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Approximations of Functions With Essential Singularities with Applications to Painlevé's First Transcendent

Nicholas Castillo

Abstract

In this work we develop an algorithmic procedure for associating a function defined on the Riemann surface of the $\log$ to given asymptotic data from a function at an essential singularity. We do this by means of rational approximations (Padé approximants) used in tandem with Borel-Écalle summation. Our method is capable of handling situations where classical methods either do not work or converge very slowly eg. \cite{DunLutz}. We provide a general outline of the procedure and then apply it to generating approximate tritronquée solutions to Painlevé's first equation ($\text{P}_\text{I}$). Our approximations (including $\text{P}_\text{I}$) are written as a finite linear combination of exponential integrals $\text{Ei}^+$. Furthermore, from arXiv:2210.17502 we have explicit rational approximations for each $\text{Ei}^+$ and thus for the approximation as a whole. In addition to rational approximations of $\text{P}_\text{I}$, we provide the first hundred or so poles of a tritronquée solution with essentially arbitrary accuracy which is dependent upon the order of Padé used.

Approximations of Functions With Essential Singularities with Applications to Painlevé's First Transcendent

Abstract

In this work we develop an algorithmic procedure for associating a function defined on the Riemann surface of the to given asymptotic data from a function at an essential singularity. We do this by means of rational approximations (Padé approximants) used in tandem with Borel-Écalle summation. Our method is capable of handling situations where classical methods either do not work or converge very slowly eg. \cite{DunLutz}. We provide a general outline of the procedure and then apply it to generating approximate tritronquée solutions to Painlevé's first equation (). Our approximations (including ) are written as a finite linear combination of exponential integrals . Furthermore, from arXiv:2210.17502 we have explicit rational approximations for each and thus for the approximation as a whole. In addition to rational approximations of , we provide the first hundred or so poles of a tritronquée solution with essentially arbitrary accuracy which is dependent upon the order of Padé used.
Paper Structure (12 sections, 5 theorems, 36 equations, 3 figures)

This paper contains 12 sections, 5 theorems, 36 equations, 3 figures.

Table of Contents

  1. Abstract
  2. The General Method
  3. Our Procedure
  4. Painlevé's First Equation
  5. Tritronqué solutions
  6. Tritronquée Pole Locations
  7. Estimate of error
  8. Appendix
  9. Padé Approximation
  10. Dyadic Expansion of $\mathrm{Ei}$
  11. More about Ei
  12. Acknowledgements

Key Result

Theorem 2

Let $h(x)$ be a solution to hEqu. Then for every $\varepsilon>0$ and compact $V \subset \widehat{\mathbb{C}}\setminus\{it:|t|\geq1\}$ where $\left \{[n_j,m_j]\right\}_{j\in\mathbb{N}}$ is a near diagonal Padé approximant.

Figures (3)

  • Figure 1: Modulus of error log-plot generated from a fifty exponential integral approximation. The $x,y$ axes are the number of steps of size $\frac{1}{10}$ in the real and imaginary directions respectively starting at $1-i$.
  • Figure 2: Modulus of error log-plot generated from a fifty exponential integral approximation. The $x,y$ axes are the number of steps of size $\frac{1}{10}$ in the real and imaginary directions respectively starting at $-1-\frac{3}{2}i$.
  • Figure 3: Pole locations of a tritronquée solution to $\text{P}_\text{I}$.

Theorems & Definitions (10)

  • Remark 1
  • Theorem 2
  • proof
  • Definition 4: Padé approximant Stahl
  • Theorem 5: Nuttall-Pommerenke theorem Nuttall, Pom
  • Theorem 6: Stahl
  • Definition 8: Stahl
  • Remark 9
  • Theorem 10: Stahl
  • Proposition 11