Physics
Scientific paper
May 1957
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1957rspta.249..597t&link_type=abstract
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Volume 249, Issue 971,
Physics
14
Scientific paper
New expansions for the Legendre functions Pn-m(z) and Qn-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 |Iz| < δ , Rz < β + δ , where δ > 0 and β = surd (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 |Rz| < β + δ , 0 >= Iz > -δ . 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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