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# Dimensional Analysis and Complex Analysis: An
Analogy via Singularity Types

Wen-Xiang Chen

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Wen-Xiang Chen. Dimensional Analysis and Complex Analysis: An Analogy via Singularity Types. 2025. hal-05161852

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Distributed under a Creative Commons Attribution 4.0 International License
Dimensional Analysis and Complex Analysis: An
Analogy via Singularity Types

Wen-Xiang Chen

School of Physics and Materials Science

Guangzhou University

wxchen4277@qq.com

July 15, 2025

Abstract

This paper establishes a conceptual correspondence between dimensional analysis in physics and singularity analysis in complex analysis. We demonstrate that translating dimensional analysis into the language of complex analysis fundamentally involves classifying singularity types. Dimensional analysis, which governs physical units and dimensionless combinations, is shown to exhibit deep structural parallels with the behavior of complex functions near singular points (poles, zeros, essential singularities, branch points). Key analogies include: (1) Physical divergences/zeros correspond to poles/zeros in complex functions, (2) Dimensionless parameters (e.g., Reynolds number) govern physical regimes similarly to Laurent series exponents controlling singularity behavior, (3) Dimensional homogeneity mirrors analyticity requirements, and (4) The rank of dimensional matrices corresponds to the order of singularities. Through examples and theoretical analysis, we prove that dimensional inconsistencies represent "non-analytic" points in physical descriptions, while dimensionless groups act as singularity classifiers. This unification provides new insights into scaling laws and critical phenomena.

Introduction

To establish that translating *dimensional analysis* into the language of *complex analysis* is essentially an analysis of singularity types, we draw an analogy between these two domains. Dimensional analysis, a tool in physics dealing with physical units and dimensionless combinations, and complex analysis, a branch of mathematics dealing with complex functions and their singular points, seem unrelated at first. However, by abstracting the concepts, we can identify deep similarities in their methods and interpretations. In what follows, we provide a step-by-step analysis to demonstrate that concepts in dimensional analysis (such as divergence, zeros, and dimensionless parameters) correspond to types of singularities (poles, zeros, essential singularities, branch points) in complex analysis. [1–6]
# 1 Basic Concepts of Dimensional Analysis

Dimensional analysis examines the *dimensions* (or fundamental units) of physical quantities and ensures consistency of units in physical equations. Common base dimensions include mass [M], length [L], and time [T]. For example, a displacement s has dimension [s] = L, velocity v has dimension [v] = LT⁻¹, and acceleration a has [a] = LT⁻². A key principle is **dimensional homogeneity**: every valid physical equation must have the same total dimension on each side. If the dimensions do not match, the equation is physically meaningless (often termed “nonsense” in dimensional analysis). As an illustration, consider Newton’s second law $F = ma$: the force has dimension [F] = MLT⁻², and the product of mass and acceleration is [m] · [a] = M · (LT⁻²) = MLT⁻², so the equation is dimensionally consistent. In contrast, an equation like $s = vt^2$ (displacement = velocity × time²) has dimensions L on the left but LT⁺¹ on the right, which are not equal, signaling a physically invalid formula.

Another central idea is the construction of **dimensionless quantities** (also called similarity parameters or Pi groups). These are combinations of variables whose overall dimension is unity (no net units). For instance, given two independent lengths $L_1$ and $L_2$, one can form the ratio $L_1/L_2$, which is dimensionless (since $[L_1/L_2] = L/L = 1$). Such dimensionless ratios often capture essential physics. A classical example is the Reynolds number in fluid mechanics, defined as [4–6]

$$ \mathrm{Re} = \frac{\rho v L}{\mu}, \qquad (1) $$

where $\rho$ is fluid density ($[\rho] = ML^{-3}$), $v$ is flow speed ($[v] = LT^{-1}$), $L$ is a characteristic length ($[L] = L$), and $\mu$ is fluid viscosity ($[\mu] = ML^{-1}T^{-1}$). Checking units,

$$ [\mathrm{Re}] = \frac{[ML^{-3}][LT^{-1}][L]}{[ML^{-1}T^{-1}]} = M^0 L^0 T^0 = 1, \qquad (2) $$

so Re is indeed dimensionless. The Reynolds number’s value predicts the nature of fluid flow: low Re indicates smooth laminar flow, whereas high Re indicates turbulent flow. In general, dimensional analysis uses such dimensionless groups to characterize regimes of behavior and to ensure that empirical or theoretical formulas capture the essence of the physics without explicit unit dependence. Importantly, any argument to transcendental functions like exponential, trigonometric, or logarithmic functions must also be dimensionless for the expression to be physically meaningful.

# 2 Singularity Types in Complex Analysis

In complex analysis, a primary concern is the behavior of complex-valued functions $f(z)$ near points where they are not analytic. These points are called **singularities** – locations at which the function ceases to be holomorphic (complex-differentiable). There are several classic types of singularities, characterized by the nature of the function’s blow-up or behavior in their vicinity:

* **Pole:** A pole is an isolated singularity where the function’s magnitude diverges to infinity at the point, typically like a finite negative power of $(z - z_0)$. Formally, $z_0$ is a pole of order $m$ if $f(z)$ behaves as $f(z) \sim (z - z_0)^{-m}$ (more precisely, $(z - z_0)^m f(z)$)
approaches a nonzero finite limit as $z \to z_0$). For example, $f(z) = 1/(z-5)^3$ has a pole of order 3 at $z_0 = 5$, since $(z-5)^3 f(z) = 1$ approaches a non-zero constant. Intuitively, near a pole the function $f(z)$ “blows up” to infinity, analogous to a physical quantity diverging. (In fact, a pole can be defined as a singularity where $f(z) \to \infty$ as $z$ approaches that point.)

* **Zero (of a function):** By contrast, a zero is not usually called a singularity (the function is still analytic there if considered as a zero of finite order), but we mention it for analogy. A zero of order $m$ at $z_0$ means $f(z_0) = 0$ and the first nonzero term in its Taylor expansion about $z_0$ is $(z-z_0)^m$. For instance, if $f(z) = (z-z_0)^2 g(z)$ with $g(z_0) \neq 0$, then $z_0$ is a zero of order 2. Zeros are the opposite of poles: the function vanishes at that point. (By the “duality” in complex analysis, a zero of $f$ is a pole of the function $1/f$.)

* **Essential singularity:** An isolated singularity $z_0$ is called essential if it is neither a pole nor a removable singularity. In this case, the Laurent series of $f$ about $z_0$ contains infinitely many negative-power terms. There is no finite power of $(z-z_0)$ that can multiply $f$ to yield a finite limit; equivalently, $(z-z_0)^m f(z)$ still diverges or oscillates for all positive integers $m$. A classic example is $f(z) = \sin(\frac{1}{z-a})$, which around $z=a$ expands as $\frac{1}{z-a} - \frac{1}{6}(z-a)^{-3} + \frac{1}{120}(z-a)^{-5} - \dots$, containing infinitely many inverse powers. No finite $m$ cancels all singular terms, so $z=a$ is an essential singularity. The behavior near an essential singularity is highly unruly: in fact, by Picard's theorem, in any neighborhood of an essential singularity, $f(z)$ attains “almost” every complex value (with at most one exception) infinitely often. This wild behavior is far more complex than a simple pole or zero.

* **Branch point:** Not all singularities are isolated; a branch point is a point where the function is multi-valued, and one cannot return to a single-valued behavior by encircling the point. In other words, upon going around a branch point, the function can end up on a different “branch” of its multi-valued definition. For example, $f(z) = \sqrt{z}$ has a branch point at $z=0$: if one travels in a loop around the origin, $f(z)$ changes sign instead of returning to the initial value. Branch points often arise from fractional powers or logarithms (e.g. $z^\alpha$ with non-integer $\alpha$ or $\log z$) and necessitate branch cuts (lines or curves emanating from the branch point) to define a single-valued branch of the function. Unlike poles or removable singularities, branch points are not isolated—the failure of single-valued analyticity occurs in any neighborhood of the point, requiring a cut in the complex plane to handle the discontinuity.

Each type of singularity is identified via the Laurent series expansion or limiting behavior of $f(z)$ around the point in question. These singularity types (poles of various orders, zeros, essential singularities, branch points) classify how a complex function can fail to be analytic at a point.

# 3 Analogy: Physical Quantities as Complex Functions

To connect the two realms, we propose to think of a physical quantity (which depends on parameters like time $t$ or position $x$) as analogous to a complex function of a complex
variable $z$. Under this analogy, the behavior of the physical quantity as some parameter approaches a critical value corresponds to the behavior of a complex function near a singular point. We can identify parallels for various cases:

*   **Divergence (blow-up):** If a physical quantity becomes unbounded under certain conditions, it mirrors a pole in complex analysis. For example, consider a simple motion where $x$ is distance and $t$ is time. The speed $v = x/t$ will grow without bound as $t \to 0$ (for fixed $x$). This unbounded growth is akin to saying $v$ has a “pole”-type behavior at $t = 0$. In the complex function analogy, $t = 0$ corresponds to a point where a function like $f(z) = \frac{x}{z}$ diverges as $z \to 0$, a pole of order 1. Indeed, a pole is characterized by the function going to infinity at that point, just as $v$ becomes infinite when the time interval vanishes.

*   **Zero behavior:** Conversely, if a physical quantity becomes zero at some limit, it is analogous to a zero of a complex function. For instance, the displacement $s$ covered in time $t$ at constant acceleration is $s = \frac{1}{2}at^2$. At $t = 0$, we have $s = 0$. One may view this as analogous to a function $f(z)$ having a zero at $z = 0$ (here $s$ vanishes as $t \to 0$ with quadratic order). In complex analysis terms, $f(z) = z^2$ has a zero of order 2 at $z = 0$. The order of the zero (2 in this case) indicates how $f$ approaches zero (quadratically). In physics, the power-law dependence of $s$ on $t$ (here $t^2$) plays a similar role – it indicates how rapidly $s$ goes to zero as $t \to 0$.

*   **Complex critical behavior:** Certain physical systems exhibit abrupt or non-analytic changes at critical parameter values (for example, a phase transition at a critical temperature, or the onset of turbulence at a critical Reynolds number). These can be abstractly compared to essential singularities or branch points, where the behavior of the system is “qualitatively different” on approaching the point from different sides. For example, near a critical point of a phase transition, physical quantities might follow power-law or logarithmic divergences. In a complex plane description, a logarithmic divergence corresponds to a branch point (since $\log(z-z_c)$ is multi-valued around $z_c$), and an infinitely sensitive dependence on initial conditions might resemble the chaos of an essential singularity. While these analogies are more qualitative, they underscore that when a physical formula breaks down or changes form (non-analytic behavior), it plays a role similar to a non-analytic singular point in a complex function.

In summary, we treat the parameter space of a physical problem like the complex plane. Points in parameter space where the physical behavior becomes singular (either unbounded, zero, or multi-valued) correspond to singular points of an analogous complex function. This sets the stage to compare dimensional analysis concepts with singularity analysis.

# 4 Dimensional Consistency vs. Analyticity

A fundamental rule in dimensional analysis is that equations must be *dimensionally consistent*: one can only equate or add quantities with the same dimensions. If an equation is not dimensionally homogeneous, it is considered invalid. This is analogous to the requirement in complex analysis that a function be *analytic* (holomorphic) in a domain for it to represent a well-behaved relationship in that domain. If a function is
not analytic at a point, that point is a singularity and the usual rules of calculus (and physics analogies) break down there. We can draw the parallel that: [1–3]

* A dimensionally inconsistent equation is like a “non-analytic” situation in physics, indicating something fundamentally wrong or undefined (just as an analytic function cannot have inconsistent values). Indeed, a dimensionally nonsensical equation is flagged as “nonsense” by physicists, much as a non-analytic point is singled out as a singularity where the usual laws (of calculus or physics) are violated.

* A dimensionally consistent equation corresponds to a well-defined rule that can hold in a continuum of conditions, analogous to a function being analytic in a region (having a convergent power series expansion there). For example, $s = vt + \frac{1}{2}at^2$ is dimensionally consistent and indeed holds analytically for all $t$ in the model of constant acceleration. If we tried to write $s = vt^2 + \frac{1}{2}at$, the left and right sides have unmatched dimensions and no analytic function could underlie such a relation (it would correspond to $s$ having two different functional dependencies with no common validity). In complex analysis, similarly, if one tried to equate $f(z)$ with a power series that has incompatible terms, it would not represent a single analytic function.

Another aspect of this analogy is the role of dimensionless inputs to functions. In physics, any argument of a transcendental function (like sin or exp) must be dimensionless. This is akin to requiring a pure (unitless) number as input – reminiscent of how complex analysis often non-dimensionalizes variables to simplify behavior near singularities. For instance, in a formula $\sin(at^2/s)$ from earlier, the combination $at^2/s$ must carry no units. If it did, the formula would be as ill-defined as a complex exponential $\exp(z)$ where $z$ carries some ill-defined “unit” – something meaningless mathematically. Thus, the condition of dimensional homogeneity mirrors the condition of analyticity and proper formulation in mathematical functions.

In summary, ensuring dimensional consistency in a physical law is analogous to ensuring analytic consistency (no singularities) in a mathematical description. If dimensions “don’t match”, the physical law is not valid – just as if a function cannot be analytic at a point, that point marks a breakdown of the usual description (a singularity).

# 5 Dimensionless Numbers and Singularity Behavior

A striking parallel emerges when we examine how dimensionless parameters characterize physical behavior and how the exponents in a function's expansion characterize singularities. In dimensional analysis, we often reduce a physical problem to a set of independent dimensionless numbers (via the Buckingham $\pi$ theorem). The values of these numbers can determine the regime of the phenomenon. For example, the Reynolds number $Re$ dictates whether flow is laminar or turbulent, the Strouhal number dictates oscillation patterns, etc. Each dimensionless number is typically a power-law combination of variables. In complex analysis, similarly, the behavior of $f(z)$ near a singular point can often be described by a leading power law or a Laurent series exponent. We can make the analogy more explicit: [4–6]

* Consider a simple power-law function in complex analysis:

$$f(z) = (z - z_0)^{\alpha}, \qquad (3)$$
where $\alpha$ is a constant (not necessarily an integer). The nature of the singularity at $z = z_0$ depends on the value of $\alpha$. If $\alpha$ is a positive integer $m$, then $f(z)$ has a zero of order $m$ at $z_0$ (it vanishes as $(z - z_0)^m$). If $\alpha$ is a negative integer $-n$, then $f(z)$ has a pole of order $n$ at $z_0$ (blowing up as $(z - z_0)^{-n}$). If $\alpha$ is not an integer (say $\alpha = 1/2$ or any non-integer), then $(z - z_0)^\alpha$ is multi-valued – $z_0$ is a branch point of $f$. Thus, the exponent $\alpha$ essentially encodes the “singularity type”: $\alpha > 0$ (integer) = zero, $\alpha < 0$ (integer) = pole, non-integer $\alpha = \text{branch point}$, etc. This classification comes directly from the power-law behavior in the Laurent or Taylor expansion.

* Now consider a physical quantity $Q$ with a dimensional formula composed of base units raised to powers:

$$[Q] = M^a L^b T^c, \qquad (4)$$

where $a, b, c$ are real exponents (which could arise, for instance, from some power-law relation between variables). The values of these exponents determine how $Q$ changes when we rescale the base units. For instance, if we examine how $Q$ behaves when time $T$ is scaled, the exponent $c$ tells us the power-law behavior in time. A negative exponent (say $c = -1$) means $Q$ increases without bound as $T \to 0$ (similar to a pole in time), a positive exponent means $Q \to 0$ as $T \to 0$ (similar to a zero), and a fractional exponent would imply a multi-valued dependence on the sign of the variable (analogous to a branch cut requirement). In this way, the dimensional exponents $\{a, b, c, \dots\}$ play a role analogous to the exponents in a Laurent series: they characterize how the quantity behaves under scaling of each fundamental dimension.

To make this more concrete, let us revisit the example of Reynolds number $Re = \rho v L / \mu$. We can interpret extreme values of this dimensionless combination in singularity terms. As viscosity $\mu \to 0$ (nearly inviscid fluid), $Re \to \infty$ – an infinite Reynolds number can be seen as a “singular” limit corresponding to the onset of turbulent (chaotic) flow. In a complex analytic sense, one might compare $\mu \to 0$ (making $Re \to \infty$) to approaching a pole in parameter space (the flow behavior becomes unboundedly sensitive, akin to a singular behavior). On the other hand, if $v \to 0$ or $\rho \to 0$, then $Re \to 0$, a calm regime analogous to approaching a zero (the flow becomes smoothly ordered, analogous to a function approaching zero). Thus, the dimensionless number’s value qualitatively determines the system’s behavior, just as the exponent in $f(z) = (z - z_0)^\alpha$ determines the singular character of the point $z_0$. In both cases, analyzing the powers (whether of units or of $(z - z_0)$) is key to understanding the nature of the solution.

Furthermore, constructing independent dimensionless groups in a problem is similar to analyzing independent directions in which a function can be singular. For a given physical system with many variables, the Buckingham $\pi$ theorem tells us the number of independent dimensionless parameters equals $n - k$, where $n$ is the number of variables and $k$ is the number of fundamental dimensions involved. This $n - k$ is analogous to the number of independent directions or “degrees” of singular behavior a complex function might have near a point. For example, a function with a pole of order $n$ has $n$ degrees of divergence in its Laurent series (from $(z - z_0)^{-1}$ up to $(z - z_0)^{-n}$). Likewise, a physical relation with $n$ variables and $k$ fundamental units yields $n - k$ dimensionless $\pi$ groups, essentially the degrees of freedom in the dimensionless analysis. If one variable’s value is changed, one of the $\pi$ groups changes, similar to how changing a coefficient in a Laurent
series principal part affects the nature of the singularity. In both settings, a matrix of exponents can be set up: in dimensional analysis we form a matrix of exponents of $M, L, T, \dots$ for each variable to solve for independent $\pi$ combinations; in complex analysis, the exponents in the Laurent series (or the pole order, zero order) tell us the structure of the singularity. Thus, the structure (rank) of the dimensional matrix has a correspondence to the structure (order, nature) of singular expansions.

# 6 Dimensional Matrix vs. Singularity Order: Structural Similarity

When performing dimensional analysis, one often arranges the exponents of each physical quantity's dimensions into a matrix (sometimes called the dimension matrix). The rank of this matrix $r$ indicates how many fundamental dimensions are involved, and consequently how many independent dimensionless groups can be formed (specifically, $n-r$ dimensionless groups for $n$ variables). For example, if we consider a simple system with variables involving only length $L$ and time $T$, the rank of the dimension matrix might be 2 (if both $L$ and $T$ appear independently). In that case, $n-r$ would be $n-2$. If we have two length variables $L_1$ and $L_2$ (with $n=2$ and $k=1$ fundamental dimension $L$), the rank $r=1$ and we get $2-1=1$ independent dimensionless combination: indeed $L_1/L_2$ is the single dimensionless ratio we can form. If we only had one length variable ($n=1, r=1$), then $n-r=0$: no non-trivial dimensionless group can be formed, consistent with the idea that one cannot non-dimensionalize a single dimensional quantity in isolation.

In complex analysis, consider an isolated pole of order $m$. This singularity has a principal part in the Laurent series with $m$ terms: $(z-z_0)^{-1}+(z-z_0)^{-2}+\cdots+(z-z_0)^{-m}$. We can say this pole has “order $m,$” which is analogous to having $m$ degrees of independent singular behavior (each power $(z-z_0)^{-k}$ for $k=1, \dots, m$ contributes a new aspect to the divergence). In a loose sense, the order $m$ of a pole is like the number of independent dimensionless parameters needed to characterize a physical system’s behavior in a certain regime. Both are expressed as integers that emerge from a linear algebraic consideration (for poles, it’s the number of negative power terms needed; for dimensional analysis, it’s the number of independent unit combinations).

Thus, a high-order pole (large $m$) is a more complex singularity, just as a physical problem requiring many dimensionless parameters ($n-r$ large) is more complex. A simple pole (order 1) is relatively simple – analogous to having a single dominant dimensionless parameter governing a phenomenon. For instance, a system described mainly by Reynolds number is simpler in similarity structure (one dimensionless group) and often has a singular perturbation at a critical Re. A pole of order 1 has a single term dominating the singular behavior. In contrast, an essential singularity can be thought of as effectively infinite order (infinitely many negative powers in the Laurent series), which has no analog in a finite-dimensional physical dimensional analysis – it represents an extremely complex behavior with infinitely many scales of variation (one might say an essential singularity is like needing an infinite number of dimensionless parameters to describe a phenomenon, which is beyond normal physical situations).

In summary, the dimensional matrix's rank and the resulting count of dimensionless groups is structurally analogous to the classification of singularities by order. Both concepts involve determining how many independent “directions” or terms are needed to describe the system's behavior near a special point (be it a critical parameter combination
in physics or a singular point in the complex plane).

# 7 Example: Reynolds Number Revisited

To solidify the analogy, let us reconsider the Reynolds number $Re$ in detail and relate it to singularity analysis. We already established that $Re$ is dimensionless: $Re = \rho vL/\mu$ with $[Re] = 1$. Here we analyze its physical significance and parallel it to singularity types:

In fluid dynamics, $Re$ encapsulates the ratio of inertial forces to viscous forces. A low Reynolds number (say $Re \ll 1$) indicates viscous forces dominate, leading to laminar (smooth) flow, whereas a high Reynolds number ($Re \gg 1$) indicates inertial forces dominate, often leading to turbulent flow. The transition between laminar and turbulent flow as $Re$ increases is analogous to a system hitting a singular point: at a critical $Re$, the nature of the solution (flow) changes qualitatively, much like crossing a singularity changes the behavior of a function.

From a dimensional analysis perspective, $Re$ is the single most important parameter governing this transition. If we treat $Re$ as a variable, the limit $Re \to \infty$ (achieved by letting $\mu \to 0$ or $v \to \infty$, etc.) is a singular limit of the fluid equations (the Navier-Stokes equations simplify to Euler's equations, and solutions can develop discontinuities like shock waves or chaotic eddies). This $Re \to \infty$ limit is akin to approaching a pole in a complex function, where the solution “blows up” or changes character dramatically (indeed, turbulence can be seen as a form of chaotic, unbounded behavior in the solution space). Likewise, $Re \to 0$ (achieved by very high viscosity or very low velocity) leads to creeping Stokes flow, which is smooth and stable – analogous to approaching a zero of a function where the value tends to a benign, small state.

We can formalize the dimensional analysis to show how $Re$ arises, and compare it to extracting a singular behavior. Starting with variables: characteristic velocity $v$, length $L$, fluid density $\rho$, and viscosity $\mu$, we have four variables and three fundamental dimensions ($M, L, T$). Buckingham's theorem predicts $4 - 3 = 1$ independent dimensionless group. By solving the dimensional matrix, we indeed find $\Pi_1 = \rho vL/\mu$ as that group (which is $Re$). In a sense, the fact that one dimensionless combination governs the system implies the system's behavior can be collapsed onto a single-parameter family – much like how a function near a simple pole can be described by one dominant term. If the problem had yielded two independent dimensionless numbers, it would be like a function with two competing singular behaviors (say two nearby poles, or a pole and a branch point), which would complicate the analysis.

Thus, the Reynolds number exemplifies how dimensional analysis distills a problem to essential parameters (here one parameter), and how the extreme limits of that parameter correspond to distinct regimes (laminar vs turbulent) much like how a singular point yields distinct local behaviors (below vs above the singularity, or approaching from different directions in the complex plane). In fact, researchers often study asymptotic expansions in $Re$ (for $Re \ll 1$ or $Re \gg 1$), which is analogous to studying the Laurent series of a function about $Re = 0$ or $Re = \infty$. A large $Re$ expansion might correspond to a singular perturbation series, reflecting the pole-like behavior at $Re = \infty$.
# 8 Conclusion

Through the above comparisons, we have demonstrated an abstract yet insightful correspondence between dimensional analysis in physics and singularity analysis in complex analysis. In summary:

* The way physical quantities behave under extreme limits (tending to infinity or zero) mirrors how complex functions behave near poles or zeros. A quantity that diverges (e.g. $v \to \infty$ as $t \to 0$) is analogous to a pole in a complex function, while a quantity that vanishes is analogous to a zero of the function. Complex and essential behaviors in physical systems (like sudden phase transitions or multi-valued dependencies) are analogous to essential singularities or branch points where normal analytic behavior breaks down.

* The requirement of dimensional homogeneity in any physical equation parallels the requirement of analyticity (holomorphicity) in a well-behaved mathematical function. An equation with inconsistent units is not physically valid, just as a function that is not analytic at a point is not valid in that neighborhood. Both indicate a fundamental breakdown: the former of physical meaning, the latter of mathematical description.

* The construction of dimensionless parameters (and the use of dimensionless analysis to simplify problems) is analogous to expanding a function in a Laurent/Taylor series and identifying leading terms that govern local behavior. Each independent dimensionless group is akin to an independent direction of singular behavior. For example, one dimensionless group (like Re) governing a system is similar to a simple pole dominating a function's local behavior; multiple dimensionless groups would be like a function with multiple singularities affecting the behavior.

* The dimensional matrix's rank (and the number of independent $\pi$ groups $n-r$) plays a role analogous to the order of a pole or the count of essential degrees of a singularity. Both are determined by solving linear constraints (unit balance for dimensional analysis, or series coefficients for analytic functions) and both determine how many free parameters characterize the local solution (be it a formula or a function).

In essence, expressing dimensional analysis in the language of complex analysis indeed becomes an analysis of singularity types. Dimensional analysis, at its core, identifies how physical formulas remain well-behaved (or break down) under scaling of units – a process quite analogous to identifying how a function remains analytic (or develops singularities) under changes of the variable. Both disciplines strive to reveal the underlying structure of a problem: dimensional analysis through units and scaling arguments, and complex analysis through singularity classification and Laurent series. The correspondence we established is not a literal one but a conceptual translation: the “singular” behaviors in physical systems (dimensional inconsistencies, infinite or zero limits, emergent scale-invariant groups) correspond to literal singular points and expansions in complex analysis. This analogy enriches our understanding of both fields by highlighting their common goal of uncovering fundamental constraints and behavior patterns – in one case through the consistency of units, and in the other through the analyticity of functions.

*(End of Proof)*
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