Terminating Poincare asymptotic expansion of the Hankel transform of entire exponential type functions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| author | Rojas, Nathalie Liezel R. Galapon, Eric A. |
| author_facet | Rojas, Nathalie Liezel R. Galapon, Eric A. |
| contents | We perform an asymptotic evaluation of the Hankel transform, $\int_0^{\infty}J_ν(λx) f(x)\mathrm{d}x$, for arbitrarily large $λ$ of an entire exponential type function, $f(x)$, of type $τ$ by shifting the contour of integration in the complex plane. Under the situation that $J_ν(λx)f(x)$ has an odd parity with respect to $x$ and the condition that the asymptotic parameter $λ$ is greater than the type $τ$, we obtain an exactly terminating Poincar{é} expansion without any trailing subdominant exponential terms. That is the Hankel transform evaluates exactly into a polynomial in inverse $λ$ as $λ$ approaches infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10948 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Terminating Poincare asymptotic expansion of the Hankel transform of entire exponential type functions Rojas, Nathalie Liezel R. Galapon, Eric A. Complex Variables Mathematical Physics We perform an asymptotic evaluation of the Hankel transform, $\int_0^{\infty}J_ν(λx) f(x)\mathrm{d}x$, for arbitrarily large $λ$ of an entire exponential type function, $f(x)$, of type $τ$ by shifting the contour of integration in the complex plane. Under the situation that $J_ν(λx)f(x)$ has an odd parity with respect to $x$ and the condition that the asymptotic parameter $λ$ is greater than the type $τ$, we obtain an exactly terminating Poincar{é} expansion without any trailing subdominant exponential terms. That is the Hankel transform evaluates exactly into a polynomial in inverse $λ$ as $λ$ approaches infinity. |
| title | Terminating Poincare asymptotic expansion of the Hankel transform of entire exponential type functions |
| topic | Complex Variables Mathematical Physics |
| url | https://arxiv.org/abs/2409.10948 |