On the coefficients that arise from Laplace's method

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• 作者：John Wojdylo
• 关键词：Laplace's method ; Asymptotic approximation of integrals ; Partial Bell polynomials ; Potential polynomials ; Faà ; di Bruno's formula ; Lagrange's reversion theorem
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2006
• 出版时间：1 November 2006
• 年：2006
• 卷：196
• 期：1
• 页码：241-266
• 全文大小：303 K

文摘

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 Erdélyi's use of Watson's Lemma to obtain an infinite asymptotic expansion valid for any Laplace-type integral, published in 1956. Erdélyi's expansion contains coefficients href=""/science?_ob=MathURL&_method=retrieve&_udi=B6TYH-4HDG9G6-6&_mathId=mml1&_user=10&_cdi=5619&_rdoc=19&_handle=V-WA-A-W-WV-MsSWYVW-UUW-U-AAZCBWVUDD-AAZWEUVYDD-VBCEUVVD-WV-U&_acct=C000050221&_version=1&_userid=10&md5=aa1d915da35c5d0ab2934d3868530079"" title=""Click to view the MathML source"">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 href=""/science?_ob=MathURL&_method=retrieve&_udi=B6TYH-4HDG9G6-6&_mathId=mml2&_user=10&_cdi=5619&_rdoc=19&_handle=V-WA-A-W-WV-MsSWYVW-UUW-U-AAZCBWVUDD-AAZWEUVYDD-VBCEUVVD-WV-U&_acct=C000050221&_version=1&_userid=10&md5=d982eb6d91701384c7de0d2155865530"" title=""Click to view the MathML source"">c_s in fact have a very simple general form. In effect, we extend Erdélyi's theorem. Our results greatly simplify calculation of the href=""/science?_ob=MathURL&_method=retrieve&_udi=B6TYH-4HDG9G6-6&_mathId=mml3&_user=10&_cdi=5619&_rdoc=19&_handle=V-WA-A-W-WV-MsSWYVW-UUW-U-AAZCBWVUDD-AAZWEUVYDD-VBCEUVVD-WV-U&_acct=C000050221&_version=1&_userid=10&md5=f7906efe3a49e1c5718239f0db76bfb2"" title=""Click to view the MathML source"">c_s in any particular application and clarify the theoretical basis of Erdélyi's expansion: it turns out that Faà di Bruno's formula has always played a central role in it.

We prove or derive the following:

l4"">href=""/science?_ob=MathURL&_method=retrieve&_udi=B6TYH-4HDG9G6-6&_mathId=mml4&_user=10&_cdi=5619&_rdoc=19&_handle=V-WA-A-W-WV-MsSWYVW-UUW-U-AAZCBWVUDD-AAZWEUVYDD-VBCEUVVD-WV-U&_acct=C000050221&_version=1&_userid=10&md5=d344d995a410862f5bf7ff33002f24a8"" title=""Click to view the MathML source"">• The correct dimensionless groups. Erdélyi's expansion is properly expressed in terms of scaled coefficients .

href=""/science?_ob=MathURL&_method=retrieve&_udi=B6TYH-4HDG9G6-6&_mathId=mml6&_user=10&_cdi=5619&_rdoc=19&_handle=V-WA-A-W-WV-MsSWYVW-UUW-U-AAZCBWVUDD-AAZWEUVYDD-VBCEUVVD-WV-U&_acct=C000050221&_version=1&_userid=10&md5=f403cab2def9a99859d5fa4ae7683a5d"" title=""Click to view the MathML source"">• Two explicit expressions for 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.

href=""/science?_ob=MathURL&_method=retrieve&_udi=B6TYH-4HDG9G6-6&_mathId=mml8&_user=10&_cdi=5619&_rdoc=19&_handle=V-WA-A-W-WV-MsSWYVW-UUW-U-AAZCBWVUDD-AAZWEUVYDD-VBCEUVVD-WV-U&_acct=C000050221&_version=1&_userid=10&md5=8436029b6a99c59ce4b77d2dab6242cc"" title=""Click to view the MathML source"">• A recursive expression for .

href=""/science?_ob=MathURL&_method=retrieve&_udi=B6TYH-4HDG9G6-6&_mathId=mml10&_user=10&_cdi=5619&_rdoc=19&_handle=V-WA-A-W-WV-MsSWYVW-UUW-U-AAZCBWVUDD-AAZWEUVYDD-VBCEUVVD-WV-U&_acct=C000050221&_version=1&_userid=10&md5=3f27b001bf4aa98e342e97f1d3d8ae55"" title=""Click to view the MathML source"">• Each coefficient can be expressed as a polynomial in href=""/science?_ob=MathURL&_method=retrieve&_udi=B6TYH-4HDG9G6-6&_mathId=mml12&_user=10&_cdi=5619&_rdoc=19&_handle=V-WA-A-W-WV-MsSWYVW-UUW-U-AAZCBWVUDD-AAZWEUVYDD-VBCEUVVD-WV-U&_acct=C000050221&_version=1&_userid=10&md5=8690343704de6f9061a108d0a312549b"" title=""Click to view the MathML source"">(α+s)/μ, where href=""/science?_ob=MathURL&_method=retrieve&_udi=B6TYH-4HDG9G6-6&_mathId=mml13&_user=10&_cdi=5619&_rdoc=19&_handle=V-WA-A-W-WV-MsSWYVW-UUW-U-AAZCBWVUDD-AAZWEUVYDD-VBCEUVVD-WV-U&_acct=C000050221&_version=1&_userid=10&md5=6778eead44863d6452a1fc2542e7c358"" title=""Click to view the MathML source"">α and href=""/science?_ob=MathURL&_method=retrieve&_udi=B6TYH-4HDG9G6-6&_mathId=mml14&_user=10&_cdi=5619&_rdoc=19&_handle=V-WA-A-W-WV-MsSWYVW-UUW-U-AAZCBWVUDD-AAZWEUVYDD-VBCEUVVD-WV-U&_acct=C000050221&_version=1&_userid=10&md5=b9687a733ab7be6c635e893325df7bd6"" title=""Click to view the MathML source"">μ are quantities in Erdélyi'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 Erdélyi's expansion—a point which Erdélyi himself alluded to.

We consider as an example an integral that occurs in a variational approach to finding the binding energy of helium dimers. We also present a three-line computer code to generate the coefficients 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 Faà di Bruno's formula is explained.