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Conformal and uniformizing maps in Borel analysis

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Abstract

Perturbative expansions in physical applications are generically divergent, and their physical content can be studied using Borel analysis. Given just a finite number of terms of such an expansion, these input data can be analyzed in different ways, leading to vastly different precision for the extrapolation of the expansion parameter away from its original asymptotic regime. Here, we describe how conformal maps and uniformizing maps can be used, in conjunction with Padé approximants, to increase the precision of the information that can be extracted from a finite amount of perturbative input data. We also summarize results from the physical interpretation of Padé approximations in terms of electrostatic potential theory.

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Notes

  1. For example, the Airy function is \(\mathrm{Ai}(x)=\frac{2 x^{5/4}}{3\sqrt{\pi }}e^{-\frac{2}{3} x^{3/2}} F\left( \frac{4}{3} x^{3/2};\frac{1}{6}, \frac{5}{6}, 1\right) \), and the Whittaker function is \(W_{\mu ,\nu }(x)=x^{1+\mu }\, e^{-x/2} F\left( x; \frac{1}{2}+\nu -\mu , \frac{1}{2}-\nu -\mu , 1\right) \).

  2. In the simple case where the singularity is not a branch point but a pole, Padé–Borel is of course optimal.

  3. Without the additional Padé approximation, the conformal map is only as effective as the \(\mathcal {PB}\) approximation described above [5].

  4. These maps can also be approximated by rapidly convergent iterations of simple maps [6].

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Acknowledgements

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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Costin, O., Dunne, G.V. Conformal and uniformizing maps in Borel analysis. Eur. Phys. J. Spec. Top. 230, 2679–2690 (2021). https://doi.org/10.1140/epjs/s11734-021-00267-x

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