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VIII.—The Asymptotic Expansion of the Confluent Hypergeometric Function, and a Fourier-Bessel Expansion
Published online by Cambridge University Press: 15 September 2014
Extract
In Whittaker and Watson's Modern Analysis, chap, xvi, the asymptotic expansion of the confluent hypergeometric function Wk.m(z) is established for the region—π < amp z < π, z ≠ 0. The object of the first part of this paper is to show that this expansion is valid in the extended region –3π/2 < amp z < 3π/2, z ≠ 0.
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- Copyright © Royal Society of Edinburgh 1923
References
page 89 note * Cf. Prof. G. A. Gibson, Proc. Edin. Math. Soc., vol. xxxviii.
page 90 note * Prof. G. A. Gibson, loc. cit.
page 90 note † Proc. Edin. Math. Soc., vol. xxxix.
page 90 note ‡ Cf. Gray and Mathews' Bessel Functions, chap. x.
page 93 note * Note.—The factor e iM(λ–γ) in the asymptotic expansion of 
is equal to e iMd cos θ – Md sin θ, when λ–γ= deiθ, so that the integral remains convergent when θ varies from 0 to π.
page 95 note * This method could also be employed in the discussion of the integral of Φ(ζ), and then it would be unnecessary to introduce the restriction 0≦n<½.