Analytic Continuation Technique

Updated 20 November 2025

Analytic continuation is a method that extends analytic functions beyond their original domain by leveraging their local power series and uniqueness properties.
Rational approximants such as Padé and barycentric methods, along with adaptive algorithms, overcome singularities and noise to provide reliable global approximations.
Modern strategies incorporating Bayesian regularization and neural network models enhance precision and stability in applications across mathematical physics.

Analytic continuation is the process of extending a given analytic function beyond its original domain of definition by exploiting the rigidity of analytic (holomorphic) functions, which are uniquely determined by their behavior on any sufficiently small nontrivial subset. In contemporary research across mathematics and mathematical physics, analytic continuation techniques are critical in the completion, evaluation, and interpretation of special functions, zeta and L-functions, Green’s functions in quantum theories, and stochastic processes. Formal and algorithmic advances connect classical construction methods—such as power series, rational (Padé) approximants, and functional equations—with modern computational, algebraic, and statistical approaches.

1. Classical and Algorithmic Foundations

At the core, analytic continuation relies on the fact that the Taylor or Laurent series of an analytic function uniquely determines the function in the function's maximal disk of convergence (identity theorem). For holomorphic functions on the complex plane, all information is encoded locally in the derivatives at a single point. However, the extension to other points can be hindered by singularities (poles, branch points, or essential singularities), and the process is generally ill-posed: analytic continuation from finite, noisy data cannot be unique unless one introduces additional constraints or regularizing assumptions (Kranich, 2014).

Effective algorithmic strategies for local analytic continuation involve the control of truncation and step size, such as computing step-wise Taylor expansions along a path, leveraging explicit Cauchy bounds on derivatives, and controlling global errors by adapting the analytic neighborhood size. For algebraic curves, for example, an analytic branch can be continued by Taylor expansion up to a prescribed accuracy, with next sampling point chosen adaptively to avoid singular loci.

2. Padé and Rational Approximants

A prevailing class of global analytic continuation methods employs rational function approximants—Padé or barycentric rational functions—to represent the analytic structure (poles and zeros) of the target function. Given data $\{G(i\omega_n)\}$ at Matsubara frequencies (in many-body quantum physics), one fits a rational function $R_{L,M}(z) = \frac{a_0 + \cdots + a_L z^L}{1 + b_1 z + \cdots + b_M z^M}$ either by direct linear regression (Wang et al., 2018), least-squares (Schött et al., 2015), or barycentric/AAA algorithms (Huang et al., 25 Dec 2024).

Padé methods approximate analytic continuation $i\omega_n \to \omega + i 0^+$ by evaluating $R(z)$ on the physical axis and extracting the spectral function via $A(\omega) = -\frac{1}{\pi} \mathrm{Im} R(\omega + i\delta)$ . The barycentric rational approach with AAA (adaptive Antoulas-Anderson algorithm) offers superior stability under intermediate noise, outperforming traditional MaxEnt and Padé (continued-fraction) methods in speed and accuracy for a wide class of Green's functions and correlators (Huang et al., 25 Dec 2024).

Critical to success is obtaining high-precision input data. Direct Matsubara summation converges slowly, but can be overcome by employing Padé decomposition of the Fermi or Bose distribution, replacing the summation with a sum over rationally-placed poles—yielding exponential acceleration and enabling the determination of high-order Padé approximants (Han et al., 2017). Conditioning and numerical precision must be carefully managed, as the linear systems involved are often highly ill-conditioned.

3. Regularized and Statistical Approaches

Because analytic continuation is severely ill-posed from finite and noisy data, modern methodologies incorporate regularization and statistical inference. The maximum entropy method (MEM) introduces a Bayesian framework, seeking the most probable spectral function under a specified prior and the data likelihood, and regularizing by maximizing entropy with respect to a default model (Tripolt et al., 2018). Alternative statistical approaches use sparse modeling and LASSO regularization in the singular-value decomposition (SVD) basis of the kernel; sparse modeling stabilizes the solution by selecting only the most singular-value-resolved directions, filtering out directions dominated by noise (Yoshimi et al., 2019).

Rational function regression with bootstrapping quantifies uncertainty: for each bootstrap sample perturbed by input noise, a rational fit is performed, and final spectral error bars are computed from the empirical variance of the resulting spectral functions. This approach is robust up to approximately 1\% relative noise in the input Green’s function, provided the number of Matsubara points and fit degrees are well-chosen (Wang et al., 2018).

4. Machine Learning and Neural Analytic Continuation

For high-dimensional inverse problems and ill-posed mappings (e.g., Green's function to spectral function in quantum Monte Carlo data), data-driven approaches recast analytic continuation as a supervised learning task. Neural networks (NN) are trained to map input data representations (e.g., Legendre expansion of $G(\tau)$ ) to spectral functions using large synthetic or physical datasets. State-of-the-art architectures such as feature-learning neural networks (FL-net) achieve at least 20\% lower normalized loss compared to the maximum entropy method and prior neural approaches, resolving multi-peak and sharp features with greater fidelity (Zhao et al., 22 Nov 2024).

Key insights include: auto-encoding spectral functions into a low-dimensional latent space (encoding peak positions, widths, weights), training an encoder from $G$ to the latent space, and controlling the trade-off between expressivity (test loss) and robustness to noise by regulating the latent-space dimension and singular value spectrum of the network mapping (Zhao et al., 22 Nov 2024, Fournier et al., 2018). Empirical analysis demonstrates that increasing the hidden dimensionality lowers test loss but reduces robustness, as measured by input noise amplification in the spectral output.

5. Function Theory, Special Functions, and Symbolic Continuation

The extension of zeta, polylogarithm, and multiple harmonic sum functions provides classical motivation and algorithmic techniques. For zeta and multiple zeta values, constant-term extensions of nested Faulhaber formulas, symbolic calculus with Bernoulli/uniform/H/V-symbols, and renormalization procedures systematically continue harmonic sums and multiple zeta functions to non-positive integer and complex arguments (Jiu et al., 2019). Analytic continuation for functions such as the Hurwitz zeta uses exact reformulations of the Euler-Maclaurin formula, replacing Bernoulli-number series by absolutely convergent integrals valid in the entire complex plane apart from isolated poles (Sousa, 2021).

The analytic continuation of the Lerch zeta function exploits gamma-integral representations, functional equations, branching analyses, and the monodromy formalism to realize multi-valued analytic functions on C³ modulo explicitly computed monodromy (arising from loops surrounding integer parameters), yielding single-valuedness on the maximal abelian cover (Lagarias et al., 2010).

For nested harmonic sums with complex indices (including those appearing in high-order QCD and superconformal gauge theory computations), symbolic decomposition and matching Laurent expansions in a small parameter around integer argument points enable algebraic analytic continuation and facilitate BFKL limit, slope, and next-to-slope calculations (Velizhanin, 2022).

6. Conformal and Nevanlinna-based Interpolations

Recent developments leverage function-theoretic constraints (causality, positivity, analytic class preservation) using conformal and Nevanlinna maps. For Green’s functions analytic in the upper half-plane, conformal maps (Cayley, branch-cut-opening) transform the problem to interpolation in the unit disk, where the Nevanlinna–Pick theorem guarantees solution existence under Pick matrix semidefiniteness (Bergamaschi et al., 2023, Nogaki et al., 2023). The entire admissible space of interpolants consistent with the data can be described via a Schur function parameterization or continued fraction; rigorous error bands for smeared spectral functions can be efficiently computed.

For bosonic analytic continuation, fermionization via the tanh kernel yields auxiliary functions amenable to Nevanlinna-class interpolation, then mapped back via analytic transformations, ensuring causality, exact enforcement of high-frequency moments, and consistent treatment of sum rules (Nogaki et al., 2023).

7. Applications in Physics and Arithmetic

Analytic continuation is central to extracting physically relevant information from lattice calculations—e.g., reconstructing the hadronic vacuum polarization function in both spacelike and timelike regions using Laplace-transform techniques and polynomial expansions, allowing model-independent evaluation and analytic differentiation (Adler function, muon $g-2$ ) (1311.0652). In many-body GW calculations, analytic continuation of the dynamically screened Coulomb interaction $W$ rather than the self-energy $\Sigma$ itself yields improved stability and accuracy, especially for deep states and lifetimes; continued-fraction fits and adaptive contour-deformation are critical (Duchemin et al., 2019).

In arithmetic statistics, the meromorphic continuation of Euler products (with constant or Frobenian coefficients) is systematically constructed by iterative factorization, pulling out known L-functions according to the local factor expansions and controlling the absolute convergence of the error Euler product. The locations and orders of all poles and branch points are explicitly tracked, underpinning Tauberian asymptotic results (Alberts, 26 Jun 2024).

8. Comparative Assessment and Regimes of Applicability

A broad comparative perspective emerges from direct benchmarks. Rational (Padé/SP) and barycentric methods offer highest resolution but require high-precision, densely sampled input and are sensitive to noise (Schött et al., 2015, Huang et al., 25 Dec 2024). Maximum Entropy methods are robust for moderate noise, especially when seeking smooth or single-peak spectra, but tend to oversmooth and lose fine structure. Linear approaches such as Backus–Gilbert excel for plateau-like transport quantities and are highly noise-tolerant but have low spectral resolution (Tripolt et al., 2018). Machine learning approaches can match or exceed MaxEnt at a fraction of computational cost and have superior noise robustness, provided training data reflect the target distribution (Zhao et al., 22 Nov 2024, Fournier et al., 2018).

Summary tables and regime diagrams (e.g., in (Tripolt et al., 2018, Schött et al., 2015)) establish that the suitability of different analytic continuation techniques depends fundamentally on the statistical quality, density, and bandwidth of the input data, as well as on the analytic complexity and physical constraints of the system under paper. Hybrid techniques, combining algorithmic, statistical, and machine-learning elements, represent a growing frontier.

References

Rational function regression for analytic continuation: (Wang et al., 2018).
Padé decomposition and high-precision Matsubara sums: (Han et al., 2017).
Barycentric/AAA rational analytic continuation: (Huang et al., 25 Dec 2024).
Averaged Padé methods and stability: (Schött et al., 2015).
Sparse modeling and SVD-based regularization: (Yoshimi et al., 2019).
Machine learning for analytic continuation: (Zhao et al., 22 Nov 2024, Fournier et al., 2018).
Nevanlinna/Schur/Hardy theory methods: (Nogaki et al., 2023, Bergamaschi et al., 2023).
Symbolic (Faulhaber/Bernoulli) continuations: (Jiu et al., 2019, Sousa, 2021).
Benchmark and comparative meta-analyses: (Tripolt et al., 2018).
Applications to lattice QCD and muon $g-2$ : (1311.0652).
Meromorphic continuation of Euler products: (Alberts, 26 Jun 2024).
Algebraic continuation of harmonic sums near integers: (Velizhanin, 2022).