Pagola Martínez, Pedro Jesús

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Pagola Martínez

Pedro Jesús

Estadística, Informática y Matemáticas

InaMat2. Instituto de Investigación en Materiales Avanzados y Matemáticas

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Open Access
The uniform asymptotic method "saddle point near an end point" revisited
(Elsevier, 2024) López García, José Luis; Pagola Martínez, Pedro Jesús; Palacios Herrero, Pablo; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertistate Publikoa
We continue the program initiated in [López & all, 2009–2011] to simplify asymptotic methods for integrals: in this paper we revise the uniform method ‘‘saddle point near an end point’’. We obtain a more systematic version of this uniform asymptotic method where the computation of the coefficients of the asymptotic expansion is remarkably simpler than in the classical method. On the other hand, as in the standard method, the asymptotic sequence is given in terms of parabolic cylinder functions. New asymptotic expansions of the confluent hypergeometric functions 𝑀(𝑐, 𝑥∕𝛼 + 𝑐 + 1, 𝑥) and 𝑈(𝑐, 𝛼𝑥 + 𝑐 + 1, 𝑥) for large 𝑥, 𝑐 fixed, uniformly valid for 𝛼 ∈ (0, ∞), are given as an illustration.