[Paper Review] Global reconstruction of analytic functions from local expansions

Ovidiu Costin, Xin Xia|arXiv (Cornell University)|Dec 5, 2006

Reservoir Engineering and Simulation Methods5 references3 citations

TL;DR

This paper introduces a novel summation method that reconstructs global analytic behavior of functions from their local Taylor coefficients using integral representations. It enables exact recovery of singularities, asymptotics, monodromy, and zero locations for a broad class of functions, with Borel summability of divergent series as a byproduct.

ABSTRACT

A new summation method is introduced to convert a relatively wide family of infinite sums and local expansions into integrals. The integral representations yield global information such as analytic continuability, position of singularities, asymptotics for large values of the variable and asymptotic location of zeros. There is a duality between the global analytic structure of the reconstructed function and the properties of the coefficients as a function of their index. Borel summability of a class of divergent series follow as a byproduct.

Motivation & Objective

• To develop a systematic method for reconstructing global analytic structure from local Taylor coefficients.
• To address the longstanding challenge of inferring global behavior—such as singularities and asymptotics—from convergent or divergent local series.
• To establish a duality between the asymptotic structure of coefficients and the global analytic properties of the reconstructed function.
• To provide a practical, integral-based reconstruction technique applicable to entire functions and functions with finite radius of convergence.
• To demonstrate that Borel summability of divergent series arises naturally from the method as a byproduct.

Proposed method

• Introduces a new summation method that converts infinite local expansions into convergent integral representations valid globally.
• Uses inverse Laplace transform techniques to relate coefficients $ c_k $ to a function $ \varphi $ defined on the right half-plane.
• Employs contour integration and complex analysis to derive integral formulas for functions like $ f_1(z) = \sum c_k^{[1]} z^k $, with explicit kernels involving $ \ln(1+t) $ and $ (1+t)^{-a-1} $.
• Applies generalized Écalle-Borel (EB) summation to handle transseries and divergent coefficient asymptotics, enabling reconstruction even when coefficients are not convergent.
• Derives closed-form integral expressions for entire functions (e.g., $ f_3(z) $) and functions with natural boundaries via contour deformation and special functions.
• Extends the method to derive known results such as the Abel-Plana formula and Borel summation of divergent series through natural generalization.

Experimental results

Research questions

• RQ1 Can global analytic structure—such as location and type of singularities—be reconstructed from the asymptotic behavior of Taylor coefficients?
• RQ2 To what extent can entire functions be analyzed globally using only their local Taylor coefficients?
• RQ3 How can divergent series with factorial or logarithmic growth in coefficients be assigned meaningful global analytic continuations?
• RQ4 What is the precise duality between the asymptotic structure of coefficients and the monodromy or Riemann surface structure of the reconstructed function?
• RQ5 Can the method systematically recover known results like the Borel-summed Stirling approximation or the Euler-Maclaurin formula?

Key findings

• For $ f_1(z) = \sum_{k=1}^\infty \frac{z^k}{(k+a)^b} $, the global integral representation reveals a single logarithmic singularity at $ z=1 $ and $ f_1(z) = o(z) $ as $ z \to \infty $.
• The function $ f_2(z) = \sum_{k=1}^\infty \frac{z^k}{k^b + \ln k} $ has a single singularity at $ z=1 $, with analytic structure involving $ \ln(1-z) $ and $ (1-z) $, as shown by the contour integral in (1.3).
• The entire function $ f_3(z) = \sum_{k=1}^\infty \frac{z^k}{k^{k+1}} $ admits an integral representation (1.4) involving the inverse of $ s - \ln s $, enabling asymptotic analysis for large negative $ z $.
• For large negative $ z $, $ f_3(z) $ behaves as a constant plus $ z^{-1/2} e^{-z/e} $ times a factorially divergent series, with coefficients calculable via the integral formula.
• The function $ f_4(z) = \sum_{k=1}^\infty e^{\sqrt{k}} z^k $ is represented via a contour integral (1.5) over $ C_1 $, a loop around the origin, involving $ p^{-3/2} e^{p + 1/(4p)} $.
• The method naturally recovers Borel summation of divergent series and the Abel-Plana formula as special cases, demonstrating broad applicability beyond the initial examples.