions with elliptic contours. Right: D-holomorphic $z^2$ yields wave-type functions with circular level sets, and zero divisor lines (red) mark the characteristic directions.</figcaption></figure>

However, unlike $\mathbb{C}$, the algebra $D$ is not a field due to the existence of zero divisors. Rather than viewing this as a mathematical failure, we recognize it as the fundamental signature of hyperbolic wave physics. The zero divisors are not obstacles—they are the characteristic curves along which wave information propagates. The "breakdown" of classical function theory is the emergence of
wave phenomena.

**Central Physical Insight:** Where complex analysis encodes elliptic phenomena (steady states, harmonic functions, circular level curves), D-holomorphic analysis *is* hyperbolic wave physics. The mathematical structures that seem problematic—zero divisors, failed residue theory, branching across characteristics—are precisely the mechanisms by which nature propagates waves, forms shocks, and transmits information along characteristic curves in supersonic flows, electromagnetic fields, and relativistic spacetimes.

## 1.1 Historical Context and Recent Breakthroughs

The split-complex numbers have appeared in various contexts throughout mathematical literature, often under different names. Yaglom [3] provided early geometric insights in *Complex Numbers in Geometry*, while Sobczyk [4] developed systematic calculus on "the hyperbolic number plane." More recently, Catoni et al. [5] explored connections to Minkowski space-time geometry, and Cruceanu, Fortuny, and Gadea [6] surveyed paracomplex geometry in differential geometry contexts.

The idempotent approach we employ follows Price [11] in *An Introduction to Multicomplex Spaces and Functions*, while our hyperfunction methods draw on the extensive work of Sato, Kawai, and Kashiwara [9, 10] on microlocal analysis and edge-of-the-wedge theorems.

Shapiro et al. [7] provided a comprehensive study of hyperbolic numbers and their elementary functions, establishing foundational results for polynomial, trigonometric, and exponential functions in the hyperbolic setting. Their work demonstrates the factorization of the wave operator and develops systematic approaches to inverse functions including roots, logarithms, and inverse trigonometric functions.

A significant breakthrough came with Luna-Elizarraras et al. [19], who proved that D-holomorphic functions satisfy a limited but meaningful version of the identity theorem. Their work reveals fundamental geometric constraints in the hyperbolic setting through the concept of hyperbolic holomorphicity hulls and establishes rigorous conditions under which uniqueness results hold. This provides the modern foundation upon which our comprehensive theory is built.

Despite this growing foundation, a systematic function theory for D-holomorphic functions that systematically addresses the zero divisor obstacles and reveals their physical interpretation has not been fully developed. Our paper aims to fill this gap by proposing multiple coordinated approaches to overcome these fundamental challenges while revealing the deep connection to hyperbolic wave physics.
# 2 The D-Holomorphic Function Problem

## 2.1 Foundations and Modern Developments

Recent work by Luna-Elizarraras et al. [8] has established rigorous foundations for D-holomorphic theory. We build on their key results as our starting point:

**Definition 1** (D-holomorphic derivative - Modern Form). Let $D = \{x + jy : x, y \in \mathbb{R}, j^2 = 1\}$ where we use $j$ instead of $k$ for consistency with our notation. For $f: \Omega \subset D \to D$ and $z_0 \in \Omega$,

$$f'(z_0) := \lim_{\substack{h \to 0 \\ h \in D^*}} \frac{f(z_0 + h) - f(z_0)}{h}, \quad D^* = \{ue_+ + ve_- : u \neq 0, v \neq 0\}.$$

Here $e_\pm = \frac{1}{2}(1 \pm j)$ are the standard idempotent projectors.

**Theorem 1** (Idempotent Structure - Luna-Elizarraras et al. [8]). A $C^1$ function $f: \Omega \to D$ is D-holomorphic if and only if

$$f(x + jy) = F(x + y)e_+ + G(x - y)e_-,$$

for some $C^1$ functions $F, G: \mathbb{R} \to \mathbb{R}$.

This immediately gives the wave equation factorization $\frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} = 4\frac{\partial}{\partial z}\frac{\partial}{\partial \bar{z}}$ and explains characteristic propagation.

## 2.2 Idempotent Structure and Basic Theory

The split-complex algebra $D = \text{Cl}(1, 0)$ admits a fundamental idempotent decomposition that clarifies all subsequent phenomena.

**Definition 2** (Idempotent Projectors). *Define the standard projectors*

$$e_{\pm} = \frac{1}{2}(1 \pm j), \quad e_{\pm}^2 = e_{\pm}, \quad e_+e_- = 0, \quad 1 = e_+ + e_-, \quad j = e_+ - e_- \quad (3)$$

Every element $z = x + jy$ decomposes as

$$z = (x + y)e_+ + (x - y)e_- := ue_+ + ve_- \quad (4)$$

where $u = x + y$ and $v = x - y$ are the characteristic coordinates.

From this point forward, we will use characteristic coordinates extensively. For wave physics applications, we identify $y = ct$ where $c$ is the wave speed, so $u = x + ct$ and $v = x - ct$ represent the characteristic coordinates along which wave information propagates.

**Theorem 2** (Invertibility Criterion). $z \in D^* \Leftrightarrow uv \neq 0$. Zero divisors occur when $u = 0$ ($y = x$) or $v = 0$ ($y = -x$).
**Definition 3** (Invertible Set and Zero Divisors). Write $z = ue_+ + ve_-$ with $u = x + y$, $v = x - y$. Then

$$D^* = \{z : u \neq 0 \text{ and } v \neq 0\}, \quad \text{ZD} = \{t(1+j) : t \in \mathbb{R}\} \cup \{t(1-j) : t \in \mathbb{R}\}. \quad (5)$$

**Definition 4** (D-Holomorphic Functions). Let $\Omega \subset D$ be open. A function $f: \Omega \to D$ is D-holomorphic if it satisfies

$$\frac{\partial f}{\partial \bar{z}} = 0, \quad (6)$$

where $\frac{\partial}{\partial z} = \frac{1}{2}(\frac{\partial}{\partial x} + j\frac{\partial}{\partial y})$ and $\frac{\partial}{\partial \bar{z}} = \frac{1}{2}(\frac{\partial}{\partial x} - j\frac{\partial}{\partial y})$.

**Theorem 3** (Wave Operator Factorization). For smooth functions $\phi$, with the standard wave operator $\square = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2}$,

$$\square = -4 \frac{\partial}{\partial u} \frac{\partial}{\partial v}. \quad (7)$$

Consequently, $\frac{\partial f}{\partial \bar{z}} = 0 \Rightarrow \square f = 0$.

This establishes that D-holomorphic functions are precisely those whose components satisfy the two-dimensional wave equation—they are the native mathematical language of hyperbolic wave physics.

**Fundamental Physical Principle:** D-holomorphic functions do not merely satisfy the wave equation as a consequence—they ARE wave solutions. The idempotent structure $f(z) = F(u)e_+ + G(v)e_-$ with $u = x + ct$, $v = x - ct$ is precisely D'Alembert's factorization [17] of wave solutions into left-moving and right-moving components. The zero divisor lines $u = 0$ and $v = 0$ are the characteristic curves along which these wave components propagate.

The "mathematical difficulties" of D-holomorphic theory are not problems to be solved—they are the mathematical manifestation of wave physics:

* **Zero divisors** = Characteristic curves where wave fronts propagate

* **Failed residue theory** = Impossibility of reversing wave flow along characteristics

* **Branching functions** = Wave mode splitting at characteristic intersections

* **Jump discontinuities** = Shock wave formation and propagation

* **Limited identity theorem** = Information can only propagate from non-characteristic initial data

Every "failure" of complex analysis in the D-setting is a success of wave physics representation.

## 2.3 Basic D-Function Toolbox

The idempotent structure provides natural formulas for fundamental operations:
**Exponential Functions:** Following the comprehensive treatment by Shapiro et al. [7]:

$$e^{j\theta} = \cosh\theta + j\sinh\theta, \quad e^{az} = e^{au}e_{+} + e^{av}e_{-} \qquad (8)$$

**Multiplicative Inverses:** For $z = x + jy$ with $uv = (x + y)(x - y) = x^2 - y^2 \neq 0$:

$$z^{-1} = \frac{x - jy}{x^2 - y^2} = \frac{1}{u}e_{+} + \frac{1}{v}e_{-} \qquad (9)$$

**Corrected Trigonometric Functions:** For the split-complex unit $j$ with $j^2 = 1$, using the power series definitions:

$$\cos(jy) = \cos(y), \quad \sin(jy) = j\sin(y) \qquad (10)$$

$$\cosh(jy) = \cosh(y), \quad \sinh(jy) = j\sinh(y) \qquad (11)$$

**Minkowski Pseudo-Norm:**

$$\|z\|_{M}^{2} = x^{2} - y^{2} = uv \qquad (12)$$

**General D-Holomorphic Functions:** Every D-holomorphic function has the canonical form

$$f(z) = F(u)e_{+} + G(v)e_{-}, \quad u = x + y, \quad v = x - y \qquad (13)$$

Let's examine concrete examples using this framework:

**Example 1: Linear Functions** $F_1(z) = z = ue_+ + ve_-$ gives $F(u) = u, G(v) = v$. Both are trivially $C^1$ and satisfy the wave equation.

**Example 2: Quadratic Functions** $F_2(z) = z^2 = (ue_+ + ve_-)^2 = u^2e_+ + v^2e_-$ since $e_+e_- = 0$. Here $F(u) = u^2, G(v) = v^2$.

**Example 3: D-Exponential** $F_3(z) = e^z = e^u e_+ + e^v e_-$ gives $F(u) = e^u, G(v) = e^v$.

**Example 4: Branching Function** $F_4(z) = \sqrt{ue_+} + \sqrt{ve_-}$ (with appropriate branch choices) demonstrates how multivaluedness naturally arises at the zero divisor lines $u = 0$ and $v = 0$.

**Key Observation:** The idempotent decomposition reveals that D-holomorphic functions are essentially pairs of univariate functions $(F(u), G(v))$ along the characteristic coordinates. This immediately explains the connection to the wave equation and wave propagation phenomena.

## 2.4 Limited Identity Theorem

Building on the Luna-Elizarraras breakthrough, we can establish when uniqueness holds:

**Theorem 4 (Limited Identity Principle - Luna-Elizarraras et al.)** If $f: D \to D$ is D-holomorphic
and $f$ vanishes on a line $y = mx + b$ with $m \neq \pm 1$ (e.g., on $y = 0$), then $f \equiv 0$ on $D$.

**Key Insight:** Lines with slopes $\pm 1$ are precisely the zero divisor lines—the characteristic curves of the wave equation where different behavior emerges.

**Corollary 1 (Exponential Uniqueness).** The D-holomorphic extension $e^x(\cos y + j \sin y)$ of the real exponential is unique, resolving Cockle's historical question.

The zero divisors in $D$ form two lines in the $(x, y)$-plane:

$$L_+ = \{(x, y) : y = x\} \quad \text{(multiples of } 1 + j\text{)} \qquad (14)$$

$$L_- = \{(x, y) : y = -x\} \quad \text{(multiples of } 1 - j\text{)} \qquad (15)$$

<figure><img src="image_3.png" /><figcaption>Figure 2: Zero divisor lines in the D-plane showing the stratification into regular regions $D^*$ and singular loci along the characteristic lines.</figcaption></figure>

These lines have profound geometric significance: they are precisely the *characteristic lines* of the two-dimensional wave equation.

## 3 The Wave Physics Essence of D-Holomorphic Theory

The mathematical structures of D-holomorphic functions directly encode fundamental wave physics. What appears as "breakdown" of complex analysis is actually the emergence of hyperbolic wave phenomena. Let's examine exactly how the algebraic structure of $D$ manifests as wave propagation.
## 3.1 Zero Divisors as Wave Characteristic Curves

The fundamental structure of $D$ directly encodes wave physics through its zero divisors:

$$ (1 + j)(1 - j) = 1 - j^2 = 1 - 1 = 0. \qquad (16) $$

**Physical Interpretation:** This is not a mathematical failure—it is the algebraic signature of characteristic curves in hyperbolic systems. In wave physics, information propagates along characteristics, and these zero divisor lines $y = \pm x$ are precisely the characteristic directions of the wave equation $\square u = 0$.

**Why Division Fails:** Consider trying to divide by $(1 + j)$, which represents a point on the characteristic line $y = x$. In wave physics, you cannot "invert" characteristic flow—information flows along characteristics but cannot flow backwards against them. The algebraic impossibility of finding $(1 + j)^{-1}$ directly reflects the physical impossibility of time-reversing wave propagation along characteristics.

This immediately reveals the deep physical meaning of several phenomena:

**Theorem 5 (Wave Characteristic Encoding).** The zero divisor structure of $D$ directly encodes the characteristic geometry of the 2D wave equation, with algebraic operations reflecting physical wave propagation constraints.

**Problem 1 (Multiplicative Inverse Problem).** Elements of the form $a(1 \pm j)$ with $a \neq 0$ have no multiplicative inverse in $D$, preventing division and invalidating rational function theory.

**Theorem 6 (Cauchy Kernel as Wave Green's Function).** The failure of the classical Cauchy kernel $(z - \zeta)^{-1}$ on characteristic sets directly reflects the physical structure of wave Green's functions, which become singular precisely along light cones and characteristic surfaces.

**Wave Physics Interpretation:** The Cauchy kernel decomposes as (valid when $u \neq \alpha$ and $v \neq \beta$):

$$ \frac{1}{z - \zeta} = \frac{1}{(u - \alpha)e_+ + (v - \beta)e_-} = \frac{1}{u - \alpha}e_+ + \frac{1}{v - \beta}e_- \qquad (17) $$

where $\zeta = \alpha e_+ + \beta e_-$. Each term becomes singular on its respective characteristic line $u = \alpha$ or $v = \beta$.

This is exactly how wave Green's functions behave: they become singular along the light cone (characteristic surfaces) where causal influence propagates. The mathematical "breakdown" is the physical representation of how waves carry information along characteristics.

## 4 Zero Divisors as Wave Information Carriers

The behavior of D-holomorphic functions on zero divisor lines reveals the fundamental physics of wave propagation. Rather than viewing this behavior as pathological, we recognize it as the
mathematical essence of hyperbolic wave phenomena.

## 4.1 Wave Characteristic Structure

Zero divisor lines are not mathematical obstacles—they are the information highways of wave physics.

Let's examine concrete examples of how D-holomorphic functions behave as they approach these characteristic lines.

**Theorem 7 (Characteristic Propagation).** If $f = F(u)e_+ + G(v)e_-$ with $F, G$ piecewise $C^1$, then jump discontinuities in $F'$ at $u = 0$ and/or $G'$ at $v = 0$ propagate along the characteristic lines $y = x$ and $y = -x$ respectively.

**Proof Sketch:** The wave equation $\Box\phi = 0$ has characteristic curves $x \pm ct$ = constant. Discontinuities in derivatives propagate along these curves, which are precisely the zero divisor lines $u = x + ct = 0$ and $v = x - ct = 0$. $\Box$

<figure><img src="image_4.png" /><figcaption>Figure 3: Behavior of D-holomorphic functions near zero divisor lines, showing branching, oscillatory approach, and information propagation along characteristic curves.</figcaption></figure>

# 5 Two-Annulus Cauchy Formula and Characteristic Line Residues

Building on the idempotent foundations, we develop a systematic Cauchy theory for D-holomorphic functions on product domains. This provides concrete tools for function reconstruction and residue calculus that exploit the characteristic coordinate structure.
Let $\mathbb{D} = \{x + jy : x, y \in \mathbb{R}, j^2 = 1\}$ with idempotents $e_{\pm} = \frac{1 \pm j}{2}$, characteristic coordinates $u = x+y, v = x-y$, and projections $\pi_{\pm}(z) = x \pm y$ so that $z = (x-y)e_{-} + (x+y)e_{+} = \pi_{-}(z)e_{-} + \pi_{+}(z)e_{+}$. We call $f : \Omega \subset \mathbb{D} \to \mathbb{D}$ D-holomorphic if $\bar{\partial}f = 0$, equivalently $f = F(u)e_{+} + G(v)e_{-}$ with real-analytic $F, G$ on their domains.

**Definition 5** (Two-Annulus Domain). Fix $z_0 = x_0 + jy_0$ and set $u_0 = x_0 + y_0$, $v_0 = x_0 - y_0$. For radii $0 < r_{\pm} < R_{\pm}$ let

$$A_{+} = \{\zeta \in \mathbb{C} : r_{+} < |\zeta - u_{0}| < R_{+}\}, \quad A_{-} = \{\eta \in \mathbb{C} : r_{-} < |\eta - v_{0}| < R_{-}\}.$$

Define the two-annulus in $\mathbb{D}$ as

$$W(z_0; r_{\pm}, R_{\pm}) = \{z \in \mathbb{D} : r_{+} < |\pi_{+}(z - z_0)| < R_{+}, r_{-} < |\pi_{-}(z - z_0)| < R_{-}\}.$$

**Lemma 1** (One-Variable Complex Extension). If $f$ is D-holomorphic on a neighborhood of $z_0$, then $F$ (resp. $G$) is real-analytic near $u_0$ (resp. $v_0$), hence extends holomorphically to some complex neighborhoods $U_+ \ni u_0$, $U_- \ni v_0$ with appropriate radii determined by the singularity structure of $F$ and $G$.

**Theorem 8** (Two-Annulus Cauchy Reconstruction and Coefficient Extraction). Assume $f$ is D-holomorphic on $W(z_0; r_{\pm}, R_{\pm})$ and that $F$ (resp. $G$) extends holomorphically to $A_+ \subset U_+$ (resp. $A_- \subset U_-$). Let $\Gamma_+ = \partial A_+$ and $\Gamma_- = \partial A_-$ be positively oriented boundaries. Then

$$f(z_0) = \frac{1}{2\pi i} \oint_{\Gamma_+} \frac{F(\zeta)}{\zeta - u_0} d\zeta e_+ + \frac{1}{2\pi i} \oint_{\Gamma_-} \frac{G(\eta)}{\eta - v_0} d\eta e_-.$$

Moreover, $F$ and $G$ admit unique Laurent expansions on $A_{\pm}$,

$$F(\zeta) = \sum_{n \in \mathbb{Z}} a_n (\zeta - u_0)^n, \quad G(\eta) = \sum_{m \in \mathbb{Z}} b_m (\eta - v_0)^m,$$

which induce the double Laurent expansion on $W$,

$$f(z) = \sum_{n \in \mathbb{Z}} a_n (\pi_+(z - z_0))^n e_+ + \sum_{m \in \mathbb{Z}} b_m (\pi_-(z - z_0))^m e_-,$$

with coefficients given by the Cauchy integrals $a_n = \frac{1}{2\pi i} \oint_{\Gamma_+} \frac{F(\zeta)}{(\zeta - u_0)^{n+1}} d\zeta$, $b_m = \frac{1}{2\pi i} \oint_{\Gamma_-} \frac{G(\eta)}{(\eta - v_0)^{m+1}} d\eta$.

*Proof.* We expand the proof to show explicitly how the push-forward via $\pi_{\pm}$ and the independence of radii work.

By Lemma 1, $F$ and $G$ are holomorphic on $A_+$ and $A_-$ respectively. The key insight is that the characteristic projections $\pi_+(z) = x + y$ and $\pi_-(z) = x - y$ allow independent complex extensions with different radii.

**Reconstruction Formula:** Since $f(z_0) = F(u_0)e_+ + G(v_0)e_-$ and $F, G$ are holomorphic on
their respective annuli, the standard complex Cauchy formula gives:

$$F(u_0) = \frac{1}{2\pi i} \oint_{\Gamma_+} \frac{F(\zeta)}{\zeta - u_0} d\zeta \qquad (18)$$

$$G(v_0) = \frac{1}{2\pi i} \oint_{\Gamma_-} \frac{G(\eta)}{\eta - v_0} d\eta \qquad (19)$$

The D-holomorphic reconstruction follows by the idempotent multiplication: $f(z_0) = F(u_0)e_+ + G(v_0)e_-$.

**Anisotropic Domain Structure:** The crucial novelty is that $r_+ \neq r_-$ and $R_+ \neq R_-$ are permitted. The domain $\mathcal{W}$ is the Cartesian product of annuli in $(u, v)$ coordinates, unlike level-set approaches that constrain $|z|^2 = |u||v|$ to hyperbolas.

**Laurent Expansions:** On each annulus, classical Laurent theory gives:

$$F(\zeta) = \sum_{n \in \mathbb{Z}} a_n(\zeta - u_0)^n, \quad G(\eta) = \sum_{m \in \mathbb{Z}} b_m(\eta - v_0)^m$$

with coefficients $a_n = \frac{1}{2\pi i} \oint_{\Gamma_+} \frac{F(\zeta)}{(\zeta - u_0)^{n+1}} d\zeta$, $b_m = \frac{1}{2\pi i} \oint_{\Gamma_-} \frac{G(\eta)}{(\eta - v_0)^{m+1}} d\eta$.

**Push-forward to D:** The substitution $\zeta - u_0 \mapsto \pi_+(z - z_0) = (x - x_0) + (y - y_0)$ and $\eta - v_0 \mapsto \pi_-(z - z_0) = (x - x_0) - (y - y_0)$ gives the double Laurent expansion on $\mathcal{W}$. $\square$

**Definition 6 (Characteristic Line Residue).** Let $f$ be D-holomorphic on $\mathcal{W}$ except for isolated poles of $F$ at $\{u_k\} \subset A_+$ and of $G$ at $\{v_\ell\} \subset A_-$. Define the $e_+$-line residue of $f$ along $u = u_k$ by

$$\mathrm{Res}_{u=u_k} f := \mathrm{Res}_{\zeta=u_k} F(\zeta) e_+,$$

and similarly $\mathrm{Res}_{v=v_\ell} f := \mathrm{Res}_{\eta=v_\ell} G(\eta) e_-$.

**Theorem 9 (Characteristic Residue Theorem on a Two-Annulus).** Under the hypotheses above, the integral of $f$ around the product boundary $\partial\mathcal{W}$ splits into the sum of characteristic line-residues:

$$\frac{1}{2\pi i} \oint_{\Gamma_+} F(\zeta) d\zeta e_+ = \sum_{u_k \in A_+} \mathrm{Res}_{u=u_k} f, \quad \frac{1}{2\pi i} \oint_{\Gamma_-} G(\eta) d\eta e_- = \sum_{v_\ell \in A_-} \mathrm{Res}_{v=v_\ell} f.$$

*Proof sketch.* This follows directly from the classical residue theorem applied to the complex variables $\zeta$ and $\eta$ in their respective annuli, combined with the idempotent projections. $\square$

**Example 10 (Double Laurent Coefficient Extraction).** We demonstrate how the anisotropic radii in the two-annulus domain enable independent coefficient extraction in each characteristic direction.

Consider the D-holomorphic function $f(z) = \frac{1}{(u-1)(u+2)}e_+ + \frac{1}{(v+1)^2}e_-$ on a two-annulus $\mathcal{W}(0; r_\pm, R_\pm)$ with: $-F(u) = \frac{1}{(u-1)(u+2)}$ (simple poles at $u = 1, -2$) $-G(v) = \frac{1}{(v+1)^2}$ (double pole at $v = -1$)

The domain requirements are: - For the $u$-channel: $r_+ < \min\{1, 2\} = 1$ and $R_+ > \max\{1, 2\} = 2$ - For the $v$-channel: $r_- < 1$ and $R_- > 1$

This allows choosing, for example, $r_+ = 0.5$, $R_+ = 3$, $r_- = 0.3$, $R_- = 1.5$.
**Laurent Expansion for $F(u)$:** Using partial fractions: $F(u) = \frac{1}{3} \left( \frac{1}{u-1} - \frac{1}{u+2} \right)$
Around $u = 0$:

$$ \frac{1}{u-1} = -1 - u - u^2 - u^3 - \dots \qquad (20) $$

$$ \frac{1}{u+2} = \frac{1}{2} - \frac{u}{4} + \frac{u^2}{8} - \frac{u^3}{16} + \dots \qquad (21) $$

Therefore: $F(u) = -\frac{1}{6} - \frac{5u}{12} - \frac{7u^2}{24} - \dots$

**Laurent Expansion for $G(v)$:** $G(v) = \frac{1}{(v+1)^2}$ around $v = 0$:

$$ G(v) = 1 - 2v + 3v^2 - 4v^3 + \dots $$

**Double Laurent Series:**

$$ f(z) = \left( -\frac{1}{6} - \frac{5u}{12} - \frac{7u^2}{24} - \dots \right) e_+ + (1 - 2v + 3v^2 - \dots) e_- \qquad (22) $$

$$ = \left( -\frac{1}{6} - \frac{5(x + y)}{12} - \frac{7(x + y)^2}{24} - \dots \right) e_+ \qquad (23) $$

$$ + (1 - 2(x - y) + 3(x - y)^2 - \dots) e_- \qquad (24) $$

**Key Point:** The radii $(r_+, R_+) = (0.5, 3)$ and $(r_-, R_-) = (0.3, 1.5)$ are completely independent, allowing different convergence regions for each characteristic direction. This anisotropy is impossible in level-set approaches that require $|z|_M^2 = uv = \text{const.}$

# 6 Causal Green's Function in D-Form

We now develop the correct retarded Green's function for the wave equation, addressing the causal structure that underlies wave propagation phenomena. We maintain consistent sign conventions throughout.

Let $c > 0$, $(x, t) \in \mathbb{R}^2$, and $u = x + ct$, $v = x - ct$. The physical wave operator is

$$ \Box = \frac{1}{c^2} \partial_t^2 - \partial_x^2 = -4 \partial_u \partial_v. $$

**Theorem 11** (Retarded Fundamental Solution and D-Split). *The distribution*

$$ G_R(x, t) = \frac{1}{2c} H(t) [\delta(x - ct) + \delta(x + ct)] $$

satisfies $\Box G_R = \delta_{(0,0)}$ in $D'(\mathbb{R}^2)$ and $\text{supp } G_R \subset \{t \ge 0, |x| = ct\}$ (future light rays). Equivalently, in $(u, v)$-coordinates

$$ G_R(u, v) = \frac{1}{2c} H\left(\frac{u-v}{2c}\right) [\delta(v) + \delta(u)]. $$
For a compactly supported source $\phi \in \mathcal{D}(\mathbb{R}^2)$, the retarded solution $w = G_R * \phi$ can be written in the D-split as

$$w(x,t) = \frac{1}{2c} H(t) (\phi(\cdot, \cdot) * \delta(v))(x,t) e_+ + \frac{1}{2c} H(t) (\phi(\cdot, \cdot) * \delta(u))(x,t) e_-, $$

which makes explicit the propagation along the two characteristics.

**Proof.** For any test function $\psi \in \mathcal{D}(\mathbb{R}^2)$, write $\langle \Box G_R, \psi \rangle = \langle G_R, \Box \psi \rangle$. Change variables to $(u, v)$ where $du dv = 2c dx dt$. Using $\partial_u \partial_v \psi = -\frac{1}{4} \Box \psi$ and the identities $\partial_u \delta(u) = \delta'(u)$, $\partial_v \delta(v) = \delta'(v)$, the action of $G_R$ evaluates $\psi$ at the average of the two null-direction traces at the origin, yielding $\psi(0, 0)$. The support statement is immediate from $H(t)\delta(x \mp ct)$. $\square$

**Remark 1** (Alternative Causal Fundamental Solution). Another causal choice is $\tilde{G}_R(u, v) = \frac{c}{2}H(u)H(-v)$, for which $-4 \partial_u \partial_v \tilde{G}_R = \delta_{(0,0)}$. Here supp $\tilde{G}_R = \{t \ge 0, |x| \le ct\}$ (closed future cone interior). The two solutions differ by the homogeneous term $\tilde{G}_R(u, v) - G_R(u, v) = \frac{c}{2}H(u)H(-v) - \frac{1}{2c}H(\frac{u-v}{2c})[\delta(v) + \delta(u)]$, which satisfies $\Box(\text{difference}) = 0$. Both are physically valid, but Theorem 11 matches the standard light-ray representation used in wave scattering.

# 7 Characteristic Residues, Interface Transfer, and Jump Laws

Having established the wave physics essence of D-holomorphic theory, we now provide concrete theorems that make this connection rigorous and computable. The key insight is that the idempotent decomposition allows us to develop residue calculus, scattering theory, and Green's function methods that directly exploit the characteristic structure.

Throughout we use the split-complex algebra $D = \{x + jy : j^2 = 1\}$ with idempotents $e_{\pm} = \frac{1}{2}(1 \pm j)$. Introduce characteristic coordinates

$$u = x + ct, \quad v = x - ct, \quad z = ue_+ + ve_-, \quad dz = e_+ du + e_- dv.$$

Differential operators factor as

$$\partial_x = \partial_u + \partial_v, \quad \partial_t = c(\partial_u - \partial_v), \quad \partial_{\bar{z}} = \frac{1}{2}(\partial_x - \frac{1}{c}j \partial_t) = e_- \partial_u + e_+ \partial_v,$$

and D-holomorphicity means $f = F(u)e_+ + G(v)e_-$ for some one-variable $F, G$.

## 7.1 Formal Residue-Jump Law

We first establish the precise distributional formulation of characteristic residues and their connection to primitive jumps.

**Definition 7** (Line Pole and Characteristic Residue). If $F(u)$ admits the decomposition $F(u) = \frac{A}{u} + h(u)$ near $u = 0$ with $h \in C^0$, we say $f$ has a line pole of order 1 along $u = 0$ with right-moving
characteristic residue

$$ \mathrm{Res}_{u=0} f := Ae_{+}. $$

Analogously, if $G(v) = \frac{B}{v} + k(v)$ near $v = 0$, we define the left-moving characteristic residue $\mathrm{Res}_{v=0} f := Be_{-}$.

**Definition 8** (Primitive Jump Across Characteristics). A distributional primitive $P(u)$ of $F(u)$ across $u = 0$ satisfies $\partial_u P = F$ in $D'(\mathbb{R})$. We call

$$ [P]_{u=0} := \lim_{\epsilon \downarrow 0} (P(\epsilon) - P(-\epsilon)) $$

the jump of $P$ across $u = 0$.

**Theorem 12** (Residue-Jump Law). If $F(u) = \frac{A}{u} + h(u)$ near $u = 0$ in the principal value sense, then there exists a primitive $P$ such that

$$ [P]_{u=0} = A, $$

and this value depends only on the coefficient of $1/u$. Equivalently, for $f = F(u)e_+ + G(v)e_-$, the following are equivalent for the $u$-channel:

1. The coefficient of $1/u$ in $F(u)$ equals $A$ (definition of $\mathrm{Res}_{u=0} f$).

2. Any causal primitive $P$ of $F$ has jump $[P]_{u=0} = A$.

*Proof sketch.* Write $F(u) = \mathrm{p.v.} \frac{A}{u} + h(u)$. Choose a distributional decomposition $F = \partial_u [AH(u) + H_0(u)]$ with $H'_0 = h$. Then a primitive is $P(u) = AH(u) + H_0(u)$, giving $[P]_{u=0} = A$. Different choices of $H_0$ (smooth) do not affect the jump. $\square$

We use the standard hyperfunction boundary value on $\mathbb{R}$:

$$ \left(\frac{1}{u-\alpha}\right)_B = \mathrm{p.v.} \frac{1}{u-\alpha} \mp i\pi \delta(u-\alpha), $$

where the sign corresponds to the approach wedge and $i$ is the standard complex imaginary unit used in boundary-value regularization for hyperfunctions (not living in $D$).

**Theorem 13** (Characteristic Residue on $u = \alpha$). Let $\Omega \subset \mathbb{R}^2$ be open. Assume in a neighborhood of $u = \alpha$,

$$ f(z) = \left(\frac{A}{u-\alpha}\right)_B e_+ + H(z), $$

with $H$ D-holomorphic. Let $\Gamma \subset \Omega$ be a small $u-v$ rectangle that crosses the line $u = \alpha$ once and no $v$-singularity. Then

$$ \oint_{\partial\Gamma} f dz = 2\pi i A e_+. $$
Proof sketch. Since $dz = e_+ du + e_- dv$ and $e_+e_- = 0$,

$$ \oint_{\partial\Gamma} f dz = e_+ \oint_{\partial\Gamma} F(u) du + e_- \oint_{\partial\Gamma} G(v) dv, $$

with $F(u) = (A/(u-\alpha))_B + \tilde{F}(u)$ and $G$ regular near $\Gamma$. The $G$-term vanishes. Write $F = \partial_u P$ in distributions, with $P(u) = A \log(u-\alpha)$ (branch chosen in a wedge). Along the two $v$-constant edges, $\int F du$ equals the boundary increment of $P$; the opposite orientations cancel the regular part but add the jump $2\pi i A$ from crossing the branch cut once. Thus $\oint_{\partial\Gamma} F du = 2\pi i A$ and the claim follows. $\square$

**Theorem 14 (Jump of a Primitive).** Under the hypotheses of Theorem 13, let $P'(u) = F(u)$ in the sense of tempered distributions on $\mathbb{R}$. Then the distributional jump across $u = \alpha$ is

$$ [P]_{u=\alpha} := \lim_{\epsilon \downarrow 0} (P(\alpha + \epsilon) - P(\alpha - \epsilon)) = 2\pi i A. $$

Equivalently, $F$ contains $(A/(u-\alpha))_B$ and $F'$ has a $\delta'$ component of weight $\pm i\pi A$.

Proof sketch. Plemelj-Sokhotski for $P(u) = A \log(u-\alpha)$ gives the stated jump; differentiating yields the $\delta'$ statement. $\square$

**Remark 2 (Physical Reading).** The $u$-channel residue injects a finite jump in a chiral primitive (a Riemann invariant). The $v$-channel is analogous with $u \leftrightarrow v$, $e_+ \leftrightarrow e_-$.

## 7.2 Interface Transfer in Idempotent Variables and Energy Balance

Consider a 1D wave medium (string/acoustics) with density $\rho$, tension/modulus $T$ and speed $c = \sqrt{T/\rho}$; define impedance $Z = \rho c$. Let $p = w_t$ and $q = c w_x$, and set chiral invariants $r := p + q$ (right-moving), $\ell := p - q$ (left-moving). Energy density and flux are

$$ \mathcal{E} = \frac{\rho}{2}(p^2 + q^2) = \frac{\rho}{4}(r^2 + \ell^2), \quad \mathcal{S} = Tw_x w_t = \rho c pq = \frac{Z}{4}(r^2 - \ell^2). $$

**Theorem 15 (Idempotent Interface Transfer).** Let $x = 0$ separate $(\rho_1, c_1, Z_1)$ for $x < 0$ and $(\rho_2, c_2, Z_2)$ for $x > 0$. Impose the interface conditions (i) $w$ is continuous, (ii) $Tw_x$ is continuous (equivalently $Zq$ continuous). Assume a right-moving incidence from the left: $r_1^{\text{in}}$ given, $\ell_2 \equiv 0$. Then

$$ \ell_1 = \mathcal{R} r_1^{\text{in}}, \quad r_2 = \mathcal{T} r_1^{\text{in}}, \quad \mathcal{R} = \frac{Z_1 - Z_2}{Z_1 + Z_2}, \quad \mathcal{T} = \frac{2Z_1}{Z_1 + Z_2}. $$

Moreover, energy-flux conservation holds pointwise in time:

$$ \frac{Z_1}{4} (r_1^{\text{in}})^2 = \frac{Z_1}{4} \ell_1^2 + \frac{Z_2}{4} r_2^2. $$

This is the standard result from 1D wave scattering theory [13], naturally expressed in idempotent variables.
Proof. At $x = 0$, write $p_i = \frac{1}{2}(r_i + \ell_i)$ and $q_i = \frac{1}{2}(r_i - \ell_i)$. Continuity of $w$ implies continuity of $p$; continuity of $Tw_x$ gives $Zq$ continuous: $r_1 + \ell_1 = r_2 + \ell_2$, $Z_1(r_1 - \ell_1) = Z_2(r_2 - \ell_2)$. With $\ell_2 = 0$ and $r_1 = r_1^{\text{in}}$, solve $r_2 = r_1 + \ell_1$ and $Z_1(r_1 - \ell_1) = Z_2(r_1 + \ell_1)$ to get the stated $\mathcal{R}, \mathcal{T}$. Energy identity follows by direct substitution into $S = \frac{Z}{4}(r^2 - \ell^2)$. $\square$

**Remark 3 (Sign Conventions).** The sign of $\mathcal{R}$ depends on which field is normalized as the amplitude (displacement/velocity/pressure). In idempotent variables $(r, \ell)$ as defined above, $\mathcal{R} = (Z_1 - Z_2)/(Z_1 + Z_2)$; for pressure-like amplitudes the familiar form $(Z_2 - Z_1)/(Z_1 + Z_2)$ appears. The flux identity is invariant to this choice.

## 7.3 Computed Example: Gaussian Pulse Scattering at an Interface

Take $Z_1 = 1$, $Z_2 = 2$ and an incident right-moving Gaussian $r_1^{\text{in}}(t) = e^{-t^2}$. From Theorem 15,

$$ \mathcal{R} = \frac{Z_1 - Z_2}{Z_1 + Z_2} = -\frac{1}{3}, \qquad \mathcal{T} = \frac{2Z_1}{Z_1 + Z_2} = \frac{2}{3}, $$

so $\ell_1(t) = -\frac{1}{3}e^{-t^2}$, $r_2(t) = \frac{2}{3}e^{-t^2}$. Instantaneous fluxes at $x = 0$ are

$$ S_{\text{in}}(t) = \frac{Z_1}{4}(r_1^{\text{in}})^2 = \frac{1}{4}e^{-2t^2}, \qquad (25) $$

$$ S_{\text{refl}}(t) = \frac{Z_1}{4}\ell_1^2 = \frac{1}{4}\cdot\frac{1}{9}e^{-2t^2} = \frac{1}{36}e^{-2t^2}, \qquad (26) $$

$$ S_{\text{trans}}(t) = \frac{Z_2}{4}r_2^2 = \frac{2}{4}\cdot\frac{4}{9}e^{-2t^2} = \frac{2}{9}e^{-2t^2}, \qquad (27) $$

and $S_{\text{in}}(t) = S_{\text{refl}}(t) + S_{\text{trans}}(t)$ holds pointwise. Integrating over time (using $\int_{\mathbb{R}} e^{-2t^2} dt = \sqrt{\pi}/2$) yields the same identity for total energies.

### Energy Flux Conservation in Interface Scattering

<figure><img src="image_5.png" /></figure>

Energy conservation: $S_{\text{in}} = S_{\text{refl}} + S_{\text{trans}}$. Verified pointwise for Gaussian pulse scattering

**Figure 4:** Energy flux conservation for Gaussian pulse scattering at an impedance interface (evaluated at $x = 0$). The incident flux (blue) exactly balances reflected (red) and transmitted (green) fluxes at all times, demonstrating the energy balance in Theorem 15.
## 7.4 Validation of the Wave Physics Framework

These concrete theorems validate the philosophical insights developed in previous sections. The two-annulus Cauchy theory (Theorems 8-9) provides the first systematic reconstruction and residue calculus for D-holomorphic functions on anisotropic domains, confirming that the characteristic coordinate structure enables genuinely new mathematical tools. The corrected causal Green's function (Theorem 11) demonstrates proper wave propagation with retarded support, while the interface transfer analysis (Theorem 15) shows how classical wave scattering emerges naturally from the idempotent decomposition.

Most significantly, these results show that the apparent "failures" of classical complex analysis in the D-setting are not mathematical defects, but rather the emergence of the correct mathematical structures for wave physics. The residues live on characteristic lines rather than isolated points, the Green's function naturally incorporates proper causal structure, and energy balance emerges from the fundamental algebraic operations. The two-annulus theory provides systematic tools that exploit rather than circumvent the characteristic geometry.

# 8 Proposed Approaches

We have established concrete theorems for characteristic residues, interface transfer, and Green's functions in the previous section. Here we outline additional strategies to complete the development of D-holomorphic function theory, building systematically on the geometric foundations established by Luna-Elizarraras et al. [8], particularly their hyperbolic holomorphicity hull construction and the insight that zero propagation along characteristic intersections reveals the deep geometric structure underlying D-holomorphic theory.

## 8.1 Stratified Regular Analysis

**Strategy:** Building on our characteristic residue theory, develop complete Cauchy reconstruction formulas on the regular part $D^* = \{z: u \neq 0 \text{ and } v \neq 0\}$, then systematically extend to handle zero divisor singularities using hyperfunction boundary values.

**Example:** Consider $F(z) = \frac{z^2}{z^2+1}$. In the complex case, this has poles at $z = \pm i$. In the D case, the denominator $z^2 + 1 = (x + iy)^2 + 1 = x^2 + y^2 + 1 + j(2xy)$ vanishes only when both $x^2 + y^2 + 1 = 0$ (impossible since $x^2 + y^2 + 1 > 0$) and $2xy = 0$. Hence $F(z)$ is well-defined on all of $D^*$ with potential branching behavior only at zero divisor lines.

## 8.2 Hyperfunction Framework

**Strategy:** Employ Sato's hyperfunction theory to handle singularities at zero divisors.
**Definition 9.** Define the space $\mathcal{B}_D(\Omega)$ of D-hyperfunctions on $\Omega$ as boundary values of D-holomorphic functions defined in appropriate wedge domains.

**Precise Construction:** The Cauchy kernel decomposes as:

$$ \frac{1}{z - \zeta} = \frac{1}{(u - \alpha)e_+ + (v - \beta)e_-} = \frac{1}{u - \alpha}e_+ + \frac{1}{v - \beta}e_- \quad (\text{when } u \neq \alpha, v \neq \beta) \quad (28) $$

where $\zeta = \alpha e_+ + \beta e_-$. Each 1D component is interpreted as a hyperfunction in the respective characteristic coordinate.

## 8.3 Sheaf-Theoretic Approach

**Strategy:** Formulate D-holomorphic functions as sections of a sheaf over appropriate base spaces.

**Connection to Concrete Results:** The sheaf-theoretic perspective beautifully unifies our concrete theorems with the geometric insights of Luna-Elizarraras et al. [8]. The "jump" captured by our characteristic residue theorem (Theorem 13) is a manifestation of non-trivial first cohomology $H^1(\Omega, \mathcal{O}_D)$ of the sheaf $\mathcal{O}_D$ of D-holomorphic function germs on a domain $\Omega$ that is "punctured" by characteristic lines.

**Sheaf Construction:**

* Base space: $D \setminus \{\text{zero divisor lines}\}$

* Fiber over each point: germs of D-holomorphic functions

* Transition functions: analytic continuation across "cuts" between zero divisor regions

**Cohomological Interpretation:** For $\Omega = D \setminus \{u = 0, v \ge 0\}$ (a characteristic slit), the cohomology $H^1(\Omega, \mathcal{O}_D) \cong \mathbb{R}$, generated by the $e_+$-cocycle corresponding to the jump of the $u$-component across the characteristic line $u = 0$. This topological invariant directly corresponds to the residue coefficient $A$ in our Theorem 13.

# 9 Applications and Physical Relevance

The development of D-holomorphic function theory has significant applications across multiple domains:

## 9.1 Wave Physics and Hyperbolic PDEs

* **D'Alembert Solutions:** New solution techniques for wave equations, Klein-Gordon equations, and nonlinear wave systems [15, 13]
* **Characteristic Methods:** Systematic approaches to initial value problems along characteristic curves

* **Shock Wave Analysis:** Mathematical tools for analyzing discontinuous solutions and their propagation [13]

## 9.2 Mathematical Physics Applications

* **Fluid Dynamics:** Analysis of supersonic flows and shock wave formation [13]

* **Electromagnetic Theory:** Wave propagation in inhomogeneous media with field discontinuities [14]

* **General Relativity:** Signature-changing spacetimes and cosmological models with phase transitions [16]

* **Quantum Field Theory:** Duality relationships between elliptic and hyperbolic systems in field theoretic contexts

## 9.3 Computational and Numerical Methods

* **Parallel Algorithms:** The idempotent structure enables natural parallelization along characteristic coordinates

* **Stable Integration:** Numerical schemes that respect the hyperbolic structure and characteristic flow

* **Boundary Conditions:** Natural handling of wave-type boundary conditions using hyperfunction methods

# 10 Conclusion and Future Directions

Building on the breakthrough identity theory results of Luna-Elizarraras et al. [8], we have established both the conceptual foundation and concrete mathematical machinery for D-holomorphic function theory as the native language of hyperbolic wave physics. Their geometric insights into hyperbolic holomorphicity hulls and characteristic line behavior provided the essential framework that guided our development of residue calculus, interface transfer theorems, and systematic wave physics applications.

**Core Theoretical Results:** We have proven characteristic residue theorems (Theorem 13) that provide a rigorous foundation for integration on zero divisor lines, jump laws (Theorem 14) that connect residues to primitive discontinuities, and interface transfer theorems (Theorem 15)
that demonstrate pointwise energy conservation in wave scattering. These results transform the intuitive wave physics interpretation into precise mathematical statements with proofs.

**Physical Applications:** The interface scattering analysis provides a concrete demonstration of how the idempotent decomposition naturally handles classical wave physics. While the scattering laws themselves are standard [13, 14], their automatic emergence from the D-holomorphic algebraic structure illustrates the deep connection between the mathematical formalism and physical wave phenomena.

**Computational Framework:** Our retarded Green kernel (Theorem 11) and the accompanying numerical implementations demonstrate that these theoretical insights translate into practical computational tools. The Gaussian pulse scattering example provides quantitative validation of the energy balance predictions.

Where complex analysis provides the natural setting for elliptic phenomena, we have established that D-holomorphic analysis *is* hyperbolic wave physics, complete with its own residue calculus, reconstruction formulas, and Green's function theory. The zero divisors encode characteristic curves, the algebraic operations reflect physical causality constraints, and every mathematical structure has a direct physical interpretation.

The most promising directions for future research include:

1. **Complete function theory:** Develop characteristic Cauchy reconstruction formulas with stability bounds, extending our residue theory to a full calculus

2. **Higher-dimensional extensions:** Generalize to Clifford algebras $Cl(p, q)$ for higher-dimensional wave systems and Yang-Mills theory

3. **Nonlinear applications:** Apply D-holomorphic methods to integrable systems, soliton equations, and shock wave formation

4. **Numerical algorithms:** Develop stable computational methods that exploit the characteristic structure for hyperbolic PDE solvers

5. **Quantum field theory:** Explore applications to signature-changing spacetimes and causal structure in curved manifolds

The ultimate significance of this work lies in demonstrating how algebraic structure can directly encode physical law. D-holomorphic function theory reveals that the split-complex numbers are not a mathematical curiosity—they are nature's algebraic language for wave phenomena, where every apparent "failure" of complex analysis manifests as a physical principle of wave propagation.
# Acknowledgments

The author is deeply grateful to Irene Sabadini for providing essential references that shaped the theoretical development of this work, and for her invaluable counsel in persuading me that I needed to dig much deeper into D-regularity before advocating a general notion of slice regularity which includes D-slices. Her insights were instrumental in focusing this research on the fundamental wave physics essence of D-holomorphic functions rather than pursuing premature generalizations.

The author also acknowledges the foundational geometric contributions of Luna-Elizarraras et al. [19] in establishing the modern geometric understanding of D-holomorphic functions through their hyperbolic holomorphicity hull construction and analysis of zero propagation along characteristic intersections. Their geometric insights provided the conceptual framework that guided the development of our residue theory and systematic wave physics interpretation.

# A Rosetta Table: Complex vs. D-Analysis

The following table provides a comprehensive comparison between classical complex analysis and D-holomorphic theory, highlighting both analogies and fundamental differences.

<table><caption>Table 1: Rosetta table: classical vs. D-holomorphic features.</caption><thead><tr><th>Complex Analysis (C)</th><th>D-Analysis (Split-Complex)</th><th>Status</th><th>Key note / formula</th></tr></thead><tbody><tr><td>Cauchy-Riemann:<br>u<sub>x</sub> = v<sub>y</sub>, u<sub>y</sub> = -v<sub>x</sub><br>Δ = 4 ∂<sub>z</sub>∂<sub>z&#773;</sub></td><td>D-CR:<br>u<sub>x</sub> = v<sub>y</sub>, u<sub>y</sub> = +v<sub>x</sub><br>□ = ∂<sub>x</sub><sup>2</sup> - ∂<sub>y</sub><sup>2</sup> = 4 ∂<sub>z</sub>∂<sub>z&#773;</sub></td><td>Analogue<br>(sign flip)<br>Analogue<br>Analogue</td><td>i<sup>2</sup> = -1 ⇒ j<sup>2</sup> = +1<br>Elliptic ↔ Hyperbolic<br>u<sub>xx</sub> - u<sub>yy</sub> = 0, v<sub>xx</sub> - v<sub>yy</sub> = 0</td></tr><tr><td>Holomorphic ⇒ harmonic</td><td>D-holomorphic ⇒ wave solutions</td><td>Analogue</td><td></td></tr><tr><td>Cauchy integral (point pole)</td><td>Characteristic jump (line pole)</td><td>Replacement</td><td>If F(u) ~ A/(u - α) then jump of primitive across u = α equals A</td></tr><tr><td>Residue theorem (sum over points)</td><td>Line-residue sum on crossed characteristics</td><td>Replacement</td><td>Channel-wise integrals in u and v</td></tr><tr><td>Identity theorem (limit point)</td><td>Limited identity</td><td>Modified</td><td>Two transverse curves (one u, one v) fix both channels</td></tr><tr><td>Liouville (bounded entire ⇒ const.)</td><td>Fails</td><td>Different</td><td>e.g. tanh(u)e<sub>+</sub> + tanh(v)e<sub>-</sub> bounded non-constant</td></tr><tr><td>Taylor/Laurent at a point</td><td>Two 1-D series in u, v</td><td>Modified</td><td>Two-annulus Cauchy, double Laurent</td></tr><tr><td>Conformal maps</td><td>Pseudo-conformal (light-cone preserving)</td><td>Analogue</td><td>Preserves characteristic structure</td></tr></tbody></table>

## Key Observations:

* **Elliptic vs. Hyperbolic:** The change from $i^2 = -1$ to $j^2 = +1$ fundamentally shifts from elliptic (harmonic) to hyperbolic (wave) behavior.
* **Point vs. Line Support:** Complex singularities are point-supported; D-holomorphic singularities are line-supported along characteristics.

* **Conservation Laws:** Energy/momentum conservation emerges naturally in D-holomorphic theory through the idempotent structure.

* **Causality Structure:** D-holomorphic functions naturally encode wave propagation and causal constraints.

* **Physical Interpretation:** Where complex analysis describes equilibrium phenomena, D-holomorphic analysis describes dynamical wave phenomena.

This comparison reveals that D-holomorphic theory is not simply "complex analysis with $j^2 = +1$"—it is a fundamentally different mathematical framework naturally suited to hyperbolic physics.

# B Computational Exploration Library

To make the wave physics essence of D-holomorphic theory accessible and interactive, we have developed a Python library that allows direct exploration of the theoretical concepts presented in this paper. The library is available at:

https://github.com/Mircus/CliffordLab/tree/main

## B.1 Library Design Philosophy

The library is designed around the central physical insight of this paper: D-holomorphic functions are wave physics, not complex analysis with complications. Every class and method embodies this perspective:

* **Zero divisors** are implemented as wave characteristic detectors

* **D-holomorphic functions** automatically factorize into left/right-moving wave components

* **Visualization tools** emphasize information propagation along characteristics

* **Failed operations** are presented as physical causal constraints rather than mathematical errors

## B.2 Core Implementation

The library consists of three main classes that directly encode the theoretical framework:
### B.2.1 SplitComplex Class

```python
class SplitComplex:
    """Split-complex number z = x + jy where j² = +1
```

This directly encodes wave physics through its algebraic structure.
Zero divisors correspond to characteristic curves where waves propagate.
`"""`

```python
def is_zero_divisor(self, tol=1e-12):
    """Check if on wave characteristic (y = ±x)"""
    return abs(self.norm squared()) < tol
```

```python
@property
def characteristic�(self):
    """Return (u, v) = (x+y, x-y) wave coordinates"""
    return self.real + self.imag, self.real - self.imag
```

### B.2.2 DHolomorphic Class

Every D-holomorphic function automatically has the wave solution form:

```python
class DHolomorphic:
    "D-holomorphic function f(z) = F(u)*e + G(v)*e
```

```python
F(u): left-moving wave component along u = x+y = constant
G(v): right-moving wave component along v = x-y = constant
"""
```

```python
def _call_(self, z):
    u, v = z characteristic�
    # Evaluate wave components
    F_val = self.F(u)  # Left-moving wave
    G_val = self.G(v)  # Right-moving wave
    # Convert back from idempotent form
    return SplitComplex((F_val + G_val)/2, (F_val - G_val)/2)
```

### B.2.3 Visualization Tools

The DPlane class provides wave-physics-focused visualizations with methods for plotting zero divisors as characteristic curves and showing wave information propagation.
## B.3 Key Explorations Enabled

The library makes the theoretical insights of this paper directly explorable through wave physics discovery, wave solution exploration, and physical intuition development.

## B.4 Educational Impact

The library transforms the theoretical insights of D-holomorphic function theory into hands-on exploration, allowing users to visualize wave characteristics, experience causal constraints, create wave solutions, and explore the physical meaning of every mathematical structure.

To make the wave physics essence of D-holomorphic theory accessible and interactive, we have developed a Python library that allows direct exploration of the theoretical concepts presented in this paper. The library is available at:

https://github.com/Mircus/CliffordLab/tree/main

## B.5 Library Design Philosophy

The library is designed around the central physical insight of this paper: D-holomorphic functions ARE wave physics, not complex analysis with complications. Every class and method embodies this perspective:

* **Zero divisors** are implemented as wave characteristic detectors

* **D-holomorphic functions** automatically factorize into left/right-moving wave components

* **Visualization tools** emphasize information propagation along characteristics

* **Failed operations** are presented as physical causal constraints rather than mathematical errors

## B.6 Core Implementation

The library consists of three main classes that directly encode the theoretical framework:

### B.6.1 SplitComplex Class

<caption>Listing 1: Split-complex core implementation</caption>

```python
class SplitComplex:
    Split-complex number z = x + jy where j² = +1

    This directly encodes wave physics through its algebraic structure.
    Zero divisors correspond to characteristic curves where waves propagate.
```
```python
def is_zero_divisor(self, tol=1e-12):
    """Check if on wave characteristic (y = ±x)
    return abs(self.norm squared()) < tol

@property
def characteristic�ers(self):
    """Return (u, v) = (x+y, x-y) wave coordinates"""
    return self.real + self.imag, self.real - self.imag
```

### B.6.2 DHolomorphic Class

Every D-holomorphic function automatically has the wave solution form:

<caption>Listing 2: D-holomorphic function implementation</caption>

```python
class DHolomorphic:
    """ D-holomorphic function f(z) = F(u)*e_+ + G(v)*e_-"
    F(u): left-moving wave component along u = x+y = constant
    G(v): right-moving wave component along v = x-y = constant
    """

    def __call__(self, z):
        u, v = z characteristic�ers
        # Evaluate wave components
        F_val = self.F(u)  # Left-moving wave
        G_val = self.G(v)  # Right-moving wave
        # Convert back from idempotent form
        return SplitComplex((F_val + G_val)/2, (F_val - G_val)/2)
```

### B.6.3 Visualization API

The DPlane class provides wave-physics-focused visualizations with methods for plotting zero divisors as characteristic curves and showing wave information propagation.

## B.7 Key Explorations Enabled

The library makes the theoretical insights of this paper directly explorable through wave physics discovery, wave solution exploration, and physical intuition development.
## B.8 Educational Impact

The library transforms the theoretical insights of D-holomorphic function theory into hands-on exploration, allowing users to visualize wave characteristics, experience causal constraints, create wave solutions, and explore the physical meaning of every mathematical structure.

## References

[1] J. Cockle, *On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra*, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 33(3) (1848), 435-439.

[2] J. Cockle, *On a new imaginary in algebra*, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 34(3) (1849), 37-47.

[3] I.M. Yaglom, *Complex Numbers in Geometry*, Academic Press, 1968.

[4] G. Sobczyk, *The hyperbolic number plane*, College Math. J. 26 (1995), 268-280.

[5] F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti, *The Mathematics of Minkowski Space-Time with an Introduction to Commutative Hypercomplex Numbers*, Birkhäuser, 2008.

[6] V. Cruceanu, P. Fortuny, P.M. Gadea, *A survey on paracomplex geometry*, Rocky Mountain J. Math. 26 (1996), 83-115.

[7] M. Shapiro, D.C. Struppa, A. Vajiac, M.B. Vajiac, *Hyperbolic numbers and their functions*, Analele Universității Oradea, Fasc. Matematica XIX (2012), 265-283.

[8] M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa, A. Vajiac, *Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers*, Frontiers in Mathematics, Birkhäuser, 2015.

[9] M. Sato, *Theory of hyperfunctions*, J. Fac. Sci. Univ. Tokyo Sect. I, 8 (1959), 139-193.

[10] M. Kashiwara, T. Kawai, T. Kimura, *Foundations of Algebraic Analysis*, Princeton University Press, 1986.

[11] G.B. Price, *An Introduction to Multicomplex Spaces and Functions*, Marcel Dekker, 1991.

[12] M. Libine, *Hyperbolic Cauchy integral formula for the split complex numbers*, arXiv:0712.0375, 2007.

[13] G.B. Whitham, *Linear and Nonlinear Waves*, John Wiley & Sons, 1974.
[14] J.D. Jackson, *Classical Electrodynamics*, 3rd Edition, John Wiley & Sons, 1999.

[15] R. Courant, D. Hilbert, *Methods of Mathematical Physics, Volume II: Partial Differential Equations*, Wiley-Interscience, 1962.

[16] S. Weinberg, *Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity*, John Wiley & Sons, 1972.

[17] J. le Rond d'Alembert, *Recherches sur la courbe que forme une corde tendue mise en vibration*, Histoire de l'Académie Royale des Sciences et Belles Lettres de Berlin, 3 (1747), 214-219.

[18] M. Luna-Elizarrarás, M. Shapiro, D. Struppa, and A. Vajiac, *Bicomplex Holomorphic Functions*, Birkh"auser, 2015.

[19] M. E. Luna-Elizarrarás, M. Shapiro, D. C. Struppa, and A. Vajiac, *Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers*, Birkh"auser, 2015. ISBN: 978-3-319-24866-0. doi:10.1007/978-3-319-24868-4.