Taylor Convergence Radius

• Taylor Convergence Radius is defined as the largest radius within which a Taylor expansion converges absolutely by measuring the distance from the expansion point to the nearest singularity.
• It plays a crucial role in numerical approximations, dynamical systems, and perturbation theory by signaling the onset of phenomena like bifurcations, phase transitions, or chaotic behavior.
• Both analytical techniques, such as the ratio and root tests, and numerical estimation methods are used to determine the convergence radius, ensuring robust evaluations of polynomial approximations.

The Taylor convergence radius is the maximal domain in which the Taylor series expansion of a function, dynamical map, or observable remains absolutely convergent. For analytic functions in one or several complex variables, this radius is precisely the distance from the expansion point to the nearest singularity of the function in the complex domain. This notion governs the fidelity of polynomial approximations, informs perturbative and numerical strategies, and encodes deep physical or dynamical information—such as the proximity to bifurcations, phase transitions, or the onset of chaos.

1. Fundamental Definition and Theoretical Foundations

Let $f(z)$ admit a Taylor expansion about $z=0$:

$f(z) = \sum_{n=0}^\infty a_n z^n.$

The radius of convergence $R$ is the supremum of all $r > 0$ such that the series converges absolutely for $|z| < r$. Classical results—Cauchy–Hadamard and Abel—establish

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

and, when limits exist,

$$R = \lim_{n\to\infty} \left| \frac{a_n}{a_{n+1}} \right|.$$

For multivariate power series or polynomial maps, such as transfer maps for differential equations, $R$ is determined by the minimum distance from the origin (in the analytically continued variable space) to the nearest singularity of the solution map (Kaltchev et al., 2012, Dolores, 2018).

In differential equations with analytic right-hand-sides, Poincaré’s theorem ensures that the flow map remains analytic in initial data and parameters for finite time, yielding a finite, nonzero convergence radius.

2. Radius of Convergence in Dynamical Systems and Differential Equations

Within dynamical systems, Taylor expansions underpin polynomial transfer maps for propagating deviations from a reference solution. Formally, for $\dot{z} = f(z, t)$ with analytic $f$, the evolute map $z(t^f) = \gamma(z^0; t^0, t^f)$ is analytic in initial conditions for finite time intervals.

Deviations $\zeta(t) = z(t) - z^{d}(t)$ satisfy complete variational equations, whose solutions yield the Taylor-transfer map:

$$\zeta^f = M(\zeta^i) = \sum_{|r| \leq N} h^r G_r(\zeta^i),$$

where $r$ is a multi-index and $G_r$ are monomials. The convergence radius $R$ is then

$$\frac{1}{R} = \limsup_{|r| \to \infty} \| h^r \|^{1/|r|},$$

in any chosen norm. $R$ measures the maximal perturbation in initial deviation space compatible with analyticity. For the Duffing oscillator stroboscopic map, as system parameters (e.g., driving frequency) approach accumulation points of period-doubling, $R$ shrinks—the domain of analyticity recedes and the system approaches chaotic dynamics (Kaltchev et al., 2012).

3. Analytical and Numerical Estimation Methodologies

Estimation of $R$ can follow either analytic or numerical pathways depending on context.

• Analytical root or ratio test: For known coefficients, directly compute the limsup or consecutive ratios.
• Transfer map ODEs: For dynamical systems, integrate the complete variational equation hierarchy, extract coefficients up to large order, and apply root/ratio tests to coefficient norms $A_k = \max_{|r|=k} \| h^r \|$, with $R_k = 1/(A_k)^{1/k}$. These convergents provide successively sharper lower bounds (Kaltchev et al., 2012).
• Physical observables and partition functions: In lattice field theories, the radius is set by the nearest Lee–Yang zero in fugacity or chemical potential space; it can be determined via direct zero search, analytic mappings, or dedicated estimators (Giordano et al., 2019, Giordano et al., 2019, Connelly et al., 2020).
• Truncated expansions: For finite series, the effective or practical convergence radius can be estimated using the last-term root test, least-squares slope estimation, or direct comparison with the original function across the domain (Christopoulos, 2013).

Tables of formulae such as those for QCD (see below) and transfer maps standardize the computational process.

Context	Radius of Convergence Formula	Reference
Standard Taylor series	$R = 1/\limsup_{n\to\infty} |a_n|^{1/n}$	(Dolores, 2018)
Multivariable transfer map	$1/R = \limsup_{|r|\to\infty} \| h^r \|^{1/|r|}$	(Kaltchev et al., 2012)
QCD pressure (finite V)	$R = \min_j |z_j|$ (nearest Lee–Yang zero)	(Giordano et al., 2019)

4. Radius of Convergence and Singularities: Physical and Mathematical Interpretation

The Taylor convergence radius universally encodes the proximity and nature of singularities in the analytically continued variable space. In statistical mechanics and QCD, the singularities correspond to phase transitions (real-axis pinch points) or crossovers (complex-zero proximity) (Giordano et al., 2019, Giordano et al., 2019, Connelly et al., 2020, Brandt et al., 2018).

• Lee–Yang zeros: In systems with polynomial (finite-volume) partition functions, the radius is set by the modulus of the nearest zero in the fugacity variable. If the phase of the leading zero approaches 0 or $\pi$, the singularity can induce a real phase transition.
• Dynamical systems: In chaotic billiards or near bifurcations, the analytic domain is bounded by emerging essential singularities, whose distance to the real phase-space domain determines $R$ (Kaltchev et al., 2012).
• Mesoscopic transport: For the transmission function of quantum conductors, the Taylor convergence radius with respect to energy is limited by the nearest complex-plane resonance or pole of the scattering matrix (Ramirez et al., 2020).

Thus, $R$ serves as a rigorous diagnostic of analytic structure, phase behavior, and the limits of perturbative expansions.

5. Applications in Physics and Mathematics

• Dynamical system approximations: The finite Taylor convergence radius governs the validity of polynomial approximations to maps and flows. Near bifurcation or chaos, shrinkage of $R$ signals reduced utility of low-order expansions (Kaltchev et al., 2012).
• QCD and statistical physics: The radius defines the maximal chemical potential or fugacity where truncated Taylor expansions of the pressure remain valid. It provides bounds on the search for the QCD critical point, with established methods for robust finite-volume estimation (Giordano et al., 2019, Giordano et al., 2019, Brandt et al., 2018, Connelly et al., 2020).
• Quantum transport: In multiterminal conductors, $R$ delimits the validity domain for polynomial representation of conductance as a function of energy. The location of complex poles is critical in interpreting resonance effects (Ramirez et al., 2020).
• Analysis in division algebras: Generalizations of the Abel and Cauchy–Hadamard theorems characterize $R$ for Taylor expansions in quaternionic or octonionic analytic frameworks, with careful treatment of noncommutativity and basis-dependent norms (Dolores, 2018).
• Functors in algebraic topology: In Goodwillie calculus, the Taylor tower’s radius of convergence determines the homotopical range of functorial equivalence between geometric and algebraic structures, with explicit analytic bounds for categories of $O$-algebras (Schonsheck, 2020).

6. Estimator Robustness, Truncation Effects, and Advanced Topics

Finite-order estimators exhibit pathologies in some contexts. In finite volumes or for particular symmetry classes, naive ratio estimates of $R$ can fail—oscillating or not converging due to the structure of the leading Taylor coefficients (e.g., oscillatory contributions from conjugate Lee–Yang zeros). Advanced estimators (such as modified Mercer–Roberts or doubled-index estimators) remedy these issues and can extract $R$ reliably even when individual coefficient errors are large or strongly correlated (Giordano et al., 2019).

Truncated series admit “effective” radii of convergence, which approximate the true $R$ for high truncation order but require practical error thresholds and norm choices (Christopoulos, 2013).

Adaptive algorithms, such as step size control in stochastic differential equation solvers, can leverage estimates of $R$ to optimize convergence and error control (Foster et al., 2019).

7. Broader Significance and Open Problems

Understanding the Taylor convergence radius is fundamental in linking analytic, combinatorial, and geometric properties of functions and systems to their practical approximation domains. It encodes precursors to transitions, sets rigorous boundaries for expansions, and structures the computational strategies across disciplines.

Major open problems include the rigorous characterization of analytic domains in multicomponent systems, the justification of saddle point arguments for complex-parameterized expansions (e.g., generalized Lambert $W$ functions (Castle, 2018)), and the extension of radius concepts to non-standard function classes (e.g., noncommutative or nonassociative settings).

The convergence radius remains a cornerstone concept in both mathematical analysis and applied fields relying on perturbation theory, numerical simulation, and analytic continuation.