Asymptotic properties of special function solutions of Painlevé III equation for fixed parameters
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910926421098496 |
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| author | Pan, Hao Prokhorov, Andrei |
| author_facet | Pan, Hao Prokhorov, Andrei |
| contents | In this paper, we compute the small and large $x$ asymptotics of the special function solutions of Painlevé-III equation in the complex plane. We use the representation in terms of Toeplitz determinants of Bessel functions obtained in arXiv:nlin/0302026. Toeplitz determinants are rewritten as multiple contour integrals using Andrèief's identity. The small and large $x$ asymptotics are obtained using elementary asymptotic methods applied to the multiple contour integral. The asymptotics is extended to the whole complex plane using analytic continuation formulas for Bessel functions. The claimed result has not appeared in the literature before. We note that Toeplitz determinant representation is useful for numerical computations of corresponding solutions of the Painlevé-III equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04852 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic properties of special function solutions of Painlevé III equation for fixed parameters Pan, Hao Prokhorov, Andrei Classical Analysis and ODEs Mathematical Physics Complex Variables Exactly Solvable and Integrable Systems 33C10, 33E17, 34E05, 34M55, 34M56 In this paper, we compute the small and large $x$ asymptotics of the special function solutions of Painlevé-III equation in the complex plane. We use the representation in terms of Toeplitz determinants of Bessel functions obtained in arXiv:nlin/0302026. Toeplitz determinants are rewritten as multiple contour integrals using Andrèief's identity. The small and large $x$ asymptotics are obtained using elementary asymptotic methods applied to the multiple contour integral. The asymptotics is extended to the whole complex plane using analytic continuation formulas for Bessel functions. The claimed result has not appeared in the literature before. We note that Toeplitz determinant representation is useful for numerical computations of corresponding solutions of the Painlevé-III equation. |
| title | Asymptotic properties of special function solutions of Painlevé III equation for fixed parameters |
| topic | Classical Analysis and ODEs Mathematical Physics Complex Variables Exactly Solvable and Integrable Systems 33C10, 33E17, 34E05, 34M55, 34M56 |
| url | https://arxiv.org/abs/2407.04852 |