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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.20 Uniform Asymptotic Approximations for Large μ

Contents
  1. §13.20(i) Large μ, Fixed κ
  2. §13.20(ii) Large μ, 0κ(1δ)μ
  3. §13.20(iii) Large μ, (1δ)μκμ
  4. §13.20(iv) Large μ, μκμ/δ
  5. §13.20(v) Large μ, Other Expansions

§13.20(i) Large μ, Fixed κ

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

13.20.1 Mκ,μ(z)=zμ+12(1+O(μ1)),

uniformly for bounded values of |z|; also

13.20.2 Wκ,μ(x)=π12Γ(κ+μ)(14x)12μ(1+O(μ1)),

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(1δ)μ

Let

13.20.3 X=4μ24κx+x2.

Then as μ

13.20.4 Mκ,μ(x)=2μxX(4μ2x2μ2κx+μX)μ(2(μκ)X+x2κ)κe12Xμ(1+O(1μ)),
13.20.5 Wκ,μ(x)=xX(2μ2κx+μX(μκ)x)μ(X+x2κ2)κe12Xκ(1+O(1μ)),

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

§13.20(iii) Large μ, (1δ)μκμ

Let

13.20.6 α =2|κμ|/μ,
13.20.7 X =|x24κx+4μ2|,
13.20.8 Φ(κ,μ,x) =(μ2ζ22κμ+2μ2x24κx+4μ2)14x,

with the variable ζ defined implicitly as follows:

(a) In the case μ<κ<μ

13.20.9 ζζ2+α2+α2arcsinh(ζα)=Xμ2κμln(X+x2κ2μ2κ2)2ln(μX+2μ2κxxμ2κ2).

(b) In the case μ=κ

13.20.10 ζ=±xμ22ln(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+μe)12(κ+μ)Φ(κ,μ,x)U(μκ,ζ2μ)(1+O(μ1lnμ)),
13.20.12 Mκ,μ(x)=(8μ)14(2μe)2μ(eκ+μ)12(κ+μ)Φ(κ,μ,x)×U(μκ,ζ2μ)(1+O(μ1lnμ)),

uniformly with respect to x(0,) and κ[(1δ)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 ix, iκ, and iμ, 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 ζζ2α2α2arccosh(ζα) =Xμ2κμln(X+x2κ2κ2μ2)2ln(κxμX2μ2xκ2μ2),
x2κ+2κ2μ2,
13.20.14 ζα2ζ2+α2arcsin(ζα) =Xμ+2κμarctan(x2κX)2arctan(κx2μ2μX),
2κ2κ2μ2x2κ+2κ2μ2,
13.20.15 ζζ2α2α2arccosh(ζα) =Xμ+2κμln(2κXx2κ2μ2)+2ln(μX+2μ2κxxκ2μ2),
0<x2κ2κ2μ2,

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+μe)12(κ+μ)Φ(κ,μ,x)×(U(μκ,ζ2μ)+envU(μκ,ζ2μ)O(μ23)),
13.20.17 Mκ,μ(x)=(8μ)14(2μe)2μ(eκ+μ)12(κ+μ)Φ(κ,μ,x)×(U(μκ,ζ2μ)+envU¯(μκ,ζ2μ)O(μ23)),

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

Also,

13.20.18 Wκ,μ(x)=(12μ)14(κ+μe)12(κ+μ)Φ(κ,μ,x)×(U(μκ,ζ2μ)+envU¯(μκ,ζ2μ)O(μ23)),
13.20.19 Mκ,μ(x)=(8μ)14(2μe)2μ(eκ+μ)12(κ+μ)Φ(κ,μ,x)×(U(μκ,ζ2μ)+envU(μκ,ζ2μ)O(μ23)),

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 ix, iκ, and iμ, 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 (1δ)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/i(0,), and κ/i[0,μ/δ], see Olver (1997b, pp. 401–403). These approximations are in terms of Airy functions.

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