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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2403.17170 |
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| _version_ | 1866911484042280960 |
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| author | Castillo, Nicholas |
| author_facet | Castillo, Nicholas |
| contents | 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. 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, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17170 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximations of Functions With Essential Singularities with Applications to Painlevé's First Transcendent Castillo, Nicholas Complex Variables 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. 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, 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. |
| title | Approximations of Functions With Essential Singularities with Applications to Painlevé's First Transcendent |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2403.17170 |