Uniformization and Constructive Analytic Continuation of Taylor Series Ovidiu Costin and Gerald V. Dunne Department of Mathematics, The Ohio State University, Columbus, OH 43210-1174, USA Department of Physics, University of Connecticut, Storrs, CT 06269-3046, USA Abstract We analyze the problem of global reconstruction of functions as accurately as possible, based on partial information in the form of a truncated power series at some point and addi- tional analyticity properties. Within the class of functions analytic on a common Riemann surface W, with a common rate of growth and a common Maclaurin polynomial, we prove an optimality result on their reconstruction at arbitrary points in W, and find a procedure to attain it. This procedure uses the uniformization theorem; the optimal reconstruction errors depend only on the conformal distance to the origin. A priori knowledge of W is rigorously available for functions often encountered in analysis (such as solutions of meromomorphic ODEs and classes of PDEs). It is also available, rigor- ously or conjecturally based on numerical evidence, for perturbative expansions in quantum mechanics, statistical physics and quantum field theory, and in other areas in physics. For a subclass of such functions, we provide the optimal procedure explicitly. These in- clude the Borel transforms of the linear special functions. We construct, in closed form, the uniformization map and optimal procedure for the Borel plane of nonlinear special functions, tronquée solutions of the Painlevé equations PI–PV. For the latter, W is the covering of C n Z by curves with fixed origin, modulo homotopies. We obtain some of the uniformization maps as rapidly convergent limits of compositions of elementary maps. Given further information about the function, such as is available for the ubiquitous class of resurgent functions, significantly better approximations are possible and we construct them. In particular, any chosen one of their singularities can be eliminated by specific linear opera- tors which we introduce, and the local structure at the chosen singularity can be obtained in fine detail. MSC codes: 30F20, 30B30, 30B10, 40A25, 41A58 arXiv:2009.01962v3 [math.CV] 14 Apr 2021 1 Introduction In problems of high complexity in mathematics and physics it is often the case that a solution can only be generated as a finite number of terms of a perturbation series at certain special points. Under these circumstances, we ask what is the optimal strategy to approximate the un- derlying function, and if optimality cannot be achieved in practice, what are the most efficient near-optimal methods? As a mathematical question this is related to, but distinct from, conven- tional approximation theory. We formulate this mathematical problem as a problem of inverse approximation theory: here the approximation is fixed, in the form of a number n of terms of a series, and the underlying function F is to be reconstructed as accurately as possible in the large 1 n limit. For this purpose, again unlike in approximation theory, the simplicity of the functions and operators used in the reconstruction plays a secondary role. On a practical level, this reconstruction question has been encountered in many problems in the literature, and dealt with in various problem-specific ways [42,8, 46, 78, 14, 16]. However, an underlying rigorous mathematical foundation and theory seem to be lacking. We prove an optimality theorem (Theorem2), which constructs the best approximant (based on partial Maclaurin sums) to F in generic classes of functions analytic on a common Riemann surface W, and sharing a common rate of growth. Optimality is achieved with the uniformization map of W, which also provides rigorous numerical access to the entire Riemann surface. Theorem 10 constructs the explicit uniformization map for the Riemann surface in Borel plane for the tronquée solutions to the Painlevé equations PI - PV. Illustrated on an example of interest, Theorem 14 provides a uniformization procedure rely- ing on a composition sequence of elementary maps. This result also enables accurate approxi- mate uniformization using elementary conformal maps. Application of the optimality results only requires information about the Riemann surface W. Such a priori information exists for resurgent functions, a ubiquitous class of functions in analysis discovered by Écalle [38]. Resurgent functions are analytic on the universal covering of C with a discrete set of punctures, the locations and types of which are rigidly predetermined from the equation (or even the type of equation) from which they originate [13, 19, 38]. An important consequence of Écalle’s pioneering theory of resurgent functions is that a considerable amount of global information is contained in their local expansions [38, 11, 20, 64, 45, 54,5]. This information is encoded in the “bridge equations” [38]. Uniformization provides a practical way (and perhaps the only way that does not require full knowledge about the function) to access all Riemann sheets, which is needed in medianization and other forms of averaging. Such averaging is a fundamental aspect of Écalle-Borel summation, the rigorous generalization of Borel summation that applies to functions with singularities along the axis of the Laplace transform [38, 59, 19]. Empirically we observe that the inclusion of even partial information about the resurgent structure leads to dramatically improved precision of the resulting analytic continuations. Some illustrative examples are given in Section6. In cases where the optimal uniformizing map is difficult to construct, the procedure in The- orem2 can be adapted to extract information using simpler maps with near-optimal precision in restricted regions of W, for example near the singularities or boundaries, see §4 and §5. Dis- crete singularities and natural boundaries are usually the most important ones to study in both mathematics and physics. We prove a singularity elimination theorem (Theorem 26), a rigorous and constructive method to transform the local behavior near a chosen singularity to be purely analytic. This leads to a novel numerical method to determine the location and nature of a sin- gularity, exploiting a highly sensitive ‘resonance’ effect that distinguishes numerically between analyticity and singularity. In physical applications, where only a finite number of terms of an expansion might be practically accessible, the goal is to extract as much and as accurate information as possible from this limited input information. This motivates the following class of questions: 1. Where are the singularities of F? 2. What is the nature of each singularity? 3. What is the local behavior, or more generally what are the associated local expansions, near each singularity? 2 4. How far can one explore the full Riemann surface of F? 5. Can one quantify the expected precision locally, especially near the singularities, as a func- tion of how much input data (and of what precision) is given? These questions have both theoretical and practical consequences. Question 1 corresponds to identifying critical points or saddle points, which typically have important mathematical and physical implications (e.g. asymptotics and phase transitions). Question 2 refers to determining whether these singularities are algebraic or logarithmic branch points (or in special simple cases, poles), or essential singularities. A common application in statistical physics and quantum field theory is the accurate determination of critical exponents [78]. Questions 3 and 4 involve for example the numerical determination of Stokes constants, or wall-crossing formulas [44], or generally the fluctuations about a given critical point. Question 5 is of particular practical value, since in nontrivial problems it is often difficult to generate many terms of the original series. In physics problems where a truncated series is the only available information, our methods provide new and precise exploratory tools. In mathematics these procedures can be used for efficient and accurate extraction of global information about solutions of meromorphic ordinary or partial differential equations, starting with the divergent series at infinity. Such series are known to be resurgent, in particular generalized Borel summable, under quite general conditions [19, 22, 23, 24, 25, 26], and the results in this paper can be applied directly in the Borel plane. A more accurate analytic continuation in the Borel plane leads to a significantly more accurate extrapolation of the function in its original domain. A mathematical implication is to produce extremely accurate extrapolations of the Painlevé transcendents PI–PV throughout their domain of existence. In Section 6.2 we give as an example the reconstruction of the tritronquée solution of PI from 200 terms of its asymptotic expansion throughout its sector of analyticity, and also in the pole sector, where we find the first 66 poles to significantly higher precision compared to existing methods. Similar reconstructions can be applied to the other Painlevé tronquée solutions.1 Such extreme precision has applications to the Painlevé project [65], and also to the spectral properties of certain Schrödinger operators [61, 63]. In §5 we combine these ideas with Stahl’s fundamental results on the theory of Padé approx- imation [73], based on capacity theory, to propose new methods for approximate uniformization, giving significantly improved analytic continuation methods for functions with branch point sin- gularities.