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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.20 Uniform Asymptotic Approximations for Large μ

Contents

§13.20(i) Large μ, Fixed κ

When μ in the sector |phμ|12π-δ(<12π), with κ() fixed

13.20.1 Mκ,μ(z)=zμ+12(1+O(μ-1)),

uniformly for bounded values of |z|; also

13.20.2 Wκ,μ(x)=π-12Γ(κ+μ)(14x)12-μ(1+O(μ-1)),

uniformly for bounded positive values of x. For an extension of (13.20.1) to an asymptotic expansion, together with error bounds, see Olver (1997b, Chapter 10, Ex. 3.4).

§13.20(ii) Large μ, 0κ(1-δ)μ

Let

13.20.3 X=4μ2-4κx+x2.

Then as μ

13.20.4 Mκ,μ(x)=2μxX(4μ2x2μ2-κx+μX)μ(2(μ-κ)X+x-2κ)κ12X-μ(1+O(1μ)),
13.20.5 Wκ,μ(x)=xX(2μ2-κx+μX(μ-κ)x)μ(X+x-2κ2)κ-12X-κ(1+O(1μ)),

uniformly with respect to x(0,) and κ[0,(1-δ)μ], where δ again denotes an arbitrary small positive constant.

§13.20(iii) Large μ, -(1-δ)μκμ

Let

13.20.6 α =2|κ-μ|/μ,
13.20.7 X =|x2-4κx+4μ2|,
13.20.8 Φ(κ,μ,x) =(μ2ζ2-2κμ+2μ2x2-4κx+4μ2)14x,

with the variable ζ defined implicitly as follows:

(a) In the case -μ<κ<μ

13.20.9 ζζ2+α2+α2arcsinh(ζα)=Xμ-2κμln(X+x-2κ2μ2-κ2)-2ln(μX+2μ2-κxxμ2-κ2).

(b) In the case μ=κ

13.20.10 ζ=±xμ-2-2ln(x2μ),

the upper or lower sign being taken according as x2μ.

(In both cases (a) and (b) the x-interval (0,) is mapped one-to-one onto the ζ-interval (-,), with x=0 and corresponding to ζ=- and , respectively.) Then as μ

13.20.11 Wκ,μ(x)=(12μ)-14(κ+μ)12(κ+μ)Φ(κ,μ,x)U(μ-κ,ζ2μ)(1+O(μ-1lnμ)),
13.20.12 Mκ,μ(x)=(8μ)14(2μ)2μ(κ+μ)12(κ+μ)Φ(κ,μ,x)U(μ-κ,-ζ2μ)(1+O(μ-1lnμ)),

uniformly with respect to x(0,) and κ[-(1-δ)μ,μ]. For the parabolic cylinder function U see §12.2.

These results are proved in Olver (1980b). This reference also supplies error bounds and corresponding approximations when x, κ, and μ are replaced by x, κ, and μ, respectively.

§13.20(iv) Large μ, μκμ/δ

Again define α, X, and Φ(κ,μ,x) by (13.20.6)–(13.20.8), but with ζ now defined by

13.20.13 ζζ2-α2-α2arccosh(ζα) =Xμ-2κμln(X+x-2κ2κ2-μ2)-2ln(κx-μX-2μ2xκ2-μ2),
x2κ+2κ2-μ2,
13.20.14 ζα2-ζ2+α2arcsin(ζα) =Xμ+2κμarctan(x-2κX)-2arctan(κx-2μ2μX),
2κ-2κ2-μ2x2κ+2κ2-μ2,
13.20.15 -ζζ2-α2-α2arccosh(-ζα) =-Xμ+2κμln(2κ-X-x2κ2-μ2)+2ln(μX+2μ2-κxxκ2-μ2),
0<x2κ-2κ2-μ2,

when μ<κ, and by (13.20.10) when μ=κ. (As in §13.20(iii) x=0 and correspond to ζ=- and , respectively). Then as μ

13.20.16 Wκ,μ(x)=(12μ)-14(κ+μ)12(κ+μ)Φ(κ,μ,x)(U(μ-κ,ζ2μ)+envU(μ-κ,ζ2μ)O(μ-23)),
13.20.17 Mκ,μ(x)=(8μ)14(2μ)2μ(κ+μ)12(κ+μ)Φ(κ,μ,x)×(U(μ-κ,-ζ2μ)+envU¯(μ-κ,ζ2μ)O(μ-23),)

uniformly with respect to ζ[0,) and κ[μ,μ/δ].

Also,

13.20.18 Wκ,μ(x)=(12μ)-14(κ+μ)12(κ+μ)Φ(κ,μ,x)(U(μ-κ,ζ2μ)+envU¯(μ-κ,-ζ2μ)O(μ-23)),
13.20.19 Mκ,μ(x)=(8μ)14(2μ)2μ(κ+μ)12(κ+μ)Φ(κ,μ,x)×(U(μ-κ,-ζ2μ)+envU(μ-κ,-ζ2μ)O(μ-23),)

uniformly with respect to ζ(-,0] and κ[μ,μ/δ].

For the parabolic cylinder functions U and U¯ see §12.2, and for the env functions associated with U and U¯ see §14.15(v).

These results are proved in Olver (1980b). Equations (13.20.17) and (13.20.18) are simpler than (6.10) and (6.11) in this reference. Olver (1980b) also supplies error bounds and corresponding approximations when x, κ, and μ are replaced by x, κ, and μ, respectively.

It should be noted that (13.20.11), (13.20.16), and (13.20.18) differ only in the common error terms. Hence without the error terms the approximation holds for -(1-δ)μκμ/δ. Similarly for (13.20.12), (13.20.17), and (13.20.19).

§13.20(v) Large μ, Other Expansions

For uniform approximations valid when μ is large, x/(0,), and κ/[0,μ/δ], see Olver (1997b, pp. 401–403). These approximations are in terms of Airy functions.

For uniform approximations of Mκ,μ(z) and Wκ,μ(z), κ and μ real, one or both large, see Dunster (2003a).