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14 Legendre and Related FunctionsReal Arguments

§14.15 Uniform Asymptotic Approximations

Contents
  1. §14.15(i) Large μ, Fixed ν
  2. §14.15(ii) Large μ, 0ν+12(1δ)μ
  3. §14.15(iii) Large ν, Fixed μ
  4. §14.15(iv) Large ν, 0μ(1δ)(ν+12)
  5. §14.15(v) Large ν, (ν+12)δμ(ν+12)/δ

§14.15(i) Large μ, Fixed ν

For the interval 1<x<1 with fixed ν, real μ, and arbitrary fixed values of the nonnegative integer J,

14.15.1 𝖯νμ(±x)=(1x1±x)μ/2(j=0J1(ν+1)j(ν)jj!Γ(j+1+μ)(1x2)j+O(1Γ(J+1+μ)))

as μ, uniformly with respect to x. In other words, the convergent hypergeometric series expansions of 𝖯νμ(±x) are also generalized (and uniform) asymptotic expansions as μ, with scale 1/Γ(j+1+μ), j=0,1,2,; compare §2.1(v).

Provided that μν the corresponding expansions for 𝖯νμ(x) and 𝖰νμ(x) can be obtained from the connection formulas (14.9.7), (14.9.9), and (14.9.10).

For the interval 1<x< the following asymptotic approximations hold when μ, with ν (12) fixed, uniformly with respect to x:

14.15.2 Pνμ(x)=1Γ(μ+1)(2μuπ)1/2Kν+12(μu)(1+O(1μ)),
14.15.3 𝑸νμ(x)=1μν+(1/2)(πu2)1/2Iν+12(μu)(1+O(1μ)),

where u is given by (14.12.10). Here I and K are the modified Bessel functions (§10.25(ii)).

For asymptotic expansions and explicit error bounds, see Dunster (2003b) and Gil et al. (2000). See also Temme (2015, Chapter 29).

§14.15(ii) Large μ, 0ν+12(1δ)μ

In this and subsequent subsections δ denotes an arbitrary constant such that 0<δ<1.

As μ,

14.15.4 𝖯νμ(x)=1Γ(μ+1)(1α2)μ/2(1α1+α)(ν/2)+(1/4)(px)1/2eμρ(1+O(1μ)),

uniformly with respect to x(1,1) and ν+12[0,(1δ)μ], where

14.15.5 α=ν+12μ(<1),
14.15.6 p=x(α2x2+1α2)1/2,

and

14.15.7 ρ=12ln(1+p1p)+12αln(1αp1+αp).

With the same conditions, the corresponding approximation for 𝖯νμ(x) is obtained by replacing eμρ by eμρ on the right-hand side of (14.15.4). Approximations for 𝖯νμ(x) and 𝖰νμ(x) can then be achieved via (14.9.7), (14.9.9), and (14.9.10).

Next,

14.15.8 Pνμ(x)=(2μπ)1/21Γ(μ+1)(1α1+α)(ν/2)+(1/4)(1α2)μ/2×(α2+η2α2(x21)+1)1/4Kν+12(μη)(1+O(1μ)),
14.15.9 𝑸νμ(x)=(π2)1/2(eμ)ν+(1/2)(1α1+α)μ/2(1α2)(ν/2)(1/4)×(α2+η2α2(x21)+1)1/4Iν+12(μη)(1+O(1μ)),

uniformly with respect to x(1,) and ν+12[0,(1δ)μ]. Here α is again given by (14.15.5), and η is defined implicitly by

14.15.10 αln((α2+η2)1/2+α)αlnη(α2+η2)1/2=12ln((1+α2)x2+1α22x(α2x2α2+1)1/2(x21)(1α2))+12αln(α2(2x21)+1+2αx(α2x2α2+1)1/21α2).

The interval 1<x< is mapped one-to-one to the interval 0<η<, with the points x=1 and x= corresponding to η= and η=0, respectively. For asymptotic expansions and explicit error bounds, see Dunster (2003b).

§14.15(iii) Large ν, Fixed μ

For ν and fixed μ (0),

14.15.11 𝖯νμ(cosθ) =1νμ(θsinθ)1/2(Jμ((ν+12)θ)+O(1ν)envJμ((ν+12)θ)),
14.15.12 𝖰νμ(cosθ) =π2νμ(θsinθ)1/2(Yμ((ν+12)θ)+O(1ν)envYμ((ν+12)θ)),

uniformly for θ(0,πδ]. For the Bessel functions J and Y see §10.2(ii), and for the env functions associated with J and Y see §2.8(iv).

For asymptotic expansions and explicit error bounds, see Olver (1997b, Chapter 12, §§12, 13) and Jones (2001). For convergent series expansions see Dunster (2004). See also Temme (2015, Chapter 29).

See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials Pn(cosθ) as n with θ fixed.

§14.15(iv) Large ν, 0μ(1δ)(ν+12)

As ν,

14.15.15 𝖯νμ(x)=β(yα21α2x2)1/4(Jμ((ν+12)y1/2)+O(1ν)envJμ((ν+12)y1/2)),
14.15.16 𝖰νμ(x)=πβ2(yα21α2x2)1/4(Yμ((ν+12)y1/2)+O(1ν)envYμ((ν+12)y1/2)),

uniformly with respect to x[0,1) and μ[0,(1δ)(ν+12)]. For α, β, and y see below.

Next,

14.15.17 Pνμ(x)=β(α2yx21+α2)1/4Iμ((ν+12)|y|1/2)(1+O(1ν)),
14.15.18 𝑸νμ(x)=1βΓ(ν+μ+1)(α2yx21+α2)1/4Kμ((ν+12)|y|1/2)(1+O(1ν)),

uniformly with respect to x(1,) and μ[0,(1δ)(ν+12)]. In (14.15.15)–(14.15.18)

14.15.19 α=μν+12(<1),
14.15.20 β=eμ(νμ+12ν+μ+12)(ν/2)+(1/4)((ν+12)2μ2)μ/2,

and the variable y is defined implicitly by

14.15.21 (yα2)1/2αarctan((yα2)1/2α)=arccos(x(1α2)1/2)α2arccos((1+α2)x21+α2(1α2)(1x2)),
x(1α2)1/2, yα2,

and

14.15.22 (α2y)1/2+12αln|y|αln((α2y)1/2+α)=ln(x+(x21+α2)1/2(1α2)1/2)+α2ln((1α2)|1x2|(1+α2)x21+α2+2αx(x21+α2)1/2),
x(1α2)1/2, yα2,

where the inverse trigonometric functions take their principal values (§4.23(ii)). The points x=(1α2)1/2, x=1, and x= are mapped to y=α2, y=0, and y=, respectively. The interval 0x< is mapped one-to-one to the interval <yy0, where y=y0 is the (positive) solution of (14.15.21) when x=0.

For asymptotic expansions and explicit error bounds, see Boyd and Dunster (1986).

§14.15(v) Large ν, (ν+12)δμ(ν+12)/δ

Here we introduce the envelopes of the parabolic cylinder functions U(c,x), U¯(c,x), which are defined in §12.2. For U(c,x) or U¯(c,x), with c and x nonnegative,

14.15.23 envU(c,x) ={(U2(c,x)+U¯2(c,x))1/2,0xXc,2U(c,x),Xcx<,
envU¯(c,x) ={(U2(c,x)+U¯2(c,x))1/2,0xXc,2U¯(c,x),Xcx<,

where x=Xc denotes the largest positive root of the equation U(c,x)=U¯(c,x).

As ν,

14.15.24 𝖯νμ(x)=1(ν+12)1/42(ν+μ)/2Γ(12ν+12μ+34)(ζ2α2x2a2)1/4×(U(μν12,(2ν+1)1/2ζ)+O(ν2/3)envU(μν12,(2ν+1)1/2ζ)),
14.15.25 𝖰νμ(x)=π(ν+12)1/42(ν+μ+2)/2Γ(12ν+12μ+34)(ζ2α2x2a2)1/4×(U¯(μν12,(2ν+1)1/2ζ)+O(ν2/3)envU¯(μν12,(2ν+1)1/2ζ)),

uniformly with respect to x[0,1) and μ[δ(ν+12),ν+12]. Here

14.15.26 a =((ν+μ+12)|νμ+12|)1/2ν+12,
α =(2|νμ+12|ν+12)1/2,

and the variable ζ is defined implicitly by

14.15.27 12ζ(ζ2α2)1/212α2arccosh(ζα)=(1a2)1/2arctanh(1x(x2a21a2)1/2)arccosh(xa),
ax<1, αζ<,

and

14.15.28 12α2arcsin(ζα)+12ζ(α2ζ2)1/2=arcsin(xa)(1a2)1/2arctan(x(1a2a2x2)1/2),
axa, αζα,

when a>0, and

14.15.29 ζ2=ln(1x2),
1<x<1,

when a=0. The inverse hyperbolic and trigonometric functions take their principal values (§§4.23(ii), 4.37(ii)).

When a>0 the interval ax<1 is mapped one-to-one to the interval αζ<, with the points x=a, x=a, and x=1 corresponding to ζ=α, ζ=α, and ζ=, respectively. When a=0 the interval 1<x<1 is mapped one-to-one to the interval <ζ<, with the points x=1, 0, and 1 corresponding to ζ=, 0, and , respectively.

Next, as ν,

14.15.30 𝖯νμ(x)=1(ν+12)1/42(ν+μ)/2Γ(12ν+12μ+34)(ζ2+α2x2+a2)1/4×U(μν12,(2ν+1)1/2ζ)(1+O(ν1lnν)),

uniformly with respect to x(1,1) and μ[ν+12,(1/δ)(ν+12)]. Here ζ is defined implicitly by

14.15.31 12ζ(ζ2+α2)1/2+12α2arcsinh(ζα)=(1+a2)1/2arctanh(x(1+a2x2+a2)1/2)arcsinh(xa),
1<x<1, <ζ<,

when a>0, which maps the interval 1<x<1 one-to-one to the interval <ζ<: the points x=1 and x=1 correspond to ζ= and ζ=, respectively. When a=0 (14.15.29) again applies. (The inverse hyperbolic functions again take their principal values.)

Since (14.15.30) holds for negative x, corresponding approximations for 𝖰νμ(x), uniformly valid in the interval 1<x<1, can be obtained from (14.9.9) and (14.9.10).

For error bounds and other extensions see Olver (1975b).