Conical functions with one or both parameters large

Abstract

SynopsisUniform asymptotic expansions are derived for conical functions, Legendre functions of order µ and degree −½ + iτ, where µ and τ are non-negative real parameters. As τ → ∞, expansions are furnished for the conical functions which involve Bessel functions of order µ. These expansions are uniformly valid for 0 ≦ µ ≦ Aτ (A an arbitrary positive constant), and are also uniformly valid for Re (z) ≧ 0 in the complex argument case, and 0 ≦ z < ∞ in the real argument case. The case µ → ∞ is also considered, and expansions are furnished which are uniformly valid in the same z regions for 0 ≦ τ ≧ Bµ (B an arbitrary positive constant); in the cases where Re(z) ≧ 0 and 1 ≦ z < ∞, the expansions involve Bessel functions of purely imaginary order iτ, and in the case where 0 ≦ z < 1 the expansions involve elementary functions.