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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.21 Uniform Asymptotic Approximations for Large κ

Contents
  1. §13.21(i) Large κ, Fixed μ
  2. §13.21(ii) Large κ, 0μ(1δ)κ
  3. §13.21(iii) Large κ, 0μ(1δ)κ (Continued)
  4. §13.21(iv) Large κ, Other Expansions

§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)κμ(J2μ(2xκ)+envJ2μ(2xκ)O(κ12)),
13.21.2 Wκ,μ(x)=xΓ(κ+12)(sin(κπμπ)J2μ(2xκ)cos(κπμπ)Y2μ(2xκ)+envY2μ(2xκ)O(κ12)),
13.21.3 Wκ,μ(xeπi)=πxΓ(κ+12)eμπi(H2μ(1)(2xκ)+envY2μ(2xκ)O(κ12)),
13.21.4 Wκ,μ(xeπi)=πxΓ(κ+12)eμπi(H2μ(2)(2xκ)+envY2μ(2xκ)O(κ12)),

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 Mκ,μ(4κx)=2Γ(2μ+1)κμ12(xζ1+x)14I2μ(4κζ12)(1+O(κ1)),
13.21.7 Wκ,μ(4κx)=8/πeκκκ12(xζ1+x)14K2μ(4κζ12)(1+O(κ1)),

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(1δ)κ

Let

13.21.8 c(κ,μ)=eμπi12π(κμκ+μ)12μ(e2κ2μ2)12κ,
13.21.9 X=|x24κx+4μ2|,
13.21.10 Ψ(κ,μ,x)=(4μ2κζx24κx+4μ2)14x,

with the variable ζ defined implicitly by

13.21.11 4μ2κζμln(2μ+4μ2κζ2μ4μ2κζ)=12X+μln(xκ2μ22μ2κx+μX)+κln(2κ2μ22κxX),
0<x2κ2κ2μ2,

and

13.21.12 κζ4μ22μarctan(κζ4μ22μ)=12(Xπμ)μarctan(xκ2μ2μX)+κarcsin(X2κ2μ2),
2κ2κ2μ2x<2κ+2κ2μ2.

Then as κ

13.21.13 Mκ,μ(x) =Γ(2μ+1)(e2κ2μ2)12μ(κμκ+μ)12κΨ(κ,μ,x)(J2μ(ζκ)+envJ2μ(ζκ)O(κ1)),
13.21.14 Wκ,μ(x) =eμπiπΓ(κ+μ+12)×Γ(κμ+12)c(κ,μ)Ψ(κ,μ,x)×(sin(κπμπ)J2μ(ζκ)cos(κπμπ)Y2μ(ζκ)+envY2μ(ζκ)O(κ1)),
13.21.15 Wκ,μ(xeπi)=c(κ,μ)Ψ(κ,μ,x)(H2μ(1)(ζκ)+envY2μ(ζκ)O(κ1)),
13.21.16 Wκ,μ(xeπi)=e2μπic(κ,μ)Ψ(κ,μ,x)×(H2μ(2)(ζκ)+envY2μ(ζκ)O(κ1)),

uniformly with respect to μ[0,(1δ)κ] and x(0,(1δ)(2κ+2κ2μ2)], 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(1δ)κ (Continued)

Let

13.21.17 c^(κ,μ)=2πκ16(κμκ+μ)12μ(e2κ2μ2)12κ,
13.21.18 X=|x24κx+4μ2|,
13.21.19 Ψ^(κ,μ,x)=(ζ^x24κx+4μ2)142x,

and define the variable ζ^ implicitly by

13.21.20 ζ^=(32κ(12X+2μarctan(xκxκ2μ22μ2μX)+κarccos(x2κ2κ2μ2)))2/3,
2κ2κ2μ2<x2κ+2κ2μ2,

and

13.21.21 ζ^=(32κ(12X+μln(xκ2μ2κx2μ2μX)+κln(2κ2μ2x2κ+X)))2/3,
x2κ+2κ2μ2.

Then as κ

13.21.22 Mκ,μ(x)=12πΓ(2μ+1)Γ(κμ+12)c^(κ,μ)Ψ^(κ,μ,x)×(sin(κπμπ)Ai(κ23ζ^)+cos(κπμπ)Bi(κ23ζ^)+envBi(κ23ζ^)O(κ1)),
13.21.23 Wκ,μ(x)=2πκ16(κ+μκμ)12μ(κ2μ2e2)12κΨ^(κ,μ,x)×(Ai(κ23ζ^)+envAi(κ23ζ^)O(κ1)),
13.21.24 Wκ,μ(xeπi)=e(κ16)πic^(κ,μ)Ψ^(κ,μ,x)×(Ai(κ23ζ^e23πi)+envBi(κ23ζ^)O(κ1)),
13.21.25 Wκ,μ(xeπi)=e(κ16)πic^(κ,μ)Ψ^(κ,μ,x)×(Ai(κ23ζ^e23πi)+envBi(κ23ζ^)O(κ1)),

uniformly with respect to μ[0,(1δ)κ] and x[(1+δ)(2κ2κ2μ2),). 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).