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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.21 Uniform Asymptotic Approximations for Large κ

Contents

§13.21(i) Large κ, Fixed μ

For the notation see §§10.2(ii), 10.25(ii), and 2.8(iv).

When κ through positive real values with μ (0) fixed

13.21.1 Mκ,μ(x)=xΓ(2μ+1)κ-μ(J2μ(2xκ)+envJ2μ(2xκ)O(κ-12)),
13.21.2 Wκ,μ(x)=xΓ(κ+12)(sin(κπ-μπ)J2μ(2xκ)-cos(κπ-μπ)Y2μ(2xκ)+envY2μ(2xκ)O(κ-12),)
13.21.3 W-κ,μ(x-π)=πxΓ(κ+12)μπ(H2μ(1)(2xκ)+envY2μ(2xκ)O(κ-12)),
13.21.4 W-κ,μ(xπ)=πxΓ(κ+12)-μπ(H2μ(2)(2xκ)+envY2μ(2xκ)O(κ-12)),

uniformly with respect to x(0,A] in each case, where A is an arbitrary positive constant.

Other types of approximations when κ through positive real values with μ (0) fixed are as follows. Define

13.21.5 2ζ=x+x2+ln(x+1+x).

Then

13.21.6 M-κ,μ(4κx)=2Γ(2μ+1)κμ-12(xζ1+x)14I2μ(4κζ12)(1+O(κ-1)),
13.21.7 W-κ,μ(4κx)=8/πκκκ-12(xζ1+x)14K2μ(4κζ12)(1+O(κ-1)),

uniformly with respect to x(0,).

For (13.21.6), (13.21.7), and extensions to asymptotic expansions and error bounds, see Olver (1997b, Chapter 12, Exs. 12.4.5, 12.4.6). For extensions to complex values of x see López (1999).

§13.21(ii) Large κ, 0μ(1-δ)κ

Let

13.21.8 c(κ,μ)=μπ12π(κ-μκ+μ)12μ(2κ2-μ2)12κ,
13.21.9 X=|x2-4κx+4μ2|,
13.21.10 Ψ(κ,μ,x)=(4μ2-κζx2-4κx+4μ2)14x,

with the variable ζ defined implicitly by

13.21.11 4μ2-κζ-μln(2μ+4μ2-κζ2μ-4μ2-κζ)=12X+μln(xκ2-μ22μ2-κx+μX)+κln(2κ2-μ22κ-x-X),
0<x2κ-2κ2-μ2,

and

13.21.12 κζ-4μ2-2μarctan(κζ-4μ22μ)=12(X-πμ)-μarctan(xκ-2μ2μX)+κarcsin(X2κ2-μ2),
2κ-2κ2-μ2x<2κ+2κ2-μ2.

Then as κ

13.21.13 Mκ,μ(x) =Γ(2μ+1)(2κ2-μ2)12μ(κ-μκ+μ)12κΨ(κ,μ,x)(J2μ(ζκ)+envJ2μ(ζκ)O(κ-1)),
13.21.14 Wκ,μ(x) =-μππΓ(κ+μ+12)Γ(κ-μ+12)×c(κ,μ)Ψ(κ,μ,x)×(sin(κπ-μπ)J2μ(ζκ)-cos(κπ-μπ)Y2μ(ζκ)+envY2μ(ζκ)O(κ-1),)
13.21.15 W-κ,μ(x-π)=c(κ,μ)Ψ(κ,μ,x)(H2μ(1)(ζκ)+envY2μ(ζκ)O(κ-1)),
13.21.16 W-κ,μ(xπ)=-2μπc(κ,μ)Ψ(κ,μ,x)(H2μ(2)(ζκ)+envY2μ(ζκ)O(κ-1)),

uniformly with respect to μ[0,(1-δ)κ] and x(0,(1-δ)(2κ+2κ2-μ2)], where δ again denotes an arbitrary small positive constant. For the functions J2μ, Y2μ, H2μ(1), and H2μ(2) see §10.2(ii), and for the env functions associated with J2μ and Y2μ see §2.8(iv).

These approximations are proved in Dunster (1989). This reference also includes error bounds and extensions to asymptotic expansions and complex values of x.

§13.21(iii) Large κ, 0μ(1-δ)κ (Continued)

Let

13.21.17 c^(κ,μ)=2πκ16(κ-μκ+μ)12μ(2κ2-μ2)12κ,
13.21.18 X=|x2-4κx+4μ2|,
13.21.19 Ψ^(κ,μ,x)=(ζ^x2-4κx+4μ2)142x,

and define the variable ζ^ implicitly by

13.21.20 ζ^=-(32κ(-12X+2μarctan(xκ-xκ2-μ2-2μ2μX)+κarccos(x-2κ2κ2-μ2)))2/3,
2κ-2κ2-μ2<x2κ+2κ2-μ2,

and

13.21.21 ζ^=(32κ(12X+μln(xκ2-μ2κx-2μ2-μX)+κln(2κ2-μ2x-2κ+X)))2/3,
x2κ+2κ2-μ2.

Then as κ

13.21.22 Mκ,μ(x)=12πΓ(2μ+1)Γ(κ-μ+12)c^(κ,μ)Ψ^(κ,μ,x)×(sin(κπ-μπ)Ai(κ23ζ^)+cos(κπ-μπ)Bi(κ23ζ^)+envBi(κ23ζ^)O(κ-1)),
13.21.23 Wκ,μ(x)=2πκ16(κ+μκ-μ)12μ(κ2-μ22)12κΨ^(κ,μ,x)(Ai(κ23ζ^)+envAi(κ23ζ^)O(κ-1)),
13.21.24 W-κ,μ(x-π)=(κ-16)πc^(κ,μ)Ψ^(κ,μ,x)(Ai(κ23ζ^-23π)+envBi(κ23ζ^)O(κ-1)),
13.21.25 W-κ,μ(xπ)=-(κ-16)πc^(κ,μ)Ψ^(κ,μ,x)(Ai(κ23ζ^23π)+envBi(κ23ζ^)O(κ-1)),

uniformly with respect to μ[0,(1-δ)κ] and x[(1+δ)(2κ-2κ2-μ2),). For the functions Ai and Bi see §9.2(i), and for the env functions associated with Ai and Bi see §2.8(iii).

These approximations are proved in Dunster (1989). This reference also includes error bounds and extensions to asymptotic expansions and complex values of x.

§13.21(iv) Large κ, Other Expansions

For a uniform asymptotic expansion in terms of Airy functions for Wκ,μ(4κx) when κ is large and positive, μ is real with |μ| bounded, and x[δ,) see Olver (1997b, Chapter 11, Ex. 7.3). This expansion is simpler in form than the expansions of Dunster (1989) that correspond to the approximations given in §13.21(iii), but the conditions on μ are more restrictive.

For asymptotic expansions having double asymptotic properties see Skovgaard (1966).

See also §13.20(v).