Uniformization and Constructive Analytic Continuation of Taylor Series
UUniformization and Constructive Analytic Continuation ofTaylor Series
Ovidiu Costin and Gerald V. Dunne
Department of Mathematics, The Ohio State University, Columbus, OH 43210-1174, USADepartment of Physics, University of Connecticut, Storrs, CT 06269-3046, USA
Abstract
We analyze the general mathematical problem of global reconstruction of a function withleast possible errors, based on partial information such as n terms of a Taylor series at apoint, say the origin, possibly also with coefficients of finite precision. We refer to this as the“inverse approximation theory” problem, because we seek to reconstruct a function from agiven approximation, rather than constructing an approximation for a given function.Within the class of functions analytic on a common Riemann surface Ω , bounded on Ω , orwith the same rate of growth in the natural metric on Ω , and a common Maclaurin series, weprove an optimality result on their reconstruction at other points on Ω , and provide a methodto attain it. The procedure uses the uniformization theorem, and the optimal reconstructionerrors depend only on the distance to the origin.We provide explicit uniformization maps for some Riemann surfaces Ω of interest in ap-plications. Some of these can also be obtained as a rapidly convergent limit of compositions ofelementary maps. One such map is the covering of C \ Z by curves with fixed origin, modulohomotopies, precisely the one needed in the analysis of the Borel plane of the tritronquée solu-tions to the Painlevé equations P I – P V . As an application we show that this uniformization mapleads to dramatic improvement in the extrapolation of the P I tritronquée solution throughoutits domain of analyticity and also into the pole sector.Given further information about the function, such as is available for the ubiquitous classof 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 infine detail. These operators involve convolutions, whose singularity nature we analyze. Moregenerally, for functions of reasonable complexity, based on the n th order truncates alone wepropose new efficient tools which are convergent as n → ∞ , and which provide near-optimalapproximations of functions globally, as well as in their most interesting regions, near singu-larities or natural boundaries. In problems of high complexity in mathematics and physics it is often the case that a solutioncan only be generated as a perturbation series at certain special points, usually with only a finitenumber of terms, and often also with coefficients known only with finite precision. Under thesecircumstances, we ask what is the optimal strategy to approximate the underlying function, andif optimality cannot be achieved in practice, what are the most efficient near-optimal methods?On a practical level, this question has been encountered in many problems in the literature, and1 a r X i v : . [ m a t h . C V ] S e p ealt with in various problem-specific ways [40, 8, 44, 74, 14, 16]. However, an underlying mathe-matical foundation and theory seems to be lacking. As a mathematical question this is related to,but distinct from, conventional approximation theory. We formulate this mathematical problemas a problem of inverse approximation theory : here the approximation is fixed, in the form of anumber n of terms of a series, and the underlying function F is to be reconstructed as accuratelyas possible in the large n limit. In contrast to other methods (such as Padé approximants) ourreconstruction methods may involve different operators for different points in the domain ofanalyticity, and can achieve rigorous pointwise convergence results rather than being limited toconvergence in capacity. This flexibly adaptive approach is motivated by the following class ofquestions that we wish to address: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 (inphysics, fluctuation functions) near each singularity?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?In physical applications, question 1 corresponds to identifying critical points or saddle points,which typically have important mathematical and physical implications (e.g. asymptotics andphase transitions). Question 2 refers to determining whether these singularities are algebraic orlogarithmic branch points (or in special simple cases, poles), or essential singularities. An im-portant application in statistical physics and quantum field theory is the accurate determinationof critical exponents [74]. Questions 3 and 4 involve for example the numerical determination ofStokes constants, or wall-crossing formulas, or generally the fluctuations about a given criticalpoint. Question 5 is of particular practical value, since in nontrivial problems it is often difficultto generate many terms of the original series. We ask how much information about the globalstructure of the underlying function can be decoded from a finite order expansion, generatedat a particular location. Here we are motivated by the theory of resurgent functions, which areubiquitous in analysis and in applications, and for which the general philosophy suggests that aconsiderable amount of global information is encoded in local expansions [37, 11, 20, 61, 51, 5].For example, for resurgent functions information concerning not just the location of the singular-ities, but also their nature, is accessible by analytic and numerical means. We develop methodsthat can exploit this additional structure.For resurgent functions, such as the tritronquée solutions of the Painlevé equations (see [39]and references therein), uniformization provides a practical way (and perhaps the only one thatdoes not require full knowledge about the function) to access the higher Riemann sheets, neededin medianization and other forms of averaging. Écalle-Borel summation (the generalization ofBorel summation that applies to functions with singularities along the axis of Laplace transform)crucially relies on such averaging [37, 56, 19].The series may be convergent or factorially divergent, and in the latter case a Borel transformrestores convergence. This is the case of asymptotic expansions in ordinary differential or differ-ence equations near singularities of nontrivial Poincaré rank, or in similar settings in differenceequations, in PDEs such as the Schrödinger equation at small or large times, and in parabolicPDEs for small times, among many examples. 2any ad hoc methods have been developed in different mathematical and physical contexts toaddress some of these problems [40, 8, 44, 74, 14, 16]. Here we develop a systematic mathematicalapproach. We also present new quantitative comparisons of some of the common methods, andsome improvements of their implementation.We begin with an optimality result, for which the construction can be made fully explicit incertain classes of problems with enough symmetry. This class includes the tritronquée solutions of Painlevé equations P I –P V and can be used in the Painlevé project [62] to calculate these solu-tions with vastly improved accuracy over the existing methods, see §6.2. Theorem 1 constructsthe best approximants (based on partial Maclaurin sums) to F in generic classes of functionsanalytic on a common Riemann surface and sharing a common bound. Optimality is achievedwith a uniformization map, which also provides rigorous numerical access to the higher sheetsof the Riemann surface. Theorem 1 also provides a benchmark with which to compare a varietyof approximate methods for situations where optimality is impractical or impossible. Comparedto existing techniques, the gain in accuracy is particularly dramatic near singular points.We then address the problem of reconstructing a given function when an exact uniformizationof the corresponding Riemann surface is not known, but when further information about thenature of the function’s singularities may be known, either analytically or approximately. Forthe wide class of resurgent functions, prevalent in applications, stronger results are obtained. In§4, §5 and §6 we develop a new set of practical approximation methods which can achieve near-optimal results, especially in the most interesting regions of the function’s underlying Riemannsurface, near the singularities.In Section 6.2 we give as an example the reconstruction with extreme precision of the P I tritronquée solution from 200 terms of its asymptotic expansion (passing through a Borel trans-form to restore convergence) throughout its sector of analyticity, and also in the opposite polesector where we find the first 66 poles to high precision. The accuracy dramatically improvesover existing methods. Similar reconstructions can be applied to the other Painlevé tritronquée solutions. Such extreme precision has applications to the Painlevé project [62], and also to thespectral properties of certain Schrödinger operators [58, 60]. In the following, D r ( a ) denotes the open disk of radius r , centered at a . The unit disk centeredat the origin appears often in the discussion, and is simply denoted D = D ( ) , with boundarythe unit circle: T = ∂ D . The Riemann sphere is written as ˆ C = C ∪ { ∞ } .We consider functions defined on Ω , a simply connected Riemann surface . An importantspecial case in applications is Ω being a simply connected domain strictly contained in C , so theuniformization map is its usual Riemann conformal map to D . Ω is assumed to contain D strictly . More precisely, if Ω is uniformized to D by ψ , then ψ is analytic in D and ψ ( ) =
0. In the special case when Ω is a covering of ˆ C \ S where S isa discrete set, this means Ω is described by equivalence classes of curves originating at zero,modulo homotopies in ˆ C \ S . We shall call such Riemann surfaces coverings with fixed origin. The set S may contain points in D , meaning functions living on Ω might be singular at pointsin D on higher sheets of Ω . By the uniformization theorem, Ω is biholomorphically equivalent Solutions with asymptotic series expansions in the largest possible sector. Using ˆ C is a standard convention, since it makes a counting difference for the analyzed functions if infinity issingular or not. For instance, ln [( − ω ) / ( + ω )] is analytic at infinity and its Riemann surface is uniformized on theplane, after a Möbius change of variable. The choice of a fixed origin is explained by the analyticity at zero of our functions on one of the Riemann sheets. z = ψ ( ω ) from the simply connected Riemann surface Ω to the unit disk D ,and its inverse ω = ϕ ( z ) .to exactly one of the following: D , C or ˆ C (see, e.g. [1, 2, 68]). Here, we mostly focus on Riemannsurfaces uniformized on D , as the latter two cases are too special [1], and also because theiranalysis would follow similar steps. See however Note 4.We denote by ψ the conformal map of Ω onto D , uniquely specified by ψ ( ) = ψ (cid:48) ( ) > ψ − = ϕ . See Figure 1. For the optimality Theorem 1, we define F Ω to be the family of functions F which are analytic on Ω . By M F we denote the Maclaurinseries of F , and M F , n will be the n th partial sum of M F . An important role in our analysis isplayed by the natural metric on Ω , induced from the Poincaré disk, an elementary function of | ψ | . Our formulas are simpler in terms of ρ = − | ψ ( ω ) | , which we call the conformal distanceto the boundary of Ω . This distance enters the convergence results in Theorem 1 for how thereconstruction depends on the number n of input terms in the input Maclaurin series M F , n . Wedefine ϕ ∗ by the right composition map: ϕ ∗ F = F ◦ ϕ .It will at times be convenient to consider shifts of sets in C in which case we write S + ζ = { ω + ζ : ω ∈ S ⊂ C } , and also to work with an inverted variable, changing the expansion pointfrom ω = ω = ∞ , in which case we write 1/ S = { ω : ω ∈ S ⊂ C } ⊂ ˆ C .In our discussion of resurgent functions, the singularities assume a simple form (cf. [19]) ( ω − ω ) α A ( ω ) + B ( ω ) ; α ∈ C \ Z ; or, for α ∈ Z , d k d ω k [( ω − ω ) α ln ( ω − ω ) A ( ω )] + B ( ω ) (1)for some α ∈ C , k ∈ N , and where A , B are analytic at ω , A ( ω ) (cid:54) =
0. In this paper we refer tosingularities of the type in (1) as elementary singularities . With this notation, important goals ofour analysis are to learn as much as possible about the singularity locations ω , the singularityexponent α , and the associated local functions A and B .Theorem 1 constructs the best approximation of a function from its truncated Taylor series,within the family of functions having a common Riemann surface of analyticity Ω , and a commonbound or rate of growth on Ω . This result only requires information about the Riemann surface Ω . Such a priori information exists for a wide class of functions encountered in analysis, theresurgent functions. In fact, for resurgent functions considerably more information, such as theirsingularity structure, is typically available. It is then possible to go well beyond the boundsprovided in Theorem 1. Resurgent functions are analytic on the universal covering of C with adiscrete set of punctures, the locations and types of which are rigidly predetermined from the4quation (or even the type of equation) from which they originate [13, 19, 37]. Explicit examplesand the application to the Painlevé equations is demonstrated in §3.In cases where the uniformizing map is difficult to construct, the procedure in Theorem 1can be adapted to extract information with near-optimal precision in restricted regions of Ω , forexample near the singularities or boundaries, using simpler maps, see §4 and §5. These regionsare usually the most important ones to study in mathematics and physics.Even when such a priori analytic information is absent, we develop techniques to extract ap-proximate information about the Riemann surface directly from the given truncated Taylor series,using standard analytical tools such as properly adapted techniques of conformal mapping, Borelsummation and Padé approximants, and which can be refined to high precision using the newmethods of singularity elimination (§4) and approximate uniformization (§5). We compare to theoptimality result in order to quantify that these approximate methods are near-optimal.In §6 we apply our methods and illustrate the main ideas on a number of examples importantin applications. Appendix 7 contains details of some maps used in our analysis. The question of the best rate of convergence is not only natural, and interesting in itself, it isof substantial practical significance. Theorem 1 gives the optimal approximation of a functionstarting from its truncated Taylor series, within the family of all functions analytic on Ω , withcommon bounds on Ω . Specifically, the question answered by Theorem 1 is: what is the optimalapproximation that can be obtained based on this general information at some point ω ∈ Ω , asa function of ω , and of the number n of input Maclaurin coefficients? It may come as a surprise that evaluating the Taylor polynomial of a function F ∈ F Ω is subopti-mal , even for points in D (cid:40) Ω close to zero, so we start with some examples illustrating variousimportant points.Let us consider first the domain Ω = C \ [ ∞ ) and let F be analytic in Ω . From the n throot test it is clear that | F ( ω ) − M F , n ( ω ) | n ∼ | ω | ( + o ( )) ; n → ∞ (2)This convergence rate can be improved in several ways. Suppose for simplicity that n is even.Then the diagonal Padé approximants P F , n of order [ n /2, n /2 ] , uniquely defined by the first n Maclaurin series coefficients, converge faster (in the sense of logarithmic capacity: see §5): | F ( ω ) − P F , n ( ω ) | n ∼ | ω | ( + o ( )) ; n → ∞ ; | ω | small (3)In fact Padé approximants converge in capacity throughout Ω at a geometric rate, as explainedin §5. For functions analytic in Ω having [ ∞ ) as a natural boundary, this gain of is optimal(this is the logarithmic capacity of the set 1/ [ ∞ ) = (
0, 1 ] , see §5). In this case, the leading orderrate of convergence of Padé approximants in capacity is also optimal at any point of Ω .The same gain is obtained by the following conformal mapping procedure. Consider thebiholomorphism ϕ = z (cid:55)→ z ( + z ) − , which maps D onto Ω , and let M ϕ be its Maclaurin5eries. Since ϕ ∗ F is analytic in D , M ϕ ∗ F , n which equals M F ◦ M ϕ truncated to o ( z n ) , also convergesin D and Theorem 1 below implies, after mapping back to Ω (see Example 1 in §3), | F ( ω ) − M ϕ ∗ F , n ◦ ϕ − ( ω ) | n ∼ | ω | ( + o ( )) ; n → ∞ ; | ω | small (4)Furthermore, the composition M ϕ ∗ F , n ◦ ϕ − converges to F uniformly on compact sets in Ω , atthe same rate as the Padé approximants do in capacity, see §5.When applied to domains in C which are not necessarily maximal domains of analyticity, werefer to this conformal map procedure as the Conformal-Taylor method, CT. It has been used inapplications [74, 14, 16]. We discuss CT and its precision in general, and in detail, in §5.3.If instead we are dealing with functions analytic on covering of ˆ C \ {
0, 1, ∞ } with fixed origin(cf. §1.1), then the constant in (4) becomes . This follows from example 8 in §3, see (18) there.For even simpler Riemann surfaces such as that of F = ω (cid:55)→ √ − ω , ϕ ( z ) = z ( + z ) − is a uniformization map, h ( z ) = ϕ ∗ F ( z ) = ( + z ) / ( − z ) is rational, and diagonal Padé ap-proximants become exact in just one step: P n = h for all n (cid:62)
1. This simple observation willbe used later as an ingredient in our singularity elimination method which uses a combination ofconvolution operators and conformal maps (see §4).
Let Ω be as in §1.1 and ω ∈ Ω . The following result constructs and characterizes ˆ R n ( ω ) ,the best approximant, in the sense stated in the Theorem, at ω within the class of functions F analytic on the same Riemann surface Ω . Let P be a polynomial of degree n −
1, our inputtruncated Maclaurin series.Recall the maps shown in Figure 1. Theorem 1 shows that the best approximants are obtainedby composing the input Maclaurin series with the series of ϕ truncated at the same order, sum-ming the composed series, and then mapping back to Ω using ψ . Part 2 of Theorem 1 showsthat optimality is sharper in the subclass of functions already well approximated by this proce-dure. Part 3 of Theorem 1 allows for functions that may grow as ρ → Ω ). Theorem 1.
Let Ω be a Riemann surface as in §1.1, ω ∈ Ω , and P n an ( n − ) -order truncation of aMaclaurin series. We denote by F P the set of bounded functions on Ω to which P n converges: F P = { F ∈ F : || F (cid:107) ∞ < ∞ , and F ( ω ) − P n ( ω ) = O ( ω n ) as ω → } Let ˆ R n be the n-th Maclaurin polynomial, of degree n − , of the composed function ϕ ∗ P.1. For F ∈ F P we have | F ( ω ) − ˆ R n ( ω ) |(cid:107) F (cid:107) ∞ (cid:54) ( − ρ ( ω )) n ρ ( ω ) = | ψ ( ω ) | n − | ψ ( ω ) | (5) For every R ∈ C and δ > there exists F δ ∈ F P so that | F δ ( ω ) − R |(cid:107) F (cid:107) ∞ (cid:62) ( − ρ ( ω )) n ( − δ ) (6)6 . For ε > let F ε = { F ∈ F P : F ( ω ) (cid:54) = and | F ( ω ) | − | F ( ω ) − ˆ R n ( ω ) | (cid:54) ε } We have sup F ∈F ε | F ( ω ) − ˆ R ( ω ) || F ( ω ) | = ε (7) Assume n is large enough so that ρ ( ω ) n < ε . Then for every R ∈ C and every δ > there existsan F δ ∈ F ε so that | F δ ( ω ) − R || F δ ( ω ) | (cid:62) − δ + ε | F δ ( ω ) − ˆ R n ( ω ) || F δ ( ω ) | (8)
3. Let W = W ◦ ( − ρ ) , where W : [
0, 1 ) → R + (a weight depending on the natural metric distance“to the boundary”). Define (cid:107) · (cid:107) W by (cid:107) F (cid:107) W = (cid:107) F / W (cid:107) ∞ , and let F W be the family of functions Fanalytic in Ω and such that (cid:107) F (cid:107) W < ∞ . Let F ∈ F W ∩ F P . Then, | F ( ω ) − ˆ R n ( ω ) |(cid:107) F (cid:107) W (cid:54) ( − ρ ( ω )) n inf r ∈ ( − ρ ( ω ) ,1 ) W ( r ) r n − ( r − ( − ρ ( ω )) (9) and for every R ∈ C and δ > there exists F δ ∈ F W ∩ F P so that | F δ ( ω ) − R |(cid:107) F (cid:107) W (cid:62) ( − ρ ( ω )) n inf r ∈ ( − ρ ( ω ) ,1 ) W ( r ) r n ( − δ ) (10) Proof.
Note first that the map ϕ ∗ is an isometric isomorphism taking the space of analytic func-tions in Ω onto the space of analytic functions in D , L ∞ ( Ω ) onto L ∞ ( D ) , and F W to F (cid:48) W – thespace of functions in D for which sup z ∈ D | F ( z ) / W ( | z | ) | is finite.Moreover, since the composition of two C n functions is C n , the first n Maclaurin coefficientsof F are in bijection to the first n Maclaurin coefficients of ϕ ∗ F . Hence the n th degree Maclaurinpolynomial of ϕ ∗ F equals Q . Finally, F ( ω ) = ( ϕ ∗ F )( ψ ( ω )) , and the optimality question is thusreduced to the case Ω = D . Hence, we can assume without loss of generality that Ω = D , andthen ϕ is the identity, and P = Q = ˆ R .The proof in D is elementary. The inequality (5) is simply obtained by taking the sup in theCauchy formula F ( ω ) − ˆ R ( ω ) = ω n π i (cid:73) s − n F ( s ) s − ω ds (11)where the contour of integration is ∂ D r ( ) with r ∈ ( | ω | , 1 ) and letting r → R ∈ C . Consider functions in F P of the form F λ ( ω ) = P ( ω ) + λω n with λ ∈ C . We have (cid:107) F λ (cid:107) − ∞ | F λ ( ω ) − R | = (cid:107) λ − P ( ω ) + ω n (cid:107) − ∞ | λ − P ( ω ) + ω n − λ − R | which, for λ → ∞ , convergesto | ω | n , proving the optimality stated in (6).To prove (7), it suffices to take F ( ω ) = ˆ R n ( ω ) + c ω n , with c = ε ˆ R ( ω ) ω − n / ( − ε ) .For (8), consider the subfamily of F ε of functions of the form F ( ω ) = ˆ R ( ω ) + ατω n , with | α | = τ = ε ˆ R ( ω ) ω − n / ( + ε ) (included in F ε for small enough ε ), for which we have | F ( ω ) − R || F ( ω ) | = | ˆ R ( ω ) + ατω n − R || ˆ R ( ω ) + ατω n | (cid:62) | ˆ R ( ω ) + ατω n − R || ˆ R ( ω ) | + | τω n | α such that arg ( ατω n ) = arg ( ˆ R ( ω ) − R ) is greater than | ˆ R ( ω ) − R | + | τω n || ˆ R ( ω ) | + | τω n | (cid:62) | τω n || ˆ R ( ω ) | + | τω n | = ε + ε Combined with (7), this implies (8).For Part 3., recalling the isomorphism at the beginning of the proof, we use (11) and note that1 (cid:107) F (cid:107) W π (cid:12)(cid:12)(cid:12)(cid:12) (cid:73) s − n F ( s ) dss − ω (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) inf r ∈ ( | ω | ,1 ) W ( r ) r n − ( r − | ω | ) where the contour of integration is ∂ D r ( ) with r ∈ ( | ω | , 1 ) . This proves (9).For (10) we proceed as in the proof of (6) and in the limit λ → ∞ we obtain | ω | n (cid:107) ω n (cid:107) W = | ω | n sup r ∈ [ ) r n W ( r ) − = | ω | n inf r ∈ [ ) W ( r ) r n Note 2.
Optimality in Theorem 1 is relative to the widest class of functions analytic on Ω andsharing a common rate of growth. Knowledge of more specific features of the functions, suchas the concrete nature of their singularities, behavior towards infinity along special curves in Ω can lead to significant further improvements: see Sections 4, 5, 6. Such detailed information isavailable for resurgent functions, including the convergent ramified expansions near singularitiesat finite distance, asymptotic expansions at infinity and so on.For instance, as will be shown in an applications paper [28], for the Borel transform of thetritronquée solution (see e.g. [17, 27]) of Painlevé P I , using ϕ and n =
200 nonzero Maclaurincoefficients, we obtain the Stokes constant µ with about 12 digits of accuracy. With the same n ,but using information about the singularity type at ω = ± Note 3.
1. Let Ω be a domain in C and K = ( Ω − ζ ) , where ζ / ∈ Ω . If K is compact, thenthe constant | ψ (cid:48) ( ) | is the logarithmic capacity (or trans-finite diameter) of K see [53, 65, 67]and §5. In this case cap ( K ) measures the improvement of the rate of convergence for small ω . For example, when Ω = C \ [ ∞ ) , then K = (
0, 1 ] , and cap ( K ) = is the factor in (3).2. From the optimality in Theorem 1 it follows that | ψ ( ω ) | < | ω | for ω ∈ Ω , and hencecap ( Ω ) <
1. This means there is always an improvement over the convergence of thetruncated series in (2).The inequality above also follows in an elementary way. Since Ω ⊃ D and ψ ( Ω ) = D , wehave | ψ | < D . Schwarz’s lemma then implies | ψ ( ω ) | < | ω | for all ω ∈ D (or else ψ would be a rotation, contradicting our assumption that Ω is a strict superset of D ).3. The approximations in Theorem 1 are expressed in terms of the conformal distance ρ ( ω ) of the point ω to the boundary of Ω . When Ω is the universal cover of ˆ C with a discreteset of punctures, then as ω approaches a puncture, the uniformization map typically hasa logarithmic singularity, since it has to locally accommodate the Riemann surface of alogarithm. Then, the Euclidean distance in C to the puncture is exponentially larger thanthe conformal distance to the boundary. The result of this exponential distance distortionis a dramatic improvement in accuracy with respect to other methods of probing generalsingularities. See §6 for illustrative examples.8 ote 4. In the case Ω is biholomorphically equivalent to C and F ∈ F Ω , the re-expanded series M F ◦ M ϕ converges in C faster than geometrically. The rate of convergence of M F ◦ M ϕ at a point ω is estimated from Cauchy’s formula by | ω | n inf R > | ω | sup | s | = R | F ( s ) s − n − | . A typical exampleis the unifomization of C \ { } on C , say for a function such as F ( ω ) = ( − ω ) a , with a ∈ C \ Q .Here ϕ ( z ) = ( − e − z ) , and ϕ ∗ F ( z ) = e − az , whose Maclaurin coefficients decrease factorially. Note 5.
1. In many applications the underlying Riemann surface is the uniformization of ˆ C with a discrete set of punctures, and moreover with special structure of the puncture posi-tions. In such cases it is often possible to construct an explicit uniformization map. In thissituation, there is a dramatic improvement over existing methods such as a Padé approxi-mation, Conformal-Taylor and also over a Padé-Conformal approximation (Padé combinedwith a conformal map, not necessarily a uniformizing map). See for example the tritronquée solution of P I in §6.2.2. Some uniformization maps have elementary approximations (conformal maps of trunca-tions of the Riemann surface, cf. §4-5), that are near-optimal and potentially simpler to use,see §3.2. We start with some known uniformization maps and their properties, and then construct newuniformization procedures needed in some applications of interest. . A simple but important case is the one-cut domain Ω = C \ [ ∞ ) , for which ψ ( ω ) = − √ − ω + √ − ω with inverse ϕ ( z ) = z ( + z ) (12)The optimal rate of convergence obtained from the Maclaurin series of a generic function F whosemaximal analyticity domain is this Ω , and is continuous up to ∂ D is, see (5), | F ( ω ) − ˆ R n ( ω ) | ∼ | ω | n (cid:12)(cid:12) √ − ω (cid:12)(cid:12) (cid:12)(cid:12) + √ − ω (cid:12)(cid:12) n − (cid:107) F (cid:107) ∞ ; ω ∈ Ω The constant C = ∂ Ω , is simply ψ (cid:48) ( ) . For the domain with two opposite cuts, Ω = C \ ( − ∞ , − ] ∪ [ ∞ ) , the maps are ψ ( ω ) = (cid:115) − √ − ω + √ − ω with inverse ϕ ( z ) = z + z (13)with ψ ( ω ) > ω ∈ (
0, 1 ) . The capacity of 1/ ∂ Ω is now C = m symmetric cuts emanating from the vertices of a regular polygon.See Appendix 7 and [48]. This example also generalizes to Ω = C \ ( − ∞ , − a ] ∪ [ b , ∞ ) : see (66) inAppendix 7. Note 6.
There is a more general principle behind (13) worth mentioning:
Lemma 7.
Let Φ by a conformal map of D to some domain D ⊂ C , and let c = Φ (cid:48) ( ) > . Then, Φ n ( z ) : = Φ ( z n ) n maps D conformally to n symmetric copies of D n , i.e., to (cid:91) (cid:54) j (cid:54) n − e π ij / n D n roof. In a neighborhood of zero, Φ n is uniquely defined by Φ n ( z ) = | c n | zH ( z n ) , where H is analytic at zero and H ( ) =
1. Since Φ (cid:54) = D \ { } , by the monodromy theorem, Φ n extends analytically to D . Since Φ is injective on D , Φ n ( z ) = Φ n ( v ) implies z n = v n . Now, Φ n ( z ) = Φ n ( v ) , written as | c n | zH ( z n ) = | c n | vH ( v n ) implies z = v , and thus Φ n is injective.Since Φ is onto D , the rest follows from injectivity.With proper adaptations, this construction extends to uniformization maps of Riemann sur-faces. For functions analytic on the universal covering Ω of ˆ C \ {−
1, 1, ∞ } the maps are (comparewith [38], p. 99) ψ ( ω ) = K (cid:0) + ω (cid:1) − K (cid:0) − ω (cid:1) K (cid:0) − ω (cid:1) + K (cid:0) + ω (cid:1) with inverse ϕ ( z ) = − + λ (cid:18) i − z + z (cid:19) (14)Here λ = θ / θ is the elliptic modular function, θ , θ are Jacobi theta functions, and K ( m ) =( π /2 ) F ( , ; 1; m ) is the complete elliptic integral of the first kind of modulus m = k [38]. Thecapacity is C = ψ (cid:48) ( ) = π − Γ (cid:0) (cid:1) ≈ R n ( ω ) in (5) converges on thewhole universal covering of ˆ C \ {−
1, 1, ∞ } ; that is, on all the Riemann sheets of the underlyingfunction. See also Figure 2 and §6.4.Note that the improvement in accuracy is particularly dramatic near singular points. Indeed,the leading order asymptotic behavior of ψ near ω = ψ ( ω ) ∼ + π / ln ( − ω ) . As aresult of this log distortion, calculating a function after uniformization in the conformal disk at z = − Ω at ω ≈ − e − × π ≈ − × − !Figure 2: Uniformization of the universal cover Ω of ˆ C \ {−
1, 1, ∞ } by the map ϕ in (14). Thetesselation in the left-hand figure is adapted from [15]. Here the points A , B , C are mapped to {− ∞ , 1 } , respectively. The gray geodesic triangle in the Poincaré disk is conformally mappedby ϕ onto the upper half plane of Ω in the middle figure, and Schwarz reflections continue ϕ tothe whole disk, with image onto Ω . The blue path in the disk is mapped to the spiral path in themiddle figure on the universal cover Ω of C \ {−
1, 1 } . The spiral is also the image of the bluecurve in the right-hand figure, obtained as a composition of elementary maps, as discussed in§3.2. 10 . The uniformization map for ˆ C \ { e − i θ , e i θ , ∞ } , another important case in applications, isgiven by ψ ( ω ) = Z ( ω ; θ ) − Z ( θ ) − ( Z ( θ )) ∗ Z ( ω ; θ ) , Z ( ω ; θ ) ≡ K (cid:0) + i (cid:0) ω sin θ − cot θ (cid:1)(cid:1) − K (cid:0) − i (cid:0) ω sin θ − cot θ (cid:1)(cid:1) K (cid:0) + i (cid:0) ω sin θ − cot θ (cid:1)(cid:1) + K (cid:0) − i (cid:0) ω sin θ − cot θ (cid:1)(cid:1) (15)The inverse is given in terms of the modular λ function by ϕ ( z ) = e i θ − i λ (cid:32) i (cid:32) K (cid:0) − i cot θ (cid:1) − K (cid:0) + i cot θ (cid:1) z K (cid:0) + i cot θ (cid:1) + K (cid:0) − i cot θ (cid:1) z (cid:33)(cid:33) (16)These maps are obtained from (14) by a suitable Möbius transformation and disk automorphism. It is straightforward to generalize the two previous examples to the universal covering ofˆ C \ { ω , ω , ∞ } , where ω , ω ∈ C . The uniformization maps are again expressed in terms of theelliptic function K and the elliptic modular function λ . The important case in the previous ex-ample, with two complex conjugate points, ω = e i θ = ω , which occurs in many applications,is discussed further in §6.3.2. For uniformization of other Riemann surfaces based on the universal covering of ˆ C \{−
1, 1, ∞ } possessing a nontrivial fundamental group see for example [38], §2.7.2. See also [48]for a collection of explicit uniformization maps of ˆ C \ S where S is a finite set of points in ˆ C ,including for example S = {−
1, 0, 1, ∞ } , S = {− −
1, 0, 1, 3, ∞ } and S = {
0, 1, e π i /3 , ∞ } , andwhen S consists of the n -th roots of unity. Algebraic functions have compact Riemann surfaces,and some explicit uniformizing maps can be found in Schwarz’s table in [38]. In certain special cases the uniformizing map produces a meromorphic or rational func-tion, in which case a subsequent Padé approximation becomes exact. For example, the function F ( ω ) = √ − ω has a compact Riemann surface Ω (as all algebraic functions do). The uni-formizing map ω = ϕ ( z ) = z / ( + z ) makes ϕ ∗ F meromorphic, ϕ ∗ F ( z ) = ( − z ) / ( + z ) ,hence analytic on the Riemann sphere, and Padé [ n , n ] is exact for n >
0. A more sophisticatedexample is the Riemann surface Ω of functions with three square root branch points, which isuniformized by ( z − ) / ( z + ) , [38], and the functions become rational; the uniformizationtheorem brings Ω to ˆ C . This is another case where Padé becomes exact. The following uniformization is described in [59], p. 323. However, we provide here amore precise mathematical description of the Riemann surface, and a shorter proof. Let Ω bethe Riemann surface with fixed origin of ˆ C \ S , with S = {
0, 1, ∞ } . Then the conformal map ψ : Ω → D is the elliptic nome function q : ψ ( ω ) = q ( ω ) = e − π K ( − ω ) K ( ω ) (17)and ϕ is the inverse elliptic nome function. We have for small z ϕ ( z ) = z − z + z + · · · (18) Proof.
We start with the well known uniformization over the upper half plane H of the universalcover of ˆ C \ {
0, 1, ∞ } by τ ( ω ) = i K ( − ω ) / K ( ω ) , see [38], p 99. Note that q = e π i τ . The map τ isalso conformal between H and the geodesic triangle ∆ , see Fig. 3. The inverse function, λ = τ − is the modular elliptic function. The set T = e π i ∆ is the curvilinear triangle T in Figure 3, wherethe upper side is an analytic curve. We aim to show that W = q − extends analytically to D bysuccessive Schwarz reflections of T , and its reflections, across their sides. We also note that with ω ∈ T we have W ( z ) = λ ( π i log z ) . Since λ is analytic in the upper half plane, W admits analytic11igure 3: Images of geodesic triangles in H through h ( ω ) = ω (cid:55)→ e π i ω . Like-colored pieces ofthe boundaries correspond to each other through h .continuation through the rays above. Using the periodicity property λ ( t + ) = λ ( t ) we see thatthe monodromy of W around zero is trivial. Furthermore, the product representation of λ for (cid:61) ω > λ ( ω ) = e π i ω ∞ ∏ k = (cid:18) + e k π i ω + e ( k − ) π i ω (cid:19) shows that lim t → i ∞ λ ( t ) =
0, hence z = W , and W ( ) =
0. EachSchwarz reflection of T mirrors Schwarz reflections of ∆ . It is clear that the set of all thesereflections cover D since e π i H = D and, as in the uniformization proof for λ , the reflections of ∆ cover H . The curves starting at zero on the universal cover of ˆ C \ {
0, 1, ∞ } correspond to curvesin D . Note 8.
1. This Ω in example 8 is distinct from the universal covering of ˆ C \ {−
1, 1, ∞ } consid-ered in example 3 above, since Ω in example 8 is a covering with fixed origin; the functionsliving on Ω are analytic at zero on one sheet and may be singular there on all others.2. The reflection of 0 across the curvilinear side of T is i , and successive reflections carry it tothe binary rational angles. Thus, zero becomes inaccessible on “higher Riemann sheets”.As an example, Fig. 4 shows the singularities on the Riemann surface of the complete ellipticintegral K ( ω ) , seen after uniformization through ϕ in (18), with 200 nonzero terms of the right-composition ϕ ∗ K . We note that the only possible singularities of K on any Riemann sheet are {
0, 1, ∞ } . The large number of singularities visible in Fig. 4 is a reflection of the large number of“deep” Riemann sheets visible after uniformization; more details are given in the caption. Let Ω Z be the set of equivalence classes of curves starting at 0, modulo homotopies in C \ Z . Then Ω Z is the universal cover for the Borel transforms of the (divergent, normalized)asymptotic series of the tritronquée solutions of the Painlevé P I − P V equations [19]. Figure 5shows the singularity structure of Y , the Borel transform of the tritronquée solution y of PainlevéP I , on its Riemann surface. Starting from a finite number of terms of the asymptotic expansion, Not ˆ C \ Z \ { ∞ } ; a curve around infinity is undefined, since Z is not compact. ϕ ∗ K on a circle of radius 0.997 using as input 200 nonzero terms of itsMaclaurin series. The marked singularity is at 0, after a clockwise loop around 1 and a clockwiseloop around ∞ . The higher the Riemann sheet, the more suppressed is the singularity. Properlydilated, the singularities exhibit a periodic structure, reflecting the simple monodromy groupof K . Compare with Fig. 5, the Riemann surface of the Borel transform of the tritronquée ofP I which has infinitely many singularities on each Riemann sheet, and exhibits a less regularstructure.the uniformization of the associated Riemann surface described in Theorem 9 permits analyticcontinuation onto the higher sheets of this Riemann surface. Theorem 9. Ω Z is uniformized by ψ = ϕ − , where ϕ = π i ln ( − q − ) , with q the elliptic nomefunction, as in (17) . See Fig. 5, and also Fig. 2.
Proof.
This result follows from example 8 of the previous section (see equations (17) and (18)),and the following lemma.
Lemma 10.
The function Φ = ω (cid:55)→ π i log ( − ω ) maps conformally the surface Ω in example 8 of theprevious section onto Ω Z .Proof. Let G be the free group with generators r k , where for k ∈ Z r k is the anticlockwise rotationaround k . An equivalence class of curves of Ω Z can be described by a word r k · · · r k m , for some m plus a piecewise linear arc in C . Let ˜ r , ˜ r the generators of the group of Ω . The result followsby noting that the element r k ∈ G is obtained through Φ from ˜ r ˜ r k . Injectivity is clear. (In words,through Φ , any “address” in Ω Z can be reached uniquely from an address on Ω .) Note 11.
1. Denoting by Y the Borel transform of the tritronquée solution y of Painlevé P I , an n -th root test on ϕ ∗ Y shows that its Maclaurin coefficients grow roughly like e √ n whichimplies exponential growth towards the boundary of the conformal disk. Importantly, Y does not have exponential growth on any particular Riemann sheet but there is substantial13igure 5: Singularities of ϕ ∗ Y along the conformal circle, where Y is the Borel transform of thetritronquée solution y of Painlevé P I , whose Riemann surface is covered by ϕ in Theorem 9.We used n =
200 nonzero terms of the asymptotic expansion of y , after Borel transform. Thesingularities depicted correspond to those of Y at −
2, reached by analytic continuation frombelow (green) − + + − ± n on the n -th sheet, [19], which results in cumulativeexponential growth on Ω , giving the two thick lines on the graph.In this sense ϕ ∗ Y can see the totality of the Riemann surface, even with 200 terms of theseries.2. The map in Theorem 9, optimal for otherwise featureless functions, is suboptimal for es-sentially any particular singularity of Y , including infinity on a given Riemann sheet. Asfor a general resurgent function, detailed information is available about each singularity.Singularity elimination (see §4) applies to Y , and is far more accurate for singularities atfinite distance, and specific methods perform much better at infinity as well.3. Resurgent functions have an infinite set of resurgence relations, so the actual Riemannsurface of Y is Ω G = R Z / G where G would be the group of relations, and in all likelihooduniformizing Ω G explicitly is beyond reach. However, these relations are contained in theshape of singularities, so singularity elimination (as discussed in §4) should achieve anoptimal, or near-optimal final result. C . We demonstrate this procedure with an example. Let ψ and ϕ be the maps in (12), and define ψ n ( ω ) = [ ψ ( ω n )] n , with inverse ϕ n ( z ) = [ ϕ ( z n )] n , for n ∈ Z , with the usual branch choices. Theorem 12.
The composition map ψ n ◦ ψ n − ◦ · · · ◦ ψ ( ω ) converges, as n → ∞ , to ψ ( ω ) in (17) , theelliptic nome function. roof. We first note that the singular points of ψ k are { { σ j } j (cid:54) k − , ∞ } , where { σ j } j (cid:54) k − are the k -th roots of unity (with zero a point of analyticity on the first Riemann sheet). Secondly, for any n ∈ N , ψ n ◦ ψ n − ◦ · · · ψ is only singular at {
0, 1, ∞ } (with zero a point of analyticity on thefirst Riemann sheet); this follows immediately by examining the singularities of ϕ k for k ∈ N .Convergence of the composition follows from the fact that ψ k = − k ω ( + O ( ω k )) (see also theproof of Lemma 13).Next, we note that the non-constant term of the Puiseux expansion of ψ n at the finite nonzerosingularities is of the form σ j + const (cid:112) ω − σ j ( + o ( )) , and 1 + const . ω − ( + o ( )) at infinity.We describe the monodromy group of ψ k in terms of the generators r σ j and r ∞ . A straightforwardcalculation shows that any of the elements r σ j , r ∞ , r ∞ r σ j of the monodromy group of the Riemannsurface of ψ k is mapped on a single r σ j or r ∞ of the monodromy group of the Riemann surface of ψ k , while r ∞ r σ j for i (cid:54) = j is mapped inside D . Injectivity of the limit map is also straightforward.An alternative proof, based on convergence to q − is given in Lemma 13 below. Lemma 13.
1. We have ϕ ◦ ϕ ◦ ϕ ◦ · · · ◦ ϕ k → q − , k → ∞ (19) uniformly in D .2. Moreover, starting the doubling iteration with ϕ n instead of ϕ , we have more generally ϕ n ◦ ϕ n ◦ ϕ n ◦ · · · ◦ ϕ k n → z (cid:55)→ [ q − ( z n )] n , k → ∞ (20) which uniformizes the covering with fixed origin of ˆ C \ S where S are the n-th roots of unity.Proof.
We note that z ( ω ) is analytic in the unit disk (see also (18)).The Landen transformation [38] implies that the map ϕ ( ω ) satisfies ϕ ( w ( z ) ) = w ( z ) ,and therefore w ( z ) = ϕ − ( w ( z )) . Iterating this identity, we find in general, for m ∈ N and z ∈ D , that w ( z m + ) m + = ϕ − m ◦ · · · ◦ ϕ − ( w ( z )) For any function analytic within the unit disk and such that f ( z ) = cz ( + o ( z )) , and for any r < n → ∞ f ( z n ) n = z uniformly in D r ( ) (since f ( z ) / z is uniformly boundedthere). This implies lim m → ∞ ϕ − m ◦ · · · ◦ ϕ − ( w ( z )) = z The result (19) follows by straightforward function inversions. The proof of (20) for general n isvery similar. The proof of (21) follows by noting that ϕ n ( ω ) = ϕ ( ω n ) n . Note 14.
1. The limit in Lemma 13 can also be expressed as an infinite iteration limit of theascending Landen transformation, L ( z ) = √ z / ( + z ) :lim k → ∞ L ◦ L ◦ · · · ◦ L ( z k ) = (cid:113) q − ( z ) (21)2. This result has an important practical implication: we can approximate a complicated (e.g.,elliptic) map by a composition of elementary (e.g., rational) maps. For example, the right-hand figure of Fig. 2, shows the result of a finite number of iterations of the elementarymaps ϕ n . The spiral path on the universal cover Ω of C \ {−
1, 1 } in the middle figureis the image of the blue curve in the right-hand figure under the six-step composition ofelementary maps ϕ ◦ ϕ ◦ ϕ ◦ · · · ◦ ϕ . Thus, these elementary rational maps enable oneto explore a finite number of Riemann sheets. See also §6.4.15 Singularity elimination.
The principal motivation underlying the analysis in this section is to develop new methods togive precise determinations of the location and nature of a chosen singularity, and informationabout the behavior of the function near the singularity. The main application of this procedure isto explore the singularities of resurgent functions, which have elementary singularities of the formin (1).For such functions, we show that there exist operators that regularize the singularities, inthe sense of transforming them into points of analyticity. This allows for a detailed and preciseanalysis of the singularity structure, type, and local expansions. In general, these operationscannot be reduced to conformal maps. For example, a function with a singularity of the typelog ( − ω ) A ( ω ) + B ( ω ) , with A , B holomorphic at ω =
1, cannot be composed with a holo-morphic map ϕ with ϕ ( ) = ϕ ( ) = ω =
1. Theproof is straightforward . Uniformizing the surface of the log at ω = N ∑ k = c k ( ω − ω k ) α (22)(with a common exponent α ) can be reduced to a rational one by the simple procedures describedbelow, but its Riemann surface is not uniformizable by any simple map for general ω k . In this section we analyze the general structure of singularities of Laplace convolution, see (23),and describe the process of elimination of elementary singularities of type (1). We begin withsome definitions and the description of the procedure. We also analyze the singularities ofconvolutions of general analytic functions F , G in the neighborhood of the singularities of F , G . Definition 15.
Laplace convolution of F and G is defined as ( F ∗ G )( ω ) = (cid:90) ω F ( s ) G ( ω − s ) ds (23) Note 16. 1.
For (cid:60) β > −
1, an important role is played by the linear operator L β of convolutionwith ω β , followed by multiplication by ω − β : ( L β F )( ω ) = ω − β (cid:90) ω F ( s )( ω − s ) β ds (24)As shown below in Lemma 17 and Lemma 23, if F is analytic in D with an elementary singularityof type (1) at ω , then L β F is also analytic in D with an elementary singularity of type (1), where α is replaced by α + β +
1. In other words, the convolution operator L β in (24) allows us to modifythe nature of the chosen singularity. Indeed, taking A = B =
0, and then A = B ( ω ) = ω , we see that ϕ must be analytic at 1, hence ϕ ( + s ) = + o ( s ) . But then log ( − ϕ ) is unbounded at 1. . Let us normalize so that the chosen singularity is at ω =
1. For singularity elimination, it isconvenient to choose β so that α + β + = k /2, for some nonnegative odd integer k , so that thesingularity of L β F becomes a square-root branch point ( L β F )( ω ) = ( − ω ) A ( ω ) + B ( ω ) (25)in a neighborhood of 1 where A , B are analytic at 1. Let ϕ denote a conformal map of the unit disk D to some Riemann surface Ω , mapping 0 to0, and 1 to 1, and such that ( ϕ − ) has a double zero at z =
1. Simple examples of such mapsare ϕ ( z ) = z − z ; ϕ ( z ) = z ( + z ) ; and ϕ ( z ) = z + z . (26)Ideally, such a map would take part of the domain of analyticity of F to the unit disk, ensuringconvergence of ϕ ∗ ( L β F )( ω ) . Replacing in (25) ω by any ϕ ( ω ) for any ϕ in (26) we see that ϕ ∗ ( L β F )( ω ) is now analytic atboth ω = ω = The inverse functions of the maps above are elementary, and inverting ϕ ∗ ( L β F ) results ina special function series representation of F . But even if, for a more complicated ϕ , invertingconvolution can only be done by some (convergent in the limit) numerical scheme, we reiteratethat the purpose here is to most accurately recover an unknown F from truncated Maclaurinseries, and not to provide economical approximations of a known function. Lemma 20 applies to much more general singularities than the elementary singularities in(1), and can be used for their elimination. However, in this paper we will not attempt to classifyin general the singularities that can be eliminated.
Lemma 17 (Preservation of Riemann surfaces by L β ) . Assume Ω is a covering with fixed origin of C \ S, and ∈ S . We generalize L β when (cid:60) β > − , by interpreting the convolution in (24) as anintegral along a smooth curve γ : [
0, 1 ] → Ω with γ ( ) = and γ ( ) = ω .Then, if F is analytic on Ω , L β F is also analytic on Ω .Proof. Analyticity in D is easily shown by replacing F by M F , and then using dominated conver-gence to integrate term by term M F . Then we calculate L β (cid:32) ∞ ∑ k = a k ω k (cid:33) = ∞ ∑ k = Γ ( β + ) Γ ( k + ) Γ ( β + k + ) a k ω k + ; | ω | < k we have Γ ( k + ) Γ ( β + k + ) = O ( k − − β ) implying that analyticity in D is preserved.Next, examining (24) we interpret F ( s ) ds as a bounded complex measure along any path from0 to ω and note that the integrand, and hence the integral over γ are manifestly analytic at allpoints in γ ([
0, 1 )) . Let ε > D ε ( ω ) ⊂ Ω , let ω ∈ D ε ( ω ) ∈ γ ([
0, 1 )) and writethe integral along γ as (cid:82) ω = (cid:82) ω + (cid:82) ωω . The first integral is analytic by the argument above. In other words, the curves in Ω start from zero and do not cross zero again unless they are homotopic to a point. ω to ω , where we changevariable s = ω − t to get (cid:90) ω − ω F ( ω − t ) t β dt which is analytic since F is analytic in D ε ( ω ) Note 18.
Convolving more general functions with singularities alters the Riemann surface, ingeneral. For instance ( − ω ) ∗ ( − ω ) , is an elementary function with a square root typesingularity at 1 and a log-type singularity at 2. We refer to [61] for the theory of the location ofsingularities generated via convolution.For analyzing the type of singularities of F ∗ G for more general F , G we restrict our attentionto star-shaped domains N ⊃ D (meaning that for each point in N , the line segment connectingit to 0 is also in N ) Lemma 19 (Analyticity of Convolution) . Let
N ⊃ D be a star-shaped domain in C . If F and G areanalytic in N , then so is F ∗ G.Proof.
This is clear if we interpret F ( s ) ds in (23) as a finite complex measure on compact sets of N . The next Lemma describes how singularities at 0 and ω interact. For simplicity of notation,we normalize ω so that ω = Lemma 20 (Calculation of singularities of convolution) .
1. Assume F is analytic in a neighbor-hood of (
0, 1 + ε ] , possibly singular at zero but F ∈ L ((
0, 1 + ε )) , G is analytic in a neighborhoodof [
0, 1 + ε ] \ [
1, 1 + ε ] with continuous lateral limits (possibly different) on the cut. Assume furtherthat there exist two functions, S analytic in D ε ( ) \ [
1, 1 + ε ] , and B is analytic in the disk D ε ( ) such that ( F ∗ G ) ( ω ) : = (cid:90) ω F ( ω − s ) G ( s ) ds = S ( ω ) + B ( ω ) (28) Then, (cid:90) ω F ( ω − s ) G ( s ) ds = S ( ω ) + B ( ω ) (29) where B ( ω ) is analytic in D ε ( ) .The result can be adapted to the case where instead G = G ( k ) for some k ∈ N , and G satisfies theassumptions of the lemma.2. The result extends to Riemann surfaces Ω as in Lemma 17 if F ( ω ) = ω β , (cid:60) β > − . Moreprecisely, recalling that we normalized ω so that its projection on C is 1, we choose a line segmentin Ω emanating from ω whose projection in C is [
1, 1 + ε ) and replace the branch jump across [
1, 1 + ε ) by the branch jump across this segment.Proof of Lemma 20.
1. We note that the jumps across the cut [
1, 1 + ε ) of (cid:82) ω F ( ω − s ) G ( s ) ds andof S must coincide, and it evidently also coincides with the jump across the cut [
1, 1 + ε ) of (cid:82) ω F ( ω − s ) G ( s ) ds (since the integrand is analytic in D . Using Lemma 19 and the assumptionsof Lemma 20, we see that B ( ω ) : = (cid:82) ω F ( ω − s ) G ( s ) ds − S ( ω ) is analytic in D ε ( ) \ [
1, 1 + ε ] andcontinuous in D ε ( ) , and Morera’s theorem implies that B is analytic in D ε ( ) . As usual, by “jump across the cut” we mean the upper limit minus the lower limit along the cut. k we write (cid:82) ω = (cid:82) + (cid:82) ω . In the second integral we integrate by parts k times to eliminate the derivatives of G and apply the first part.2. Follows in the same way, noting that only a neighborhood of ω is involved in both thestatement and in the proof of 1. Note 21.
Lemma 20 is a “localization” lemma, whose point is that S ( ω ) can be calculated fromlocal expansions of F and G at 0 and 1 respectively, whenever such expansions exist , since (cid:82) ω F ( ω − s ) G ( s ) ds only depends on these local expansions. Note 22.
1. Clearly, for the elementary singularities of type (1), the local expansion assumed in(28) always exists. (Note that other singularities of F and G , including subtler singularitieson the second Riemann sheet, may mean that the local expansion of F at zero and that of G at 1 have radius of convergence < L β is important in applications, as it gives a simple way to trans-form the nature of the singularity. Lemma 23 below shows that the effect of this operatorcan be implemented directly on the original expansion coefficients.3. The transformation of singularities can also be understood in terms of fractional derivatives,as is clear from the representation in (24). See [5] for an application to Borel transforms. Lemma 23.
Assume F is analytic in D , and at ω = it has an elementary singularity of type (1) . Then,for (cid:60) β > − the covolution L β F in (24) has an elementary singularity of type (1) with α replaced by α + β + .Proof. We rely on Lemma 20. Take first α / ∈ Z . In view of the second part, we may assume (cid:60) α > −
1. In our case, for small enough t , S ( ω ) = − i sin ( πα )( − ω ) β + α + (cid:90) ( − t ) β t α A ( + t ( − ω )) + B ( ω ) (30)as seen by the change of variable s = t ω . Writing near 1, the local expansion A ( ω ) = ∑ ∞ k = a k ( − ω ) k and inserting in (30), we see that S ( ω ) = ( − ω ) β + α + ∞ ∑ k = a k sin ( απ ) Γ ( β + ) Γ ( α + k + ) Γ ( β + α + k + ) sin (( β + α ) π ) ( − ω ) k (31)proving the assertion in this case. (Note the obvious analogy to singularities of hypergeometricfunctions.)For a logarithmic singularity, F ( ω ) = ln ( − ω ) A ( ω ) + B ( ω ) , we have ( L β F )( ω ) = ω − β (cid:90) ζ ( ζ − u ) β ln ( − u ) A ( + u ) d u , ζ = ω − L β F is S ( ω ) = π i ω − β (cid:90) ζ ( ζ − u ) β A ( + u ) d u (33) Very generally, even in the absence of local expansions, Plemelj’s formulas [2] give a local representation S ( ω ) inthe form π i (cid:82) + ε ( ω − τ ) − (cid:82) λ F ( λ − s ) ∆ G ( s ) dsd τ , where ∆ G is the jump across the cut of G . A ( ω ) = ∑ ∞ k = a k ( − ω ) k near 1, one can verify that, in order to achieve the same branchjump with an S having a cut [ ε ) we define S ( ω ) = π i ω − β ( ω − ) β + ( − e π i β ) − ∑ ∞ k = b k ( ω − ) k where ( ω − ) β + is defined to be positive on the upper part of the cut [ ∞ ) and b k = Γ ( β + ) ( − ) k Γ ( k + ) Γ ( β + k + ) a k (34)proving the statement when α =
0. Using the second part of Lemma 20, the general case followsfrom it by integration by parts in the branch jump formula. An explicit example is shown in§6.
Theorem 24 (Singularity Elimination) . Let Ω be a covering with fixed origin of C \ S where ∈ S,assume F is analytic on Ω , that ω is on the boundary of Ω and that F has a singularity of type (1) at ω .Without loss of generality, we can assume that the projection of ω on C is . Then the singularity can beeliminated by a combination of an appropriate L β and a composition with a rational map such as those in (26) .Proof of Theorem 24. The proof follows from Lemmas 19, 20, and items 1, 3 and 4 of Note 16 atthe beginning of §4.1.
Note 25 (Comments on Theorem 24) .
1. Theorem 24 yields a practical method to apply sim-ple convolution and conformal maps to make the local behavior near a singularity purelyanalytic. Since analyticity and singularity are highly sensitive to being distinguished nu-merically, this therefore provides a numerical mechanism to refine both the location of thesingularity and also to refine the convolution parameter β , in order to determine the powerexponent α which characterizes the nature of the original singularity. This is particularlyuseful when empirical analysis is the only option available. See examples in §6.2. After eliminating a singularity (say at ω ) of F , if we uniformize the Riemann surface of thenew function F , then z = ψ ( ω ) ∈ D , F ◦ ϕ is analytic at z and can be calculated conver-gently and with rigorous bounds. This means that the complete information about the sin-gularity of F (such as the functions A , B if the singularity is of type (1)) follows. Uniformiza-tion of the new surface may be impractical, in which case an appropriate Conformal-Taylorexpansion (see §5.3) would provide, with sub-optimal rate of convergence, the same infor-mation (or, non-rigorously, using Padé approximants).3. In practical computations we noticed that the precision of the local information obtainedfrom singularity elimination is usually significantly better than what is obtained numer-ically from an explicit uniformization map. See for example the Painlevé I computationdescribed in Note 2.4. There are important special cases in which all singularities are eliminated, for examplearrays of pure singularities of the type ∑ nk = ( ω − ω k ) α , which can be transformed by anappropriate L β into a sum of logs, which becomes rational after differentiation.5. Still for empirical analysis, for all three maps in (26) analytic continuation past 1 of ϕ ∗ F leads to the second Riemann sheet of F . For example, the map ϕ takes the origin ofthe second Riemann sheet of F to ω = ϕ ∗ F . Therefore,singularity elimination provides access to higher Riemann sheets. This will be an importantelement of the tools developed in §5 to refine approximate data about the Riemann surface.20 ote 26 (Important special cases of singularity transformation and elimination) . For example,we can manipulate and eliminate the log singularity at ω = K ( ω ) . See §6.5. Both steps of the analytic operations described above for singularity elimination have a simpleand explicit counterpart as operations at the level of the series coefficients, taking Maclaurin poly-nomials to Maclaurin polynomials. This is important in applications, since series compositionswith many coefficients is computer-algebra time-expensive. Recall first the explicit expression(27) for the action of the convolution operator L β on series. Here we consider the second step,that of singularity elimination, and derive explicit formulas for the composition with the elimi-nation maps ϕ , ϕ and ϕ listed in equation (26) of item 3 of Note 16. Lemma 27.
On the level of series, the operator ϕ ∗ of composition with the conformal map ϕ is given by1. ϕ ∗ ( ∑ nk = a k ω k ) = ∑ nk = b k ω k whereb = a ; b k = (cid:98) ( k + ) /2 (cid:99) ∑ l = U k , l a l (35) where U k , l is the coefficient of x l in the kth Chebyshev polynomial of the second kind U k ( x ) , explicitly,U k , l = ( − ) l l + (cid:18) k + + l k − − l (cid:19) , k odd ( − ) l (cid:18) k + l k − l (cid:19) , k even (36) ϕ ∗ acts by ϕ ∗ ( ∑ nk = a k ω k ) = ∑ nk = b k ω k whereb = a ; b k = k ∑ l = ˜ U k , l a l (37) where ˜ U k , l is the coefficient of x l in the kth Chebyshev polynomial U k ( x − ) .3. ϕ ∗ acts by ϕ ∗ ( ∑ nk = a k ω k ) = ∑ nk = b k ω k , where the new series coefficients b k are related to theoriginal series coefficients a k viab = a ; b k = k ∑ l = U k , l a l + ; ( k odd), and b k = k ∑ l = U k , l a l ; ( k even ) (38) Proof.
We prove only part 3., since all three proofs are very similar. Define f ( ω ) = ω ( − x ω ) .Then, ( ϕ ∗ f )( ω ) = ω − x ω + ω (39)which is, up to multiplication by ω , the known generating function of U k ( x ) : ∞ ∑ k = U k ( x ) ω k = − x ω + ω and the rest follows easily by comparing coefficients.21 ote 28 (Remarks on numerical accuracy) .
1. Since the calculation of the b k from the a k in-volves summations, there can be cancellations, which could potentially become significantfor large k . The following lemma addresses the question of the accuracy of the b k , oreven how many coefficients can be meaningfully retained. This is relevant also for selectingwhich map results in the minimal loss of accuracy. This is an interesting question for whichexamples will be given in an accompanying paper [28], and for which rigorous estimatesare under investigation. Lemma 29.
For fixed large k, | U k , l | reaches its maximum value M k at l ∼ √ k + (cid:16) √ − (cid:17) M k ∼ − √ π k (cid:16) + √ (cid:17) k + , k → ∞ Proof.
This result can be derived by asymptotically solving the equation U k , l / U k , l − = l = l ( k ) , and using Stirling’s formula for large k in U k , l ( k ) .For example, using ϕ ∗ , taking into account the position of the maximum, for large k thecoefficient b k involves cancellations of terms a m , with m (cid:54) k weighted by (cid:16) + √ (cid:17) m ≈ m .2. In general, the accuracy needed can be calculated similarly, from the capacity C . In this Section we address the question of how to extrapolate the function F when the only inputis a finite number n of terms of its Maclaurin series about some point. Evidently nothing rigorouscan be said if n is fixed, and we focus on methods that are efficient and convergent as n → ∞ . Wepresent methods to determine approximate information about the singularity structure of F , andmethods to refine and corroborate this approximate information. We also adapt known resultsto provide precise rates of the convergence of these methods. Diagonal and near-diagonal Padé approximation is one of the most frequently used methods forempirical reconstruction.
Definition 30.
The [ m / n ] Padé approximant of F at ω = is the unique rational function A m / B n , withA m a polynomial of degree at most m, and B n a polynomial of degree at most n, for which we haveF ( ω ) − A m ( ω ) B n ( ω ) = O (cid:16) ω m + n + (cid:17) , ω → If we normalize B n ( ) = , then A m and B n are also unique. Since it can be calculated directly from theMaclaurin series M F of F, we also say that [ m / n ] is the Padé approximant of M F .A sequence of Padé approximants { [ n / n ] } n ∈ N is called diagonal, and { [ m j / n j ] } j ∈ N is near-diagonalif n j → ∞ and m j / n j → as j → ∞ .
22n spite of their simplicity (they are rational functions with the same Maclaurin series as F , inasmuch as their degree permits) Padé approximants are, in most applications, uncannilyaccurate and able to detect poles and branch points in the whole complex domain (in principle).However, except for special types of functions such as Riesz-Markov ones (see [73, 29] andreferences therein, and Note 31), they do not generally converge pointwise, but only in a weakersense, in the sense of capacity theory. For this reason Padé approximants can only be used as anexploratory tool. Nevertheless, we will explain how these exploratory findings can be backed uprigorously, in the limit n → ∞ , see Note 34, part 3.We briefly describe some important results (both negative and positive) concerning the con-vergence of Padé approximants, which seem to be little known to the applied community, outsidethe specialized literature. We also propose practical methods to overcome some of the limitationsof Padé approximants, see Notes 34, parts 1 and 4.An intrinsic limitation is immediately clear: as any sequence of rational approximations, theycan only converge in some domain of single-valuedness of their associated function.In fact, even for single-valued functions, uniform convergence of some diagonal Padé subse-quence to general meromorphic functions, the Baker-Gammel-Wills conjecture [9], was settledin the negative in a remarkable paper of Lubinsky in 2003 [54]. The phenomenon that preventspointwise convergence are the so-called spurious poles, or Froissart doublets, appearing at pointsunrelated to the properties of the associated function. In practice however, most often spuriouspoles appear infrequently and their exploratory value is largely unaffected.Diagonal (and near-diagonal) Padé approximations do converge in a weaker sense, namelyin capacity, and in this sense they “choose” a maximal domain of single-valuedness where theyconverge, maximizing also the rate of convergence near ω =
0. This choice however also comeswith a drawback: points of interest of F may be hidden in their boundary of convergence. Thisis actually a common occurrence in applications, and we propose methods to detect such hiddensingularities: see Note 34. Convergence in capacity is also a very difficult question, only elucidated in 1997, in the funda-mental paper of Stahl [70]. It is interesting to note that convergence in capacity is established atthis time only for functions analytic on Riemann surfaces which are universal covers of C \ E ,for sets E of zero logarithmic capacity or in domains in C bounded by piecewise analytic arcsunder a stringent symmetry condition [70]. We focus on the first type of functions, the only onesof interest here.The general theory of Padé approximants summarized below is based on [70]. For furtherdevelopments and refinements, see [6, 57].The theory is best described by doing an inversion and placing the point of expansion atinfinity rather than at ω =
0. It is shown in [70] that there exists a domain
D ⊂ ˆ C , uniqueup to a capacity zero set, whose boundary has minimal logarithmic capacity, which contains ∞ and where F is analytic and single valued. This D is the domain where near-diagonal Padéapproximants converge in capacity to F . The rate of convergence is controlled by the Green’sfunction g D (see, e.g., [72, 65, 67]) relative to infinity as follows. Define G D = e − g D . We have G D ∈ [
0, 1 ) and G D > D \ { ∞ } (in the case of interest, where cap( D ) > ε > V ⊂ D \ { ∞ } we havelim n → ∞ cap { ω ∈ V | ( F − [ m j / n j ])( ω ) > ( G D ( ω ) + ε ) m j + n j } = F has branch points, which occurs iff G D (cid:54) =
0, then for any compact set V ⊂ D \ { ∞ } andany 0 < ε (cid:54) inf ω ∈ V G D ( ω ) we havelim n → ∞ cap { ω ∈ V | ( F − [ m j / n j ])( ω ) < ( G D ( ω ) − ε ) m j + n j } = Note 31.
1. In the rather generic case when D is simply connected, then G D = | ψ ∞ | , where ψ ∞ is a conformal map from D to D , with ψ ∞ ( ∞ ) =
0. Comparing with Theorem 1, wenote that if the maximal domain of analyticity of F happens to be this D , then the leadingorder rate of convergence in capacity of Padé would be optimal.2. When D is simply connected, in view of 1. above and (41), we see that Padé effectively“creates its own conformal map” of a single-valuedness domain for F , D , that can be re-covered, in the limit n → ∞ , from the harmonic function | G D | , obtained by taking the n -throot of the convergence rate (41).3. In very special cases, such as Riesz-Markov functions, under some further restrictions, theconvergence of Padé approximants is uniform on compact sets (cf. [29] and referencestherein). A Riesz-Markov is a function that can be written in the form F ( ω ) = (cid:90) ba d µ ( x ) x − ω where µ is a positive measure. Riesz-Markov functions occur frequently in certain appli-cations, but general functions cannot be brought to this form. For example, a commonsituation in applications, discussed in more detail in §6.3.2 below, is the situation of twocomplex conjugate singularities in C . This is not a Riesz-Markov function, and Padé pro-duces curved arcs of poles (see Figure 8) which do not relate to the properties of the func-tion.4. As mentioned, for more general functions, Padé approximants may place spurious poles(“Froissart doublets”) on sets of zero capacity, “random” pairs of a pole and a nearby zero,unrelated to the function they approximate.5. The numerators and denominators of Padé approximants are orthogonal polynomials, ina generalized sense, along arcs in the complex domain, but therefore without a bona-fideHilbert space structure. According to [70], this is the ultimate source of capacity-onlyconvergence, and of the appearance of Froissart doublets.6. If F has only isolated singularities on the universal cover of ˆ C with finitely many punctures,then ∂ D is a set of piecewise analytic arcs joining branch points of F , and some accessorypoints (similar to those of the Schwarz-Christoffel formula) associated with junctions ofthese analytic arcs. For an example see Figure 8. Padé represents actual poles of F by poles,and branch points by lines (either straight or curved arcs) where poles accumulate. Note 32 ( Potential Theory and Physical Interpretation of Padé Approximants).
There is a re-markable and intuitively useful physical interpretation, which can be derived from [70, 67], ofthe domain D and of the placement of poles of Padé.1. Take any set D (cid:48) of single-valuedness of F and let E (cid:48) = ∂ D (cid:48) be its boundary. Thinking of E (cid:48) as an electrical conductor we place a unit charge on E (cid:48) , and normalize the electrostaticpotential V ( x , y ) = V ( ω ) , ω = x + iy (always constant along a conductor) by V ( E (cid:48) ) = E (cid:48) is cap ( E (cid:48) ) = V ( ∞ ) .24. The domain boundary E = ∂ D of the domain of convergence of Padé is obtained by de-forming the shape (keeping the singularity locations fixed) of the conductor E (cid:48) (defined initem 1 of this Note) until it has minimal capacity.3. The equilibrium measure µ on E is the equilibrium density of charges on E in the settingabove. As j → ∞ the poles of the near diagonal Padé approximants place themselves(except for a set of zero capacity) close to E , and Dirac masses placed at these poles convergein measure to µ [70].4. For ω ∈ D , we have e − g D ( ω ) = | G D ( ω ) | = e − V ( ω ) . We compare the accuracy of two methods that have been used in the physics literature. Whilethey have been used rather infrequently and without convergence analysis, they can be quiteuseful to reach points outside of D . In the Conformal-Taylor (CT) method (as defined above in§2.1) a domain D ⊂ C of analyticity of F is chosen, and then one proceeds as in Theorem 1,with D in guise of Ω . The Conformal-Padé method (CP) [74, 14, 27] consists of applying Padéapproximants to CT, which typically results in a significant increase in accuracy, at the price ofhaving convergence in capacity only. Surprisingly, the CP method appears to have been usedeven less frequently than CT. Note 33.
1. If the domain D happens to be the maximal domain of analyticity, then of course,Theorem 1 shows that CT is optimal.2. The error control of each of CT and CP approximation is obtained from the map ψ as inTheorem 1.3. Also as a consequence of Theorem 1, CT can be improved by choosing a domain D withcap( D ) as small as possible within the class of domains having explicit conformal maps ψ .4. To expand the class of explicit maps, we note that one only needs a map ϕ : D → D whichis surjective (at the price of a slower rate of reconstruction of F ). Note 34 (Some Improvements to Conformal-Taylor (CT) and Conformal-Padé (CP)) . (Detecting singularities hidden in the analytic arcs of poles of Padé) .This is not an unusual situation. All resurgent functions coming from differential equationshave their singularities along half-lines starting from the origin, and symmetry reasonsgenerally make those rays part of the capacitor. This is the case, for instance, for the tritronquée Painlevé transcendents, P I –P V . In general the leading singularity is a branch-point, and, to ensure single-valuedness, Padé “creates” a cut, a curve from the point toinfinity where poles accumulate, thereby obscuring any genuine singularities along suchcuts. Two ways to detect such “hidden” singularities are described here. The simplestmethod is to place an artificial probe singularity near the arc. By the potential theoryinterpretation of Padé we know that this additional singularity will distort the minimalcapacitor, but it cannot move the genuine singularities. This simple procedure can beimplemented as follows: assume that J is an analytic arc of the Padé approximants ofthe function F ∈ G . Define F ( ω ) = F ( ω ) + ( ω − ω ) α where α / ∈ Z (a negative power istypically more effective), and ω is a point in the proximity of the arc J . Clearly, the Padé25pproximants of F determine the values of F as well, simply by subtracting out ( ω − ω ) α .The capacitor of F is necessarily different from that of F , since the probe singularity ω must be part of the new capacitor. Generically, the arc J moves when ω is chosen near anypoint of J which is a point of analyticity of F . Evidently too, points in J which are branchedsingularities of F cannot move.2. The second method consists of applying a form of CT. Any nontrivial conformal map ofdomains in C will typically distort all the arcs of Padé, exposing hidden singularities, andpossibly hiding ones that were visible before, and exposing domains that lie on the secondRiemann sheet relative to the cut ∂ D ( D being the domain of convergence of Padé).3. In the large n limit, CT provides a rigorous way to check the information inferred from aPadé analysis. Indeed, conformally mapping a mistaken domain (or parts of a Riemann sur-face), results in singularities in D , seen in an n -th root test of CT. This provides a practicalmethod to refine the conjectured domain (for example by tuning the conjectured locationsof the singularities) to remove these spurious singularities in D .4. The singularities of F that lie on the boundary ∂ D are mapped onto the unit circle T , theboundary of D , and can therefore be resolved using a discrete Fourier transform of theproperly normalized Maclaurin coefficients. Note 35.
Other numerical and analytic methods to enhance the accuracy of Padé approximants,for the purpose of discovering precise information about the Riemann surface of a function fromits Maclaurin approximants, will be described in an applications paper [28].
In this Section we present some examples that illustrate the effectiveness of the uniformizingmaps, and the methods of singularity elimination.
Écalle’s resurgent functions [37] are ubiquitous as solutions of equations in analysis. This ubiq-uity is fundamentally due to the closure of the family of resurgent functions under virtually alloperations used in solving analytic equations. A partial list of natural applications is the follow-ing: (i) solutions of generic systems of linear or nonlinear ODEs with meromorphic coefficients[19]; (ii) solutions of difference and q-difference equations [13, 64]; (iii) saddle point expansionsin finite dimensional exponential integrals [11, 51, 30, 31]; (iv) solutions of special classes ofPDEs [22, 23, 24, 25, 26]. There is also significant evidence, both numerical and analytic, forresurgent behavior in some infinite-dimensional physical systems, for example in quantum fieldtheory, statistical field theory and string theory [75, 55, 42, 43, 3, 35, 36, 32, 4, 47, 51].A resurgent function f possesses in its Borel plane, the plane of a properly normalized inverseLaplace transform F of f , a relatively simple Riemann surface, Ω with a fundamental group havinga rich structure of relations, encoded in Écalle’s alien derivations and bridge equations. Thebehavior of the Borel transform near its singularities determines asymptotic and global propertiesof the original function f . The Riemann surface Ω of the Borel transform function F is theuniversal cover of C with a discrete set of punctures, typically equally spaced. The singularitiesof F are generically of the elementary form in (1). In this paper we only partially exploit this rich26lgebraic structure provided by Écalle’s bridge equations. Further implications of this algebraicstructure will be described in a future publication.The Riemann surface of a resurgent function F , its precise form of singularities, as well as apriori bounds for F in Ω , are predetermined from the equation from which it originates. Con-sider, for example, a function F with an asymptotic series at infinity, and satisfying a generic N -th order system of linear or nonlinear ordinary differential equations with meromorphic co-efficients (see [19] for precise conditions). After normalization of the variables, the system canbe presented in the standard form, y (cid:48) ( x ) = Λ y + x − By + g ( y ; x ) , where Λ and B are diagonalmatrices with constant coefficients, and g contains the higher order terms in the linearization, aswell as the nonlinear terms. The data Λ and B from the linearized problem determines the basicBorel plane structure of the Borel transform function F : the Borel transform is analytic on theuniversal cover of ˆ C \ ∪ j (cid:54) N λ j Z , and at zero, and the nature of the singularities λ j is determinedby B . Furthermore, the nature of the singularities at all integer multiples, ∪ j (cid:54) N λ j Z , is determinedby the ODE. These singularities are all of the form indicated in (1). Difference and q-differenceequations of roughly the same form are also known to have resurgent solutions [13, 64], andsimilar comments and methods apply.For certain PDEs, such as the time-periodic one-particle Schrödinger equation, i ψ t = ( H + V ( x , t ) ψ , where H = − ∆ + V ( x ) is the time-independent part and V is time-periodic, the Borelsingularities Σ B have the following structured form: a periodic array of typically square rootbranch points (see e.g. [26]; exceptions are long range potentials such as Coulomb, see [21]),originating from the bottom of the continuous spectrum of H , and periodic arrays of poles,which come from the eigenvalues and resonances (“dressed states”) of H [23]. The solutioncan be extrapolated using the techniques described in this paper, providing far superior ap-proximation to the time evolution, as well as providing access to higher Riemann sheets. Theuniformization and conformal mapping methods described here give a new numerical approachto this well known difficult problem. Similar methods apply to the heat equation, or the waveequation with potential.Exponential integrals provide another important context for studying extrapolation and resur-gent asymptotics. Consider functions defined as exponential integrals, f ( ¯ h ) = (cid:90) Γ e − g ( x ) /¯ h d n x where f is a holomorphic function on an n -dimensional complex manifold, and Γ is a steepestdescent integration cycle. In the Borel plane, there are isolated singularities at g ( x c ) , associatedwith the critical points x c of the exponent function. The critical points x c can be classified ac-cording to their Arnold normal forms [7], and their location and hessian determine the buildingblocks from which a full trans-series expansion of f may be generated. The methods presentedin this paper provide substantial improvements to extrapolations of formal expansions of suchfunctions. The tritronquée
Painlevé transcendents are resurgent functions. The Painlevé equations, P I - P V ,have a common and simple Borel singularity structure, which can be arranged as integer-spacedsingularities along the real line (excluding the origin). Even the Conformal-Padé method, basedon the simple two-cut conformal map in (13), leads to a remarkably accurate extrapolation ofthe formal solution generated at infinity, throughout the complex plane: see [27] for a detailedanalysis of the tritronquée solution of P I . But with the Painlevé uniformizing maps of section §3.127 - - [ x ]- - [ x ] Figure 6: The poles of the tritronquée solution of the Painlevé 1 equation, which lie only in thewedge π ≤ arg ( x ) ≤ π of the complex plane, obtained by extrapolation of the asymptoticexpansion about x → + ∞ , using 200 input coefficients. The smaller black dots show the resultsusing the Conformal-Padé method in the Borel plane, as described in [27], while the largerred dots are obtained by replacing the conformal map with the Painlevé uniformizing map inTheorem 9. With exactly the same input data, the uniformizing map leads to a significantly betterextrapolation. There is comparable accuracy throughout the domain of analyticity, | arg ( x ) | ≤ π .significantly better extrapolation can be achieved. For example, as mentioned already in Note 11,the Stokes constant (which is known analytically from isomonodromy [49] and from asymptotics[19]) can be determined numerically to extraordinary precision with surprisingly little input data[27].A much stronger demonstration of the power of the uniformization map is to reconstruct the P I tritronquée solution throughout its domain of analyticity, and in its pole sector π ≤ arg ( x ) ≤ π , using only input from its asymptotic expansion about the opposite direction, as x → + ∞ .Recall that the solution of the P I equation, y (cid:48)(cid:48) ( x ) = y − x , is meromorphic throughout thecomplex plane, and the special tritronquée solution has poles only in the wedge, π ≤ arg ( x ) ≤ π [33, 18]. Figure 6 shows as black dots 44 P tritronquée poles found using our Conformal-Padéapproach of [27], starting with 200 terms of the asymptotic expansion generated at x → + ∞ ,while the red dots show the first 66 P tritronquée poles found simply by adapting the analysis of[27] to use the uniformizing map in Theorem 9 instead of the conformal map in (13), and withexactly the same input data. The gain in precision in the pole sector, and also throughout thedomain of analyticity, is quite dramatic. These numerical poles, even the first few ones, fit veryprecisely the asymptotic Boutroux structure [49, 27].In addition, the uniformized extrapolation yields high-precision fine structure of the poleregion. In the vicinity of a moveable pole, any P I solution y ( x ) has a Laurent expansion of thefollowing form y ( x ) ≈ ( x − x pole ) + x pole ( x − x pole ) + ( x − x pole ) + h pole ( x − x pole ) + x ( x − x pole ) + x pole ( x − x pole ) + . . . (43)All higher coefficients of this expansion are expressed as polynomials in the two parameters x pole and h pole . Thus, any P I solution y ( x ) is completely determined by two constants, x pole and28 pole , in the vicinity of any one of its poles . It is natural to characterize the tritronquée solution bythe two constants, x pole and h pole , at the closest pole to the origin. From the uniformized mapextrapolation, we obtain the following high-precision values at this first pole: x = − h = tritronquée solutions, providing new methods to obtain high-precision com-putations for the Painlevé project [62], and also to compute high-precision spectral properties ofcertain Schrödinger operators [58, 60]. The generic improvement of analytic continuations based on uniformization maps, comparedwith other common methods such as Padé or Conformal-Padé (CP) [as illustrated in the previoussection], can often be traced to some elementary properties of these maps, especially near thesingularities. Here we illustrate this with some examples.
For Ω = C \ [ ∞ ) , the conformal map is in (12), and the uniformizing map of ˆ C \ { ∞ } is ω = − e − z , with inverse z = − log ( − ω ) . The uniformizing map pushes the singularityat ω = z = ∞ . A Padé approximant of the truncated composed map F ◦ ϕ , with eitherthe conformal map or the uniformizing map, each mapped back to Ω , produces dramaticallyimproved extrapolations throughout Ω . For example, for the one-branch-cut function F ( ω ) =( − ω ) − , beginning with just 10 terms of a Maclaurin expansion at ω =
0, Figure 7 showsthe ratio of the extrapolated to the exact function, as the singularity is approached: ω → − .The uniformization map is vastly superior. This is due to the typical exponential distortion ofdistance near the singularity. Ordinary Padé, without composition with either map, is not at allcompetitive. Another common case in applications is the two-cut complex plane, Ω = C \ ( − ∞ , − ] ∪ [ ∞ ) .The conformal and uniformizing maps are given in (13) and (14), respectively, in §3. For example,applying this to the two-branch-cut function F ( ω ) = ( − ω ) − , with branch points at ω = ± ω planeto the interior of the unit disk in the z plane, while the uniformizing map sends the cut ω plane tothe interior of the symmetric geodesic quadrilateral with boundaries given by orthogonal circlesintersecting the unit disk at z = ± ± i . See the left plot in Figure 2.Even when these are not the exact conformal or uniformizing maps, such as in nonlinearproblems where the singularities at ω = ± tritronquée Painlevé I-V solutions,29 .992 0.994 0.996 0.998 1.000w0.60.70.80.91.0ratio
Figure 7: Ratio of the approximate to the exact one-branch-cut function F ( ω ) = ( − ω ) − , fora Padé approximation [blue], a Padé-Conformal approximation [red], and a Padé-Uniformizedapproximation [black]. Note the dramatically superior behavior of the uniformized approxima-tion as the singularity is approached. These approximations are each generated starting with just10 input coefficients of the series expansion of F ( ω ) at ω = [ w ]- - [ w ] Figure 8: Arcs of Padé poles [blue points] for a pair of complex conjugate singularities, here at ω = e ± i π /3 , for the function F ( ω ) = ( − ω cos ( π /3 ) + ω ) − . Padé generates arcs of poles,including unphysical poles on the positive real axis, while the natural radial cuts [red lines] areassociated with the conformal map (46).singularities at ω = e ± i θ . This type of configuration occurs in physical applications involvingsymmetry breaking [71, 12, 66, 69], and in differential equations arising in holographic models[41]. In this case, without a conformal or uniformizing map, Padé produces curved arcs of poles(see Figure 8) as well as artificial poles along the positive real axis. Problems due to these artifi-cial poles can be mitigated by using a conformal or uniformizing map. The conformal map forthis configuration is 30 .5 1.0 1.5 θ [ θ ] Figure 9: Capacity plot, showing C conf. ( θ ) [blue curve] and C unif. ( θ ) [red curve] from (47) and(48), respectively, as a function of θ ∈ [ π /2 ] , for the configuration with two complex conju-gate singularities at e ± i θ . Lower capacity corresponds to a superior extrapolation. The ratio ofcapacities gives the improvement of the geometric rate of convergence. ω = c ( θ ) z ( + z ) (cid:18) + z − z (cid:19) θ / π , c ( θ ) = (cid:18) θπ (cid:19) θ / π (cid:18) − θπ (cid:19) − θ / π (46)and the uniformizing map is given by (15). The capacity factor for the two cases areconformal : C conf. ( θ ) = (cid:18) θπ (cid:19) − θπ (cid:18) − θπ (cid:19) θπ − (47)uniformizing : C unif. ( θ ) = i π sin θ (cid:0) K (cid:0) + i cot θ (cid:1)(cid:1) + (cid:0) K (cid:0) − i cot θ (cid:1)(cid:1) (48)As shown in Figure 9, the uniformizing map has lower capacity, indicating a superior extrap-olation. The uniformizing map has lower capacity, for all θ , and also has an explicit inversion.The conformal map is a simpler elementary function, but has no explicit inversion except for afew special cases of rational θ / π . A simple Padé approximation, which produces the minimalcapacitor (for which there exist implicit transcendental expressions for the capacity [52, 46]), isinferior to both the conformal and uniformizing maps. With the exact uniformizing map, one can then analytically continue beyond the Riemann sheet,in principle onto all higher sheets. But even without the exact map, some degree of analyticcontinuation onto higher sheets can be achieved using approximate maps. (Access to higherRiemann sheets can also be achieved using singularity elimination (see §4 and §6.5). Considerthe uniformization of the universal cover Ω of ˆ C \ {−
1, 1, ∞ } . This is achieved exactly by the mapin (14). See Figure 2. The points A , B , C are mapped to {− ∞ , 1 } . The gray geodesic triangle inthe Poincaré disk is conformally mapped by ϕ onto the upper half plane, and successive Schwarz31eflections across their circular sides continue ϕ to the whole disk, with image onto Ω (see [2],p. 379). Thus, the exact uniformization map permits systematic analytic continuation to higherRiemann sheets. The union of all reflected triangles is the unit disk D . The image through ϕ of a curve in D crossing a reflection of ( A , B ) , ( B , C ) , ( C , A ) crosses the real line between ( − ∞ , − ) , ( ∞ ) , ( −
1, 1 ) resp. The collection, modulo homotopies, of images through ϕ of allthe curves in D (each of them traceable using this geometric description) represents the universalcovering of ˆ C \ {−
1, 1, ∞ } .In the left figure of Figure 2, the blue path inside the disk is mapped to the spiral pathin the middle figure on the universal cover Ω of C \ {−
1, 1 } . The improvement of the rate ofconvergence is determined by the conformal distance to the boundary of the unit disk, and is ∼ n near zero, and the rate is about 0.83 n at the other end of the spiral.However, even without the exact uniformization map, an accurate approximate uniformizationof the same universal cover can be achieved by the following iterative application of elementaryconformal maps. We apply the construction of iterated maps in Theorem 12. The conformalmap z = ψ n ( z ) from Theorem 12 maps a surface with n symmetric radial cuts to the unit disk.Applying ψ to ˆ C \ {−
1, 1, ∞ } produces 4 symmetric cuts emanating from ± ± i . Subsequentapplication of ψ produces 8 symmetric cuts emanating from e π ij /8 , j =
0, . . . , 7. Each of thesemaps is an elementary conformal map. Theorem 12 implies that iterating this to all orders, weobtain a full uniformization of the universal cover Ω of ˆ C \ {−
1, 1, ∞ } . But even a finite iterationof this procedure permits an accurate extrapolation and analytic continuation onto higher sheetsof Ω , all in terms of elementary conformal maps. For example, in the center plot of Figure 2, thespiral path is the image of the blue path in the left plot, through the exact uniformizing map,but it is also the image of the blue curve on the right, through the elementary map ϕ ◦ · · · ◦ ϕ n ,iterated up to n =
6. The rate of improvement is ∼ n near zero and ∼ n , at the end,which are comparable to the exact uniformization results quoted above. In this section we illustrate the general procedure of singularity elimination by using a suitableconvolution operator L β in (24) to transform the logarithmic singularity of the elliptic integralfunction K ( ω ) into a square root singularity, and then eliminating this singularity by compositionwith a suitable conformal map. We choose this example because of its practical interest and alsobecause we can compare the general expressions in §4 with analytic results for the transformationof hypergeometric functions. We define F ( ω ) = K ( ω ) = π F (cid:18)
12 , 12 , 1; ω (cid:19) (49)The expansions as ω → + and ω → − are: F ( ω ) ∼ ∞ ∑ k = (cid:32) Γ (cid:0) k + (cid:1) Γ ( k + ) (cid:33) ω k , ω → + (50) F ( ω ) ∼ A ( ω ) ln ( − ω ) + B ( ω ) , ω → − (51)32here at the logarithmic singularity the regular functions A ( ω ) and B ( ω ) behave as A ( ω ) = − π K ( − ω ) = − π ∞ ∑ k = (cid:32) Γ (cid:0) k + (cid:1) Γ ( k + ) (cid:33) ( − ω ) k , ω → − (52) B ( ω ) = − ∞ ∑ k = (cid:32) Γ (cid:0) k + (cid:1) Γ ( k + ) (cid:33) (cid:18) ψ (cid:18) k + (cid:19) − ψ ( k + ) (cid:19) ( − ω ) k , ω → − (53)The convolution operator (24), with β / ∈ Z , acting on F leads to ( L β F )( ω ) = π ( + β ) ω F (cid:18)
12 , 12 , 2 + β ; ω (cid:19) (54) = Γ ( + β ) ∞ ∑ k = Γ (cid:0) k + (cid:1) Γ ( k + ) Γ ( + k + β ) ω k + , ω → + (55) ∼ ˜ A ( ω )( − ω ) + β + ˜ B ( ω ) , ω → − (56)where ˜ A ( ω ) and ˜ B ( ω ) are analytic at ω =
1. Equation (55) confirms the general expression(27) relating the original expansion coefficients of F ( ω ) with those of the convolved function ( L β F )( ω ) . To transform the original logarithmic singularity at ω = ω = β = − , to obtain ( L − F )( ω ) = π ω F (cid:18)
12 , 12 , 32 ; ω (cid:19) = π √ ω arcsin ( √ ω ) (57)(Of course, in this special case composition with the series of sin results in factorial convergenceof the composed series, far superior to the generic rate in the optimality theorem.) L β F is analytic at zero and singular at { ∞ } , and at at {
0, 1, ∞ } on higher Riemann sheets.Its Maclaurin series is √ π ∞ ∑ k = Γ (cid:0) k + (cid:1) Γ ( k + ) Γ (cid:0) k + (cid:1) ω k + , ω ∈ D (58)and its singularity structure near ω = A ( ω )( − ω ) + ˜ B ( ω ) (59)where ˜ A ( ω ) = − π √ ω arcsin ( √ − ω ) (cid:112) − ω ) = − π √ ω ∞ ∑ k = ( − ) k Γ (cid:0) k + (cid:1) Γ ( k + ) Γ (cid:0) k + (cid:1) ( − ω ) k (60)˜ B ( ω ) = π √ ω (61)We see from (60) that the expansion coefficients of ˜ A ( ω ) , the function in (59) multiplying thesquare root behavior, match the general expression in (34). In the practical situation wherewe only have (a finite number of) the original expansion coefficients, we simply transform theexpansion coefficients according to the convolution results in §4.4.The final step of the singularity elimination is to make a composition map that transformsthe square root behavior in (59) into analytic behavior, using the map ϕ in (26). The Riemann33urface of the new function, after composition with z (cid:55)→ iz is uniformized by (18) which bringsthe original singular point − A , ˜ B ) thesecould be calculated with extremely high accuracy.A similar comparison can be made for a general hypergeometric function F ( ω ) = F ( a , b , c ; ω ) (62) ∼ A ( ω )( − ω ) c − a − b + B ( ω ) , ω → − (63)for which the convolved function becomes a generalized hypergeometric function with a differentsingularity exponent at ω = ( L β F )( ω ) = ( + β ) Γ ( + β ) ω F ( a , b , c , 2 + β ; ω ) (64) ∼ ˜ A ( ω )( − ω ) c − a − b + + β + ˜ B ( ω ) , ω → − (65)For a given original singularity exponent, c − a − b , a suitable choice of β transforms the singu-larity into a square root singularity, which can then be eliminated by composition with one ofthe conformal maps in (26). Here we record a few examples of conformal maps in frequently encountered cases of resurgentfunctions, and which can also serve as guides in more complicated arrangements of singulari-ties. General conformal maps can in principle be derived from Schwarz-Christoffel, as describedbelow, but this procedure is rather tedious, and in cases of symmetry the resulting maps can bequite elementary. For a comprehensive list of known conformal maps, see [50].The maps ϕ and ϕ in (26) take D to C \ [ ∞ ) and C \ ( − ∞ , − ] ∪ [ ∞ ) , respectively.The latter case generalizes to two more general cuts on the real line, C \ ( − ∞ , − a ] ∪ [ b , ∞ ) with a , b ∈ R + , for which ω = ϕ ab ( z ) = ab za ( + z ) + b ( − z ) ↔ z = − (cid:113) a ( b − ω ) b ( a + ω ) + (cid:113) a ( b − ω ) b ( a + ω ) (66)For a symmetric set of n singularities on the unit circle, C \ (cid:83) n − j = e π ij / n [ ∞ ) , the conformal mapproducing symmetric radial cuts is ω = ϕ n ( z ) = n z ( + z n ) n (67)used in §3.2. For two complex conjugate radial cuts, C \ e i θ [ ∞ ) ∪ e − i θ [ ∞ ) , the conformal mapis as in (46), and this extends to a symmetric set of n such paired cuts as: z = ϕ n , θ ( z ) = c n ( θ ) z ( + z n ) n (cid:18) + z n − z n (cid:19) θ / π , c n ( θ ) = n (cid:18) n θπ (cid:19) θ / π (cid:18) − n θπ (cid:19) n − θ / π (68)For a general finite set of branch points, Padé produces a conformal map in the infinite orderlimit, and this map corresponds to the minimal capacitor. Recall the discussion in §5. The analyticdescription of this minimal capacitor conformal map is as follows [46]. Let S = { ω , ..., ω n } be34ranch points, and C the minimal capacitor, with the point of analyticity placed at infinity. Theconformal map ϕ that takes C \ D to C \ C , with + ∞ (cid:55)→ + ∞ , has the Taylor expansion at infinity ϕ ( ζ ) = C B ζ + ∞ ∑ k = b k ζ − k (69)where C B is the capacity. There is a set { a , ..., a n − } ⊂ C of auxiliary parameters such that ϕ satisfies the equation log ζ = (cid:90) ϕ ( ζ ) (cid:118)(cid:117)(cid:117)(cid:116) ∏ n − j = ( s − a j ) ∏ nj = ( s − ω j ) ds (70)These auxiliary parameters are the intersection points of the set of analytic arcs of C , the limitinglocation set of the poles of the diagonal Padé approximation P n [ F ] for any function F having S as the set of branch points, and being analytic in the complement of C . (For example, in theinfinite n limit, in Figure 8 the point on the positive real axis near ω ≈ γ j ( γ (cid:48) j , resp) joining a with ω j , j =
1, ..., n − a to a j , j =
2, ..., n −
2, resp.) are given by (cid:60) (cid:90) ωγ k (cid:118)(cid:117)(cid:117)(cid:116) ∏ n − j = ( s − a j ) ∏ nj = ( s − ω j ) ds = (cid:60) (cid:90) ωγ (cid:48) m (cid:118)(cid:117)(cid:117)(cid:116) ∏ n − j = ( s − a j ) ∏ nj = ( s − ω j ) ds = k =
1, ..., n − m =
2, ..., n −
2. In cases of symmetrically distributed branch points,these integrals can be expressed in terms of elementary or elliptic functions [52], and in moregeneral cases the minimal capacitor produced by Padé can be found numerically [46]. This con-struction benefits from physical intuition arising from the interpretation of the minimal capacitorin terms of potential theory (see §5).
Acknowledgements
We thank R. Costin for numerous helpful discussions and comments. This work is supported inpart by the U.S. Department of Energy, Office of High Energy Physics, Award DE-SC0010339.
References [1] W. Abikoff, The Uniformization Theorem, The American Mathematical Monthly, v. 88, No.8, pp. 574–592 (1981).[2] M. J. Ablowitz and A. S. Fokas, Complex variables, 2nd. ed., Cambridge University Press(2003).[3] I. Aniceto, R. Schiappa and M. Vonk, “The Resurgence of Instantons in String Theory,”Commun. Num. Theor. Phys. , 339 (2012), arXiv:1106.5922.[4] I. Aniceto, “The Resurgence of the Cusp Anomalous Dimension,” J. Phys. A , 065403(2016), arXiv:1506.03388.[5] I. Aniceto, G. Basar and R. Schiappa, “A Primer on Resurgent Transseries and Their Asymp-totics,” Phys. Rept. , 1 (2019), arXiv:1802.10441.[6] A. Aptekarev and M. L. Yattselev, Padé approximants for functions with branch points –strong asymptotics of Nuttall–Stahl polynomials, Acta Math. , 217–280 (2015).357] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of Differentiable Maps (Mon-odromy and Asymptotics of Integrals) (Birkhauser, Berlin 1988).[8] G. A. Baker, and P. Graves-Morris, Padé Approximants , (Cambridge University Press, 2009).[9] G. A. Baker, J. L. Gammel, and J. G. Wills, An investigation of the applicability of the Padéapproximant method, J. Math. Anal. Appl. 2, 405–418. (1961).[10] M. Beneke, “Renormalons,” Phys. Rept. , 1-142 (1999), arXiv:hep-ph/9807443.[11] M. V. Berry, C. Howls, Hyperasymptotics, Proc. Roy. Soc. Lond A, , 653-668 (1990); Hy-perasymptotics for integrals with saddles, Proc. Roy. Soc. A, , 657-675 (1991).[12] C. Bertrand, S. Florens, O. Parcollet, and X. Waintal, “Reconstructing NonequilibriumRegimes of Quantum Many-Body Systems from the Analytical Structure of PerturbativeExpansions”, Phys. Rev. X , 041008 (2019), arXiv:1903.11646.[13] B. L. J. Braaksma, Transseries for a class of nonlinear difference equations, J. Difference Eq.and Appl., 5, (2001).[14] E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov and U. D. Jentschura, “From usefulalgorithms for slowly convergent series to physical predictions based on divergent pertur-bative expansions,” Phys. Rept. , 1 (2007), arXiv:0707.1596.[15] Cannon, J.W., Dicks, W. On hyperbolic once-punctured-torus bundles II: fractal tessellationsof the plane. Geom Dedicata 123, 11–63 (2006).[16] I. Caprini, J. Fischer, G. Abbas and B. Ananthanarayan, “Perturbative Expansions in QCDImproved by Conformal Mappings of the Borel Plane,” in Perturbation Theory: Advances inResearch and Applications , (Nova Science Publishers, 2018), arXiv:1711.04445.[17] Costin, O.; Costin, R. D.; Huang, M. Tronquée solutions of the Painlevé equation PI. Constr.Approx. 41 (2015), no. 3, 467–494.[18] Costin, O.; Huang, M.; Tanveer, S. Proof of the Dubrovin conjecture and analysis of thetritronquée solutions of PI. Duke Math. J. (2014), no. 4, 665–704.[19] O. Costin, On Borel Summation and Stokes Phenomena for Rank-1 Nonlinear Systems ofOrdinary Differential Equations, Duke Math. J., , no. 2, 289-344 (1998).[20] O. Costin, Asymptotics and Borel summability , (Chapman and Hall/CRC, 2008).[21] Costin, O.; Lebowitz, J. L.; Tanveer, S. Ionization of Coulomb systems in R by time periodicforcings of arbitrary size. Comm. Math. Phys. 296 (2010), no. 3, 681–738.[22] O. Costin and S. Tanveer, Nonlinear evolution PDEs in R + × C d : existence and uniquenessof solutions, asymptotic and Borel summability properties. Ann. Inst. H. Poincaré Anal. NonLinéaire, , 795-823 (2007).[23] O. Costin, M. Huang, Gamow vectors and Borel summability in a class of quantum systems.J. Stat. Phys. , no. 4., 846-871 (2011).[24] Costin O.; Luo, G.; Tanveer, S. Integral formulation of 3D Navier-Stokes and longer timeexistence of smooth solutions. Commun. Contemp. Math. (2011), no. 3, 407–462.3625] Costin, O.; Park, H.; Takei, Y. Borel summability of the heat equation with variable coeffi-cients. J. Differential Equations 252 (2012), no. 4, 3076–3092.[26] Costin, Ovidiu; Costin, Rodica D.; Lebowitz, J.L; Nonperturbative time dependent solutionof a simple ionization model, Comm. Math. Phys. 361 (2018), no. 1, 217–238, (2018).[27] O. Costin and G. V. Dunne, “Resurgent extrapolation: rebuilding a function from asymptoticdata. Painlevé I,” J. Phys. A , no. 44, 445205 (2019), arXiv:1904.11593.[28] O. Costin and G. V. Dunne, to appear.[29] D. Damanik and B. Simon, Jost functions and Jost solutions for Jacobi matrices, I. A neces-sary and sufficient condition for Szegö asymptotics, Invent. Math. , 1-50 (2006).[30] E. Delabaere and F. Pham, “Resurgent Methods in Semiclassical Asymptotics”, Ann. de l’I.Henri Poincaré, , 1-94 (1999).[31] E. Delabaere and C. Howls, Global asymptotics for multiple integrals with boundaries, DukeMath. J., v. 112, 199-264 (2002).[32] D. Dorigoni and Y. Hatsuda, “Resurgence of the Cusp Anomalous Dimension,” JHEP ,138 (2015), arXiv:1506.03763.[33] B. Dubrovin, T. Grava, and C. Klein, “On universality of critical behavior in the focusingnonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution tothe Painlevé-I equation”, J. Nonlinear Sci. , 57-94 (2009).[34] G. V. Dunne, “Heisenberg-Euler effective Lagrangians: Basics and extensions,” in Fromfields to strings, vol. 1, 445-522, M. Shifman et al (eds.) et al, (World Scientific, Singapore,2005), arXiv:hep-th/0406216.[35] G. V. Dunne and M. Ünsal, “Resurgence and Trans-series in Quantum Field Theory: TheCP(N-1) Model,” JHEP , 170 (2012), arXiv:1210.2423.[36] G. V. Dunne and M. Ünsal, “New Nonperturbative Methods in Quantum Field Theory:From Large-N Orbifold Equivalence to Bions and Resurgence,” Ann. Rev. Nucl. Part. Sci. , 245 (2016), arXiv:1601.03414.[37] J. Écalle, Fonctions Resurgentes, Publ. Math. Orsay 81, Université de Paris–Sud, Departe-ment de Mathématique, Orsay, (1981).[38] A. Erdélyi, Higher Transcendental Functions, The Bateman Manuscript Project, vol 1., NewYork–London (1953), https://authors.library.caltech.edu/43491/[39] Fokas, Athanassios S.; Its, Alexander R.; Kapaev, Andrei A.; Novokshenov, Victor Yu.PainlevÃl’ transcendents: The Riemannâ ˘A ¸SHilbert approach, Mathematical Surveys andMonographs, 128, Providence, R.I.: American Mathematical Society, (2006).[40] M. E. Fisher, “Critical Point Phenomena - the role of series expansions”, Rocky Mount. J.Math. , 181-201 (1974).[41] W. Florkowski, M. P. Heller and M. Spalinski, “New theories of relativistic hydrodynamicsin the LHC era,” Rept. Prog. Phys. , no.4, 046001 (2018), arXiv:1707.02282.3742] D. Gaiotto, G. W. Moore and A. Neitzke, “Wall-crossing, Hitchin Systems, and the WKBApproximation,” Adv. Math. , 239-403 (2013), arXiv:0907.3987.[43] S. Garoufalidis, A. Its, A. Kapaev and M. Marino, “Asymptotics of the instantons of PainlevéI,” Int. Math. Res. Not. , no. 3, 561 (2012), arXiv:1002.3634.[44] D. S. Gaunt and A. J. Guttmann, “Asymptotic Analysis of Coefficients”, in Phase Transitionsand Critical Phenomena, Vol. 3 , C. Domb and M. S. Green (Eds) (Academic Press, 1974).[45] A. Gopal, L. N. Trefethen, “Representation of conformal maps by rational functions”, Nu-mer. Math. , 359-382 (2019), arXiv:1804.08127.[46] E. G. Grassmann and J. Rokne, An explicit calculation of some sets of minimal capacity,SIAM J. Math. Anal. , 242-249 (1975).[47] S. Gukov, M. Mariño and P. Putrov, “Resurgence in complex Chern-Simons theory,”arXiv:1605.07615.[48] J. A. Hempel, On the uniformization of the n -punctured sphere, Bull. London Math. Soc. ,97-115 (1980).[49] See A. V. Kitaev, “Elliptic asymptotics of the first and the second PainlevÃl’ transcendents”,Uspekhi Mat. Nauk, 49:1(295) (1994), 77–140; Russian Math. Surveys, 49:1 (1994), 81–150,and references therein.[50] H. Kober, Dictionary of Conformal Representations, Dover (1957).[51] M. Kontsevich, Y. Soibelman, Airy structures and symplectic geometry of topological recur-sion, arXiv:1701.09137.[52] G.V. Kuz’mina, Estimates for the transfinite diameter of a family of continua and coveringtheorems for univalent functions, Proc. Steklov Inst. Math. , 53-74 (1969).[53] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York–Heidelberg (1972).[54] D. S. Lubinsky, Rogers-Ramanujan and the Baker-Gammel-Wills (Padé) conjecture, Annalsof Mathematics, 157, 847–889 (2003).[55] M. Mariño, “Nonperturbative effects and nonperturbative definitions in matrix models andtopological strings,” JHEP , 114 (2008), arXiv:0805.3033.[56] Fréderic Menous. Les bonnes moyennes uniformisantes et une application à la resommationrélle. Annales de la Faculté des sciences de Toulouse : Math ´matiques, 6e série, 8(4):579–628,(1999).[57] A. Martinez-Finkelshtein, E. A. Rakhmanov, S. P. Suetin, Heine, Hilbert, Pade, Riemann, andStieltjes: John Nuttall’s work 25 years later, Contemporary Mathematics 578, 165–193 (2012).[58] D. Masoero, “Poles of Integrale Tritronquee and Anharmonic Oscillators. Asymptotic local-ization from WKB analysis,” Nonlinearity , 2501 (2010), arXiv:1002.1042; “Poles of Inte-grale Tritronquee and Anharmonic Oscillators. A WKB Approach”, J. Phys. A: Math. Theor.43 095201 (2010), arXiv:0909.5537. 3859] Z. Nehari, Conformal Mapping, Dover (1952).[60] V. Yu. Novokshenov, “Poles of Tritronquée Solution to the Painlevé I Equation and CubicAnharmonic Oscillator”, Reg. Chaotic Dyn. , 390 - 403 (2010).[61] C. Mitschi, D. Sauzin, Divergent Series, Summability and Resurgence I, II, Springer (2016).[62] initiated by F. Bornemann, P. Clarkson, P. Deift, A. Edelman, A. Its, and D. Lozier,https://math.nist.gov/ DLozier/PainleveProject/[63] Ch. Pommerenke, Boundary Behavior of Conformal Maps, Springer-Verlag (1992).[64] J.-P. Ramis, J. Sauloy, C. Zhang, Local analytic classification of q-difference equations,arXiv:0903.0853, Astérisque Volume: 355, (2013).[65] T. Ransford, Potential theory in the complex plane, vol. 28 of London Mathematical SocietyStudent Texts, Cambridge University Press, Cambridge (1995).[66] R. Rossi, T. Ohgoe, K. Van Houcke and F. Werner, “Resummation of diagrammatic serieswith zero convergence radius for strongly correlated fermions,” Phys. Rev. Lett. , no. 13,130405 (2018), arXiv:1802.07717.[67] E. B. Saff, Logarithmic Potential Theory with Applications to Approximation Theory,arXiv:1010.3760 (2010)[68] W. Schlag, A Course in Complex Analysis and Riemann Surfaces, American MathematicalSociety, Graduate Studies in Mathematics, vol. 154 (2014).[69] M. Serone, G. Spada and G. Villadoro, “ λϕ theory II. The broken phase beyondNNNN(NNNN)LO,” JHEP , 047 (2019), arXiv:1901.05023.[70] H. Stahl, The Convergence of Padé Approximants to Functions with Branch Points, Journalof Approximation Theory , 139–204 (1997).[71] M. A. Stephanov, “QCD critical point and complex chemical potential singularities,” Phys.Rev. D , 094508 (2006), arXiv:hep-lat/0603014.[72] G. Szegö, Orthogonal Polynomials , (American Mathematical Society, 1939); U. Grenander andG. Szegö,
Toeplitz forms and their applications , (Univ. California Press, Berkeley, 1958).[73] H. S. Wall, General Theorems on the Convergence of Sequences of Pade Approximants,TAMS, Vol. 34, No. 2 (1932).[74] J. Zinn-Justin,
Quantum Field Theory and Critical Phenomena , Int. Ser. Monogr. Phys. , 1(2002).[75] J. Zinn-Justin and U. D. Jentschura, “Multi-instantons and exact results I: Conjectures,WKB expansions, and instanton interactions,” Annals Phys. , 197 (2004), arXiv:quant-ph/0501136; “Multi-instantons and exact results II: Specific cases, higher-order effects, andnumerical calculations,” Annals Phys.313