The Maclaurin Series With Analytic Continuation

Abstract

We analyze the problem of global reconstruction of functions as accurately as possible, based on partial information in the form of a truncated power series at some point, and additional analyticity properties. This situation occurs frequently in applications. The question of the optimal procedure was open, and we formulate it as a well-posed mathematical problem. Its solution leads to a practical method which provides dramatic accuracy improvements over existing techniques. Our procedure is based on uniformization of Riemann surfaces. As an application, we show that our procedure can be implemented for solutions of a wide class of nonlinear ODEs. We find a new uniformization method, which we use to construct the uniformizing maps needed for special functions, including solution of the Painlevé equations $P_\mathrm{I}$–$P_{\mathrm{V}}$. We also introduce a new rigorous and constructive method of regularization, elimination of singularities whose position and type are known. If these are unknown, the same procedure enables a highly sensitive resonance method to determine the position and type of a singularity. In applications where less explicit information is available about the Riemann surface, our approach and techniques lead to new approximate, but still much more precise reconstruction methods than existing ones, especially in the vicinity of singularities, which are the points of greatest interest.

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Notes

Non-simply connected Riemann surfaces are uniformized on ${\mathbb {C}}/\Gamma $ or ${\mathbb {D}}/\Gamma $, where $\Gamma $ is a discrete group of automorphisms, see e.g. [70]. At this stage it is unclear to us how to take advantage of the factorization and in such a case we take instead $\Omega $ to be the universal covering.
Using ${\hat{{\mathbb {C}}}}$ is a standard convention, since it makes a counting difference for the analyzed functions if infinity is singular or not. For instance, $\ln [(1-\omega )/(1+\omega )]$ is analytic at infinity and its Riemann surface is uniformized on the plane, after a Möbius change of variable.
Analyticity follows from the fact that $\varphi ({\mathbb {D}})=\Omega $ and F is analytic in $\Omega $.
Some properties of this map, other than its uniformizing features, are described in [63], p. 323.
Not $\hat{{\mathbb {C}}}{\setminus }{\mathbb {Z}}{\setminus }\{\infty \}$; a curve around infinity is undefined, since ${\mathbb {Z}}$ is not compact.
For some of these ODE solutions, e.g. the tronquées of $P_{\mathrm{I}}$, the Riemann surface $\Omega $ is "larger", and not simply connected (all odd integers are square root branch points); we pass instead to the universal cover of $\Omega $ which is indeed $\Omega _{{\mathbb {Z}}}$. See also footnote 1 on p. 4.
There is nothing intrinsically special about the order 471; it relates to the degree of a large-order Maclaurin polynomial already stored on the computer.
Indeed, taking $A=1, B=0$, and then $A=1$ and $B(\omega )=\omega $, we see that $\varphi $ must be analytic at 1, hence $\varphi (1+s)=1+o(s)$. But then $\log (1-\varphi )$ is unbounded at 1.
As usual, by "jump across the cut" we mean the upper limit minus the lower limit along the cut.
Very generally, even in the absence of local expansions, Plemelj's formulas [2] give a local representation $S(\omega )$ in the form $ \frac{1}{2\pi i}\int _0^{1+\varepsilon }(\omega -\tau )^{-1}\int _1^\lambda F(\lambda -s)\Delta G(s)dsd\tau $, where $\Delta G$ is the jump across the cut of G.

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Acknowledgements

We thank R. Costin for numerous helpful discussions and comments. We also thank A. Voros for interesting discussions, and Jean Écalle for detailed correspondence, and for valuable comments on an earlier draft. This work is supported in part by the U.S. Department of Energy, Office of High Energy Physics, Award DE-SC0010339 (GD).

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Authors and Affiliations

Department of Mathematics, The Ohio State University, Columbus, OH, 43210-1174, USA

Ovidiu Costin
Department of Physics, University of Connecticut, Storrs, CT, 06269-3046, USA

Gerald V. Dunne

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Correspondence to Ovidiu Costin.

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Communicated by A. Giuliani.

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Appendix: Some Relevant Conformal Maps

In this Appendix we record some conformal maps relevant for frequently encountered cases of resurgent functions, and which can also serve as guides in more complicated arrangements of singularities. General conformal maps can in principle be derived from Schwarz-Christoffel, but this procedure is rather tedious, and in cases of symmetry the resulting maps can be quite elementary. For a comprehensive list of many known conformal maps, see [54].

A simple but important case is the one-cut domain $\Omega ={\mathbb {C}}{\setminus } [1,\infty )$, for which

$$\begin{aligned} z= \psi (\omega )=\frac{1-\sqrt{1-\omega }}{1+\sqrt{1-\omega }} \qquad {\text {with inverse }}\qquad \omega = \varphi (z)=\frac{4z}{(1+z)^2} \end{aligned}$$

(69)

The optimal rate of convergence obtained from the Maclaurin series of a generic function F whose maximal analyticity domain is this $\Omega $, and is continuous up to $\partial {\mathbb {D}}$ is, see (7),

$$\begin{aligned} |F(\omega _0)- {\hat{R}}_n(\omega _0) |\sim \frac{|\omega _0|^n}{2\left| \sqrt{1-\omega _0}\right| \left| 1+\sqrt{1-\omega _0}\right| ^{2n-1}} \Vert F\Vert _{\infty };\qquad \omega _0\in \Omega \end{aligned}$$

The constant $C=1/4$ in (23), the capacity of $1/\partial \Omega $, is simply $\psi '(0)$.

For the domain with two opposite cuts, $\Omega ={\mathbb {C}}{\setminus } (-\infty ,-1]\cup [1,\infty )$, the maps are

$$\begin{aligned} z= \psi (\omega )=\sqrt{\frac{1-\sqrt{1-\omega ^2}}{1+\sqrt{1-\omega ^2}}} \qquad {\text {with inverse }}\qquad \omega = \varphi (z)=\frac{2z}{1+z^2} \end{aligned}$$

(70)

with $\psi (\omega )>0$ for $\omega \in (0,1)$. The capacity of $1/\partial \Omega $ is now $C=1/2=\psi '(0)$. This construction generalizes straightforwardly to m symmetric cuts emanating from the vertices of a regular polygon. See Appendix 7 and [49]. This example also generalizes to $\Omega ={\mathbb {C}}{\setminus } (-\infty ,-a]\cup [b,\infty )$: see (75) in Appendix 7.

Note 43

There is a more general principle behind (70) worth mentioning:

Lemma 44

Let $\Phi $ by a conformal map of ${\mathbb {D}}$ to some domain ${\mathcal {D}}\subset {\mathbb {C}}$, and let $c=\Phi '(0)>0$. Then, $\Phi _n(z):=\Phi (z^n)^{1/n}$ maps ${\mathbb {D}}$ conformally to n symmetric copies of ${\mathcal {D}}^{1/n}$, i.e., to

$$\begin{aligned} \bigcup _{0\le j\le n-1}e^{2\pi i j/n}{\mathcal {D}}^{1/n} \end{aligned}$$

Proof

In a neighborhood of zero, $\Phi _n$ is uniquely defined by $\Phi _n(z)=|c^{1/n}|z H(z^n)$, where H is analytic at zero and $H(0)=1$. Since $\Phi \ne 0$ on ${\mathbb {D}}{\setminus }\{0\}$, by the monodromy theorem, $\Phi _n$ extends analytically to ${\mathbb {D}}$. Since $\Phi $ is injective on ${\mathbb {D}}$, $\Phi _n(z)=\Phi _n(v)$ implies $z^n=v^n$. Now, $\Phi _n(z)=\Phi _n(v)$, written as $|c^{1/n}|z H(z^n)=|c^{1/n}|v H(v^n)$ implies $z=v$, and thus $\Phi _n$ is injective. Since $\Phi $ is onto ${\mathcal {D}}$, the rest follows from injectivity. $\square $

With proper adaptations, this construction extends to uniformization maps of Riemann surfaces.

1.:

For functions analytic on $\Omega \big ({\hat{{\mathbb {C}}}}{\setminus } \{-1,1, \infty \}\big )$ the maps are (compare with [40], p. 99)

$$\begin{aligned}&z= \psi (\omega )=\frac{{\mathbb {K}}\left( \frac{1+\omega }{2}\right) -{\mathbb {K}}\left( \frac{1-\omega }{2}\right) }{{\mathbb {K}}\left( \frac{1-\omega }{2}\right) +{\mathbb {K}}\left( \frac{1+\omega }{2}\right) } \qquad {\text {with inverse }}\nonumber \\&\omega = \varphi (z)=-1+2 \lambda \left( i\, \frac{1-z}{1+z}\right) \end{aligned}$$

(71)

Here $\lambda =\theta _2^4/\theta _3^4$ is the elliptic modular function, $\theta _2,\theta _3$ are Jacobi theta functions, and ${\mathbb {K}}(m)=(\pi /2) \,_2F_1(\tfrac{1}{2},\tfrac{1}{2};1;m)$ is the complete elliptic integral of the first kind of modulus $m=k^2$ [40]. The capacity is $C=\psi '(0)=\pi ^{-2}\Gamma \left( \frac{3}{4}\right) ^4\approx 0.2285$, more than a factor of two better than the capacity of the conformal map in Eq. (70) of Example 2 above. Furthermore, ${\hat{R}}_n(\omega _0)$ in (7) converges on the whole universal covering of ${\hat{{\mathbb {C}}}}{\setminus } \{-1,1, \infty \}$; that is, on all the Riemann sheets of the underlying function. See Fig. 13. Note that the improvement in accuracy is particularly dramatic near singular points. Indeed, the leading order asymptotic behavior of $\psi $ near $\omega =1$ is $\psi (\omega )\sim 1+2\pi /\ln (1-\omega )$. The function $\psi $ in (71) maps the points A,B,C to $\{-1,\infty ,1\}$. The gray geodesic triangle in the Poincaré disk is conformally mapped by $\varphi $ onto the upper half plane, and successive Schwarz reflections across their circular sides continue $\varphi $ to the whole disk, with image onto $\Omega $ (see [2], p. 379). The union of all reflected triangles is the unit disk ${\mathbb {D}}$. The image through $\varphi $ of a curve in ${\mathbb {D}}$ crossing a reflection of (A,B), (B,C), (C,A) crosses the real line between $(-\infty ,-1), (1,\infty ), (-1,1)$ resp. The collection, modulo homotopies, of images through $\varphi $ of all the curves in ${\mathbb {D}}$ (each of them traceable using this geometric description) represents $\Omega \big ({\hat{{\mathbb {C}}}}{\setminus } \{-1, 1, \infty \}\big )$. In the left figure of Fig. 13, the blue path inside the disk is mapped to the spiral path in the middle figure on $\Omega \big ({\hat{{\mathbb {C}}}}{\setminus } \{-1, 1, \infty \}\big )$. The improvement of the rate of convergence is determined by the conformal distance to the boundary of the unit disk, and is $\sim 0.228^n$ near zero, and the rate is about $0.83^n$ at the other end of the spiral.

2.:

The uniformization map for ${\hat{{\mathbb {C}}}}{\setminus } \{e^{-i\theta },e^{i\theta },\infty \}$, another important case in applications, is given by (we consider $0\le \theta \le \frac{\pi }{2}$, and extend by symmetry)

$$\begin{aligned} z=\psi (\omega )=\frac{Z(\omega ; \theta ) -Z(0; \theta )}{1-\left( Z(0; \theta )\right) ^*\, Z(\omega ; \theta )} \end{aligned}$$

(72)

where $Z(\omega ; \theta )$ is defined as

$$\begin{aligned} Z(\omega ; \theta )\equiv \frac{{\mathbb {K}}\left( \frac{1}{2}+\frac{i}{2}\left( \frac{\omega }{\sin \theta }-\cot \theta \right) \right) -{\mathbb {K}}\left( \frac{1}{2}-\frac{i}{2}\left( \frac{\omega }{\sin \theta }-\cot \theta \right) \right) }{{\mathbb {K}}\left( \frac{1}{2}+\frac{i}{2}\left( \frac{\omega }{\sin \theta }-\cot \theta \right) \right) +{\mathbb {K}}\left( \frac{1}{2}-\frac{i}{2}\left( \frac{\omega }{\sin \theta }-\cot \theta \right) \right) } \end{aligned}$$

(73)

The inverse map is given in terms of the modular $\lambda $ function by

$$\begin{aligned} \omega = \varphi (z)= e^{i\, \theta } -2i\,\sin (\theta )\, \lambda \left( i\left( \frac{{\mathbb {K}}\left( \frac{1}{2}+\frac{i}{2} \cot \theta \right) -{\mathbb {K}}\left( \frac{1}{2}-\frac{i}{2} \cot \theta \right) z}{{\mathbb {K}}\left( \frac{1}{2}-\frac{i}{2} \cot \theta \right) +{\mathbb {K}}\left( \frac{1}{2}+\frac{i}{2} \cot \theta \right) z}\right) \right) \end{aligned}$$

(74)

These maps are obtained from (71) by a suitable Möbius transformation and disk automorphism.

3.:

It is straightforward to generalize the two previous examples to $\Omega \big ({\hat{{\mathbb {C}}}}{\setminus } \{\omega _1, \omega _2, \infty \}\big )$, where $\omega _1, \omega _2 \in {\mathbb {C}}$. The uniformization maps are again expressed in terms of the elliptic function ${\mathbb {K}}$ and the elliptic modular function $\lambda $. The important case in the previous example, with two complex conjugate points, $\omega _1=e^{i\theta }=1/\omega _2$, which occurs in many applications, is discussed further in Sect. 6.2.2.

4.:

For uniformization of other Riemann surfaces based on that of ${\hat{{\mathbb {C}}}}{\setminus } \{-1,1, \infty \}$, possessing a nontrivial fundamental group, see for example [40], §2.7.2. See also [49] for a collection of explicit uniformization maps of $\Omega \big ({\hat{{\mathbb {C}}}}{\setminus } S\big )$, where S is a finite set of points in ${\hat{{\mathbb {C}}}}$, including for example $S=\{-1,0,1, \infty \}$, $S=\{-3,-1,0,1,3, \infty \}$ and $S=\{0,1, e^{\pi i/3}, \infty \}$, and when S consists of the n-th roots of unity. Uniformizing maps for the four-punctured torus are analyzed in [51]. Algebraic functions have compact Riemann surfaces, and some explicit uniformizing maps can be found in Schwarz's table in [40].

5.:

In certain special cases the uniformizing map produces a meromorphic or rational function, in which case a subsequent Padé approximation becomes exact. For example, the function $F(\omega )= \sqrt{1-\omega }$ has a compact Riemann surface $\Omega $ (as all algebraic functions do). The uniformizing map $\omega =\varphi (z)=4z/(1+z)^2$ makes $F\circ \varphi $ meromorphic, $(F\circ \varphi )(z)=(1-z)/(1+z)$, hence analytic on the Riemann sphere, and Padé [n,n] is exact for $n>0$. A more sophisticated example is the Riemann surface $\Omega $ of functions with three square root branch points, which is uniformized by $(z^2-1)^2/(z^2+1)^2$, [40], and the functions become rational; the uniformization theorem brings $\Omega $ to $\hat{{\mathbb {C}}}$. This is another case where Padé becomes exact.

The maps $\varphi _1$ and $\varphi _2$ in (29) take ${\mathbb {D}}$ to ${\mathbb {C}}{\setminus } [1,\infty )$ and ${\mathbb {C}}{\setminus } (-\infty ,-1]\cup [1,\infty )$,respectively. The latter case generalizes to two more general cuts on the real line, ${\mathbb {C}}{\setminus } (-\infty ,-a]\cup [b,\infty )$ with $a, b\in {\mathbb {R}}^+$, for which

$$\begin{aligned} \omega = \varphi _{ab}(z)= \frac{4 a b\, z}{a (1+z)^2+b (1-z)^2} \quad \leftrightarrow \quad z=\frac{1-\sqrt{\frac{a (b-\omega )}{b (a+\omega )}}}{1+\sqrt{\frac{a (b-\omega )}{b (a+\omega )}}} \end{aligned}$$

(75)

For a symmetric set of n singularities on the unit circle, ${\mathbb {C}}{\setminus } \bigcup _{j=0}^{n-1} e^{2 \pi i j/n}[1,\infty )$, the conformal map producing symmetric radial cuts is

$$\begin{aligned} \omega =\varphi _n(z)= \frac{2^{2/n} z}{(1+z^n)^{2/n}} \end{aligned}$$

(76)

used in Sect. 3.2. For two complex conjugate radial cuts, ${\mathbb {C}}{\setminus } e^{ i\theta }[1,\infty )\cup e^{ -i\theta }[1,\infty )$, the conformal map is as in (66), and this extends to a symmetric set of n such paired cuts as:

$$\begin{aligned} z= & {} \varphi _{n, \theta }(z)= c_n(\theta )\frac{z}{(1+z^n)^{2/n} }\, \left( \frac{1+z^n}{1-z^n}\right) ^{2\theta /\pi } \quad , \quad \nonumber \\ c_n(\theta )= & {} 2^{2/n}\left( \frac{n\, \theta }{\pi }\right) ^{\theta /\pi }\left( 1-\frac{n\, \theta }{\pi }\right) ^{1/n-\theta /\pi } \end{aligned}$$

(77)

For a general finite set of branch points, Padé produces a conformal map in the infinite order limit, and this map corresponds to the minimal capacitor. Recall the discussion in Sect. 5. The analytic description of this minimal capacitor conformal map is as follows [46]. Let $S=\{\omega _1,\ldots ,\omega _n\}$ be branch points, and ${\mathcal {C}}$ the minimal capacitor, with the point of analyticity placed at infinity. The conformal map $\varphi $ that takes ${\mathbb {C}}{\setminus } {\mathbb {D}}$ to ${\mathbb {C}}{\setminus } {\mathcal {C}}$, with $+\infty \mapsto +\infty $, has the Taylor expansion at infinity

$$\begin{aligned} \varphi (\zeta )=C_B\zeta +\sum _{k=0}^\infty b_k \zeta ^{-k} \end{aligned}$$

(78)

where $C_B$ is the capacity. There is a set $\{a_1,\ldots ,a_{n-2}\}\subset {\mathcal {C}}$ of auxiliary parameters such that $\varphi $ satisfies the equation

$$\begin{aligned} \log \zeta =\int ^{\varphi (\zeta )}\sqrt{\frac{\prod _{j=1}^{n-2}(s-a_j)}{\prod _{j=1}^{n}(s-\omega _j)}}ds \end{aligned}$$

(79)

These auxiliary parameters are the intersection points of the set of analytic arcs of ${\mathcal {C}}$, the limiting location 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 ${\mathcal {C}}$. (For example, in the infinite n limit, in Fig. 11 the point on the positive real axis near $\omega \approx 4.5$ would tend to the single intersection point for this configuration.) The analytic arcs $\gamma _j$ ($\gamma '_j$, resp) joining $a_1$ with $\omega _j, j=1,\ldots ,n-1$ ($a_1$ to $a_j, j=2,\ldots ,n-2$, resp.) are given by

$$\begin{aligned} \Re \int _{\gamma _k}^{\omega }\sqrt{\frac{\prod _{j=1}^{n-2}(s-a_j)}{\prod _{j=1}^{n}(s-\omega _j)}}ds=0\ \quad {\text {and}} \quad \Re \int _{\gamma '_m}^{\omega }\sqrt{\frac{\prod _{j=1}^{n-2}(s-a_j)}{\prod _{j=1}^{n}(s-\omega _j)}}ds=0 \end{aligned}$$

(80)

where $k=1,\ldots ,n-1$ and $m=2,\ldots ,n-2$. In cases of symmetrically distributed branch points, these integrals can be expressed in terms of elementary or elliptic functions [56], and in more general cases the minimal capacitor produced by Padé can be found numerically [46]. This construction benefits from physical intuition arising from the interpretation of the minimal capacitor in terms of potential theory (see Sect. 5).

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Costin, O., Dunne, G.V. Uniformization and Constructive Analytic Continuation of Taylor Series. Commun. Math. Phys. 392, 863–906 (2022). https://doi.org/10.1007/s00220-022-04361-6

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Received: 12 November 2021
Accepted: 19 February 2022
Published: 24 March 2022
Issue Date: June 2022
DOI : https://doi.org/10.1007/s00220-022-04361-6

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