Saddle point approximation to higher order
Abstract
Introduction/purpose: Saddle point approximation has been considered in the paper.
Methods: The saddle point method is used in several different fields of mathematics and physics. Several terms of the expansion for the factorial function have been explicitely computed.
Results: The integrals estimated in this way have values close to the exact one.
Conclusions: Higher order corrections are not negligible even when requiring moderate levels of precision.
References
Brillouin, L. 1926. La mécanique ondulatoire de Schrödinger: une méthode générale de resolution par approximations successives. Comptes Rendus de l’Académie des Sciences, 183, pp.24-26.
Daniels, H.E. 1954. Saddlepoint Approximations in Statistics. The Annals of Mathematical Statistics, 25(4), pp.631-650 [online]. Available at: http://www.jstor.org/stable/2236650 [Accessed: 15 January 2022].
Dirac, P.A.M. 1933. The Lagrangian in Quantum Mechanics. Physikalische Zeitschrift der Sowjetunion, 3(1), pp.64–72.
Erdelyi, A. 1956. Asymptotic expansions. New York: Dover Publications, Inc.
Feynman, R.P. 1965. Nobel Lecture. The Development of the Space-Time View of Quantum Electrodynamics. The Nobel Prize, December 11 [online]. Available at: https://www.nobelprize.org/prizes/physics/1965/feynman/lecture/ [Accessed: 15 January 2022].
Kramers, H.A. 1926. Wellenmechanik und halbzahlige Quantisierung. Zeitschrift für Physik, 39(10-11), pp.828-840. Available at: https://doi.org/10.1007/BF01451751
Laplace, P.S. 1986. Memoir on the Probability of the Causes of Events. Statistical Science, 1(3), pp.364-378 [online]. Available at: http://www.jstor.org/stable/2245476 [Accessed: 15 January 2022].
Moivre, A. de. 1730. Miscellanea analytica de seriebus et quadraturis. London: Tonson & J. Watts.
Moivre, A. de. 1756. The Doctrine of Chances: Or, a Method of Calculating the Probabilities of Events in Play, the third edition. London: A. Millar.
Parisi, G. 1988. Statistical field theory. Boston, MA, USA: Addison-Wesley Pub.Co. ISBN-13: 978-0201059854.
Ramond, P. 1989. Field theory: A modern primer (Frontiers in physics). Boston, MA, USA: Addison-Wesley Pub. Co. ISBN-13: 978-0201157727.
Stirling, J. 1764. Methodus differentialis: sive tractatus de summatione et in- terpolatione serierum infinitarum. London: Prostat apud J. Whiston & B. White.
Wentzel, G. 1926. Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik. Zeitschrift für Physik, 38(6-7), pp.518-529. Available at: https://doi.org/10.1007/BF01397171
Whittaker, E.T. & Watson, G.N. 1927. A Course of Modern Analysis, 4th edition. Cambridge University Press. ISBN-13: 978-0521091893.
Proposed Creative Commons Copyright Notices
Proposed Policy for Military Technical Courier (Journals That Offer Open Access)
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).