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The Asymptotic Expansions of the Spherical Harmonics

Published online by Cambridge University Press:  20 January 2009

T. M. MacRobert
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Associated Legendre Functions as Integrals involving Bessel Functions. Let

,

where C denotes a contour which begins at −∞ on the real axis, passes positively round the origin, and returns to −∞, amp λ=−π initially, and R(z)>0, z being finite and ≠1. [If R(z)>0 and z is finite, then R(z±)>0.] Then if I−m (λ) be expanded in ascending powers of λ, and if the resulting expression be integrated term by term, it is found that

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , February 1922 , pp. 82 - 93
Copyright
Copyright © Edinburgh Mathematical Society 1922

References

* cf. Barnes, , Quart. Journ. of Math., Vol. 39, p. 120.Google Scholar The notation employed for the Associated Legendre Functions is that of Hobson, , Phil. Trans., Vol. 187, A.Google Scholar

* The existence of a relation between the asymptotic expansions of the Bessel Functions and those of the Spherical Harmonics was suggested by DrDougall, John, Proc. of the. Edin. Math. Soc., Vol. 18, p. 52.Google Scholar

* Phil. Trans., 187 A. (1896), 1 p. 485–489.Google Scholar

† Quart. Journ. of Maths., XXXIX (1908), pp. 143–174.Google Scholar

‡ Proc. Camb. Phil. Soc., XXII (1918), pp. 277–308.Google Scholar

* Cf. Whittaker and Watson's Analysis, Chap. XVI.

† Cf. ProfGibson, G. A., Proc. Edin. Math. Soc., Vol. XXXVIII.Google Scholar

* Jour, de Math. (3) 1. 1875, p. 394.Google Scholar