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Published online by Cambridge University Press: 20 January 2009
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±
* 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
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