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14 Legendre and Related FunctionsReal Arguments

§14.15 Uniform Asymptotic Approximations

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§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 Pν-μ(±x)=(1x1±x)μ/2(j=0J-1(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 Pν-μ(±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 μ-ν 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 Pν-μ(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).

§14.15(ii) Large μ, 0ν+12(1-δ)μ

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

As μ,

14.15.4 Pν-μ(x)=1Γ(μ+1)(1-α2)-μ/2(1-α1+α)(ν/2)+(1/4)(px)1/2e-μρ(1+O(1μ)),

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

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

and

14.15.7 ρ=12ln(1+p1-p)+12αln(1-αp1+αp).

With the same conditions, the corresponding approximation for Pν-μ(-x) is obtained by replacing e-μρ 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 Pν-μ(x)=(2μπ)1/21Γ(μ+1)(1-α1+α)(ν/2)+(1/4)(1-α2)-μ/2(α2+η2α2(x2-1)+1)1/4Kν+12(μη)(1+O(1μ)),
14.15.9 Qνμ(x)=(π2)1/2(eμ)ν+(1/2)(1-α1+α)μ/2(1-α2)-(ν/2)-(1/4)(α2+η2α2(x2-1)+1)1/4Iν+12(μη)(1+O(1μ)),

uniformly with respect to x(1,) and ν+12[0,(1-δ)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+1-α2-2x(α2x2-α2+1)1/2(x2-1)(1-α2))+12αln(α2(2x2-1)+1+2αx(α2x2-α2+1)1/21-α2).

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 Pν-μ(cosθ) =1νμ(θsinθ)1/2(Jμ((ν+12)θ)+O(1ν)envJμ((ν+12)θ)),
14.15.12 Qν-μ(cosθ) =-π2νμ(θsinθ)1/2(Yμ((ν+12)θ)+O(1ν)envYμ((ν+12)θ)),

uniformly for θ(0,π-δ]. 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 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(1-δ)(ν+12)

As ν,

14.15.15 Pν-μ(x)=β(y-α21-α2-x2)1/4(Jμ((ν+12)y1/2)+O(1ν)envJμ((ν+12)y1/2)),
14.15.16 Qν-μ(x)=-πβ2(y-α21-α2-x2)1/4(Yμ((ν+12)y1/2)+O(1ν)envYμ((ν+12)y1/2)),

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

Next,

14.15.17 Pν-μ(x)=β(α2-yx2-1+α2)1/4Iμ((ν+12)|y|1/2)(1+O(1ν)),
14.15.18 Qνμ(x)=1βΓ(ν+μ+1)(α2-yx2-1+α2)1/4Kμ((ν+12)|y|1/2)(1+O(1ν)),

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

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

and the variable y is defined implicitly by

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

and

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

where the inverse trigonometric functions take their principal values (§4.23(ii)). The points x=(1-α2)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 f(x)=U(-c,x) or U¯(-c,x), with c and x nonnegative,

14.15.23 envf(x)={((U(-c,x))2+(U¯(-c,x))2)1/2,0xXc,2f(x),Xcx<,

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

As ν,

14.15.24 Pν-μ(x)=1(ν+12)1/42(ν+μ)/2Γ(12ν+12μ+34)(ζ2-α2x2-a2)1/4×(U(μ-ν-12,(2ν+1)1/2ζ)+O(ν-2/3)envU(μ-ν-12,(2ν+1)1/2ζ)),
14.15.25 Qν-μ(x)=π(ν+12)1/42(ν+μ+2)/2Γ(12ν+12μ+34)(ζ2-α2x2-a2)1/4×(U¯(μ-ν-12,(2ν+1)1/2ζ)+O(ν-2/3)envU¯(μ-ν-12,(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)|ν-μ+12|)1/2ν+12,
α =(2|ν-μ+12|ν+12)1/2,

and the variable ζ is defined implicitly by

14.15.27 12ζ(ζ2-α2)1/2-12α2arccosh(ζα)=(1-a2)1/2arctanh(1x(x2-a21-a2)1/2)-arccosh(xa),
ax<1, αζ<,

and

14.15.28 12α2arcsin(ζα)+12ζ(α2-ζ2)1/2=arcsin(xa)-(1-a2)1/2arctan(x(1-a2a2-x2)1/2),
-axa, -αζα,

when a>0, and

14.15.29 ζ2=-ln(1-x2),
-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 Pν-μ(x)=1(ν+12)1/42(ν+μ)/2Γ(12ν+12μ+34)(ζ2+α2x2+a2)1/4×U(μ-ν-12,(2ν+1)1/2ζ)(1+O(ν-1lnν)),

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).