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The Asymptotic Expansion of Legendre Functions of Large Degree and Order

Thorne, R. C. (1957) The Asymptotic Expansion of Legendre Functions of Large Degree and Order. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 249 (971). pp. 597-620. ISSN 1364-503X. doi:10.1098/rsta.1957.0008.

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New expansions for the Legendre functions P_n^(-m)(z) and Q_n^(-m)(z) are obtained; m and n are large positive numbers, 0 < m < n and α = m/(n +1/2) is kept fixed as n→ ∞; z is an unrestricted complex variable. Three groups of expansions are obtained. The first is in terms of exponential functions. These expansions are uniformly valid as n→ ∞ with respect to z for all z lying in Rz ≥ 0 except for the strips given by | lz |< ∂, Rz < β + ∂, where ∂ > 0 and β = √(1 - α^2). The second set of expansions is in terms of Airy functions. These expansions are uniformly valid with respect to z throughout the whole z plane cut from +1 to -∞ except for a pear-shaped domain surrounding the point z = -1 and a strip lying immediately below the real z axis for which |Rzl < β + ∂, 0 ≥ lz> -∂. The third group of expansions is in terms of Bessel functions of order m. These expansions are valid uniformly with respect to z over the whole cut z plane except for the pear-shaped domain surrounding z = -1. No expansions have been given before for the Legendre functions of large degree and order.

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Additional Information:© 1957 The Royal Society. (Communicated by Sir Harold Jeffreys, F .R.S. - Received 28 August 1956) Based partly on research prepared under contract Nonr-220(11) between the U.S. Office of Naval Research and the California Institute of Technology, reference no. NR 043-121. I should like to thank Mr F. W. J. Olver, Mr F. Ursell and Professor A. Erdélyi for helpful discussions. I also wish to acknowledge the generous support of Trinity College, Cambridge, and the University of Cambridge for scholarships and grants during the tenure of which most of the work described above was carried out. I also wish to thank the U.S. Office of Naval Research for the sponsorship of a research fellowship at the California Institute of Technology. This present paper has also been produced as Technical Report number 13, Contract Nonr-220(11), Department of Mathematics, California Institute of Technology.
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Office of Naval Research (ONR)Nonr-220(11)
University of CambridgeUNSPECIFIED
Issue or Number:971
Record Number:CaltechAUTHORS:20150817-162154164
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Official Citation:The Asymptotic Expansion of Legendre Functions of Large Degree and Order R. C. Thorne Phil. Trans. R. Soc. Lond. A 1957 249 597-620; DOI: 10.1098/rsta.1957.0008. Published 30 May 1957
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:59670
Deposited By: George Porter
Deposited On:18 Aug 2015 15:24
Last Modified:10 Nov 2021 22:23

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