Digital Library of Mathematical Functions
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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.19 Asymptotic Expansions for Large Argument

As x

13.19.1 Mκ,μ(x)Γ(1+2μ)Γ(12+μ-κ)12xx-κs=0(12-μ+κ)s(12+μ+κ)ss!x-s,
μ-κ-12,-32,.

As z

13.19.2 Mκ,μ(z)Γ(1+2μ)Γ(12+μ-κ)12zz-κs=0(12-μ+κ)s(12+μ+κ)ss!z-s+Γ(1+2μ)Γ(12+μ+κ)-12z±(12+μ-κ)πzκs=0(12+μ-κ)s(12-μ-κ)ss!(-z)-s,
-12π+δ±phz32π-δ,

provided that both μκ-12,-32,. Again, δ denotes an arbitrary small positive constant. Also,

13.19.3 Wκ,μ(z)-12zzκs=0(12+μ-κ)s(12-μ-κ)ss!(-z)-s,
|phz|32π-δ.

Error bounds and exponentially-improved expansions are derivable by combining §§13.7(ii) and 13.7(iii) with (13.14.2) and (13.14.3). See also Olver (1965).

For an asymptotic expansion of Wκ,μ(z) as z that is valid in the sector |phz|π-δ and where the real parameters κ, μ are subject to the growth conditions κ=o(z), μ=o(z), see Wong (1973a).