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Exponentially small expansions related to the parabolic cylinder function

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R B Paris

Classical Analysis and ODEs · Submitted 30 Dec 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Sci.} {\bf 7} (2013) 6601--6609 is employed to derive the correct exponentially small contribution to the asymptotic expansion for the even and odd solutions of a second-order differential equation related to Weber's equation.

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arXiv ID1912.12974 (version 1)
Pith ID3EZGJ4JC, Pith's permanent record of this paper and its review
CitationR B Paris. "Exponentially small expansions related to the parabolic cylinder function." arXiv:1912.12974 (2019). https://pith.science/paper/3EZGJ4JC
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,674 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.15 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.96 · mentions code or data availability 0.03.

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arXiv listing abstract v1. Results and the main claim are underlined; .

The refined asymptotic expansion of the confluent hypergeometric function $M(a,b,z)$ on the Stokes line $\arg\,z=\pi$ given in {\it Appl. Math. Sci.} {\bf 7} (2013) 6601--6609 is employed to derive the correct exponentially small contribution to the asymptotic expansion for the even and odd solutions of a second-order differential equation related to Weber's equation. It is demonstrated that the standard asymptotics of the parabolic cylinder function $U(a,z)$ yield an incorrect exponentially small contribution to these solutions. Numerical results verifying the accuracy of the new expansions are given.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The refined asymptotic expansion of the confluent hypergeometric function $M(a,b,z)$ on the Stokes line $\arg\,z=\pi$ given in {\it Appl. Math. background 0.64 0.16 0.07 asserted 0.62
s1 Sci.} {\bf 7} (2013) 6601--6609 is employed to derive the correct exponentially small contribution to the asymptotic expansion for the even and odd solutions of a second-order differential equation related to Weber's equation. method 0.49 0.85 0.43 asserted 1.00
s2 It is demonstrated that the standard asymptotics of the parabolic cylinder function $U(a,z)$ yield an incorrect exponentially small contribution to these solutions. result 0.92 0.69 0.50 asserted 1.00
s3 Numerical results verifying the accuracy of the new expansions are given. result 0.61 0.90 0.07 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

The refined asymptotic expansion of the confluent hypergeometric function $M(a,b,z)$ on the Stokes line $\arg\,z=\pi$ given in {\it Appl. Math. Sci.} {\bf 7} (2013) 6601--6609 is employed to derive the correct exponentially small contribution to the asymptotic expansion for the even and odd solutions of a second-order differential equation related to Weber's equation. It is demonstrated that the standard asymptotics of the parabolic cylinder function $U(a,z)$ yield an incorrect exponentially small contribution to these solutions. Numerical results verifying the accuracy of the new expansions are given.

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