On the coefficients that arise from Laplace's method

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    Abstract

    Laplace's method is one of the best-known techniques in the asymptotic approximation of integrals. The salient step in the technique's historical development was Erdelyi's use of Watson's Lemma to obtain an infinite asymptotic expansion valid for any Laplace-type integral, published in 1956. Erdelyi's expansion contains coefficients c(s) that must be calculated in each application of Laplace's method, a tedious process that has traditionally involved the reversion of a series. This paper shows that the coefficients c(s) in fact have a very simple general form. In effect, we extend Erdelyi's theorem. Our results greatly simplify calculation of the c(s) in any particular application and clarify the theoretical basis of Erdelyi's expansion: it turns out that Fail di Bruno's formula has always played a central role in it.We prove or derive the following:The correct dimensionless groups. Erdelyi's expansion is properly expressed in terms of scaled coefficients c(s)*.Two explicit expressions for c(s)* in terms of combinatorial objects called partial ordinary Bell polynomials. This form is probably computationally optimal and makes checking for correctness a relatively straightforward process.A recursive expression for c(s)*.Each coefficient c(s)* can be expressed as a polynomial in (alpha + s)/mu, where alpha and mu are quantities in Erdelyi's formulation.The main insight that emerges is that the traditional approach to Laplace's method, involving reversion of a series, is less efficient and need only be invoked if one is interested in the role of the reversion coefficients in Erdelyi's expansion-a point which Erdelyi himself alluded to.We consider as an example an integral that occurs in a variational approach to finding the binding energy of helium dinners. We also present a three-line computer code to generate the coefficients c(s)* exactly in the general case. In a sequel paper (to be published in SIAM Review), a new representation for the gamma function is obtained, and the link with Faa di Bruno's formula is explained. (c) 2005 Elsevier B.V. All rights reserved.
    Original languageEnglish
    Pages (from-to)241-266
    JournalJournal of Computational and Applied Mathematics
    Volume196
    Issue number1
    DOIs
    Publication statusPublished - 2006

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    Laplace's Method
    Coefficient
    Polynomials
    Bell Polynomials
    Binding energy
    Gamma function
    Series
    Binding Energy
    Asymptotic Approximation
    Variational Approach
    Helium
    Laplace
    Dimensionless
    Asymptotic Expansion
    Lemma
    Correctness
    Simplify
    Valid
    Partial
    Polynomial

    Cite this

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    title = "On the coefficients that arise from Laplace's method",
    abstract = "Laplace's method is one of the best-known techniques in the asymptotic approximation of integrals. The salient step in the technique's historical development was Erdelyi's use of Watson's Lemma to obtain an infinite asymptotic expansion valid for any Laplace-type integral, published in 1956. Erdelyi's expansion contains coefficients c(s) that must be calculated in each application of Laplace's method, a tedious process that has traditionally involved the reversion of a series. This paper shows that the coefficients c(s) in fact have a very simple general form. In effect, we extend Erdelyi's theorem. Our results greatly simplify calculation of the c(s) in any particular application and clarify the theoretical basis of Erdelyi's expansion: it turns out that Fail di Bruno's formula has always played a central role in it.We prove or derive the following:The correct dimensionless groups. Erdelyi's expansion is properly expressed in terms of scaled coefficients c(s)*.Two explicit expressions for c(s)* in terms of combinatorial objects called partial ordinary Bell polynomials. This form is probably computationally optimal and makes checking for correctness a relatively straightforward process.A recursive expression for c(s)*.Each coefficient c(s)* can be expressed as a polynomial in (alpha + s)/mu, where alpha and mu are quantities in Erdelyi's formulation.The main insight that emerges is that the traditional approach to Laplace's method, involving reversion of a series, is less efficient and need only be invoked if one is interested in the role of the reversion coefficients in Erdelyi's expansion-a point which Erdelyi himself alluded to.We consider as an example an integral that occurs in a variational approach to finding the binding energy of helium dinners. We also present a three-line computer code to generate the coefficients c(s)* exactly in the general case. In a sequel paper (to be published in SIAM Review), a new representation for the gamma function is obtained, and the link with Faa di Bruno's formula is explained. (c) 2005 Elsevier B.V. All rights reserved.",
    author = "John Wojdylo",
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    language = "English",
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    On the coefficients that arise from Laplace's method. / Wojdylo, John.

    In: Journal of Computational and Applied Mathematics, Vol. 196, No. 1, 2006, p. 241-266.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - On the coefficients that arise from Laplace's method

    AU - Wojdylo, John

    PY - 2006

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    N2 - Laplace's method is one of the best-known techniques in the asymptotic approximation of integrals. The salient step in the technique's historical development was Erdelyi's use of Watson's Lemma to obtain an infinite asymptotic expansion valid for any Laplace-type integral, published in 1956. Erdelyi's expansion contains coefficients c(s) that must be calculated in each application of Laplace's method, a tedious process that has traditionally involved the reversion of a series. This paper shows that the coefficients c(s) in fact have a very simple general form. In effect, we extend Erdelyi's theorem. Our results greatly simplify calculation of the c(s) in any particular application and clarify the theoretical basis of Erdelyi's expansion: it turns out that Fail di Bruno's formula has always played a central role in it.We prove or derive the following:The correct dimensionless groups. Erdelyi's expansion is properly expressed in terms of scaled coefficients c(s)*.Two explicit expressions for c(s)* in terms of combinatorial objects called partial ordinary Bell polynomials. This form is probably computationally optimal and makes checking for correctness a relatively straightforward process.A recursive expression for c(s)*.Each coefficient c(s)* can be expressed as a polynomial in (alpha + s)/mu, where alpha and mu are quantities in Erdelyi's formulation.The main insight that emerges is that the traditional approach to Laplace's method, involving reversion of a series, is less efficient and need only be invoked if one is interested in the role of the reversion coefficients in Erdelyi's expansion-a point which Erdelyi himself alluded to.We consider as an example an integral that occurs in a variational approach to finding the binding energy of helium dinners. We also present a three-line computer code to generate the coefficients c(s)* exactly in the general case. In a sequel paper (to be published in SIAM Review), a new representation for the gamma function is obtained, and the link with Faa di Bruno's formula is explained. (c) 2005 Elsevier B.V. All rights reserved.

    AB - Laplace's method is one of the best-known techniques in the asymptotic approximation of integrals. The salient step in the technique's historical development was Erdelyi's use of Watson's Lemma to obtain an infinite asymptotic expansion valid for any Laplace-type integral, published in 1956. Erdelyi's expansion contains coefficients c(s) that must be calculated in each application of Laplace's method, a tedious process that has traditionally involved the reversion of a series. This paper shows that the coefficients c(s) in fact have a very simple general form. In effect, we extend Erdelyi's theorem. Our results greatly simplify calculation of the c(s) in any particular application and clarify the theoretical basis of Erdelyi's expansion: it turns out that Fail di Bruno's formula has always played a central role in it.We prove or derive the following:The correct dimensionless groups. Erdelyi's expansion is properly expressed in terms of scaled coefficients c(s)*.Two explicit expressions for c(s)* in terms of combinatorial objects called partial ordinary Bell polynomials. This form is probably computationally optimal and makes checking for correctness a relatively straightforward process.A recursive expression for c(s)*.Each coefficient c(s)* can be expressed as a polynomial in (alpha + s)/mu, where alpha and mu are quantities in Erdelyi's formulation.The main insight that emerges is that the traditional approach to Laplace's method, involving reversion of a series, is less efficient and need only be invoked if one is interested in the role of the reversion coefficients in Erdelyi's expansion-a point which Erdelyi himself alluded to.We consider as an example an integral that occurs in a variational approach to finding the binding energy of helium dinners. We also present a three-line computer code to generate the coefficients c(s)* exactly in the general case. In a sequel paper (to be published in SIAM Review), a new representation for the gamma function is obtained, and the link with Faa di Bruno's formula is explained. (c) 2005 Elsevier B.V. All rights reserved.

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    JF - Journal of Computational and Applied Mathematics

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