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14 Legendre and Related FunctionsReal Arguments

§14.20 Conical (or Mehler) Functions


§14.20(i) Definitions and Wronskians

Throughout §14.20 we assume that ν=-12+iτ, with μ0 and τ0. (14.2.2) takes the form

14.20.1 (1-x2)d2wdx2-2xdwdx-(τ2+14+μ21-x2)w=0.

Solutions are known as conical or Mehler functions. For -1<x<1 and τ>0, a numerically satisfactory pair of real conical functions is P-12+iτ-μ(x) and P-12+iτ-μ(-x).

Another real-valued solution Q^-12+iτ-μ(x) of (14.20.1) was introduced in Dunster (1991). This is defined by

14.20.2 Q^-12+iτ-μ(x)=(eμπiQ-12+iτ-μ(x))-12πsin(μπ)P-12+iτ-μ(x).


14.20.3 Q^-12+iτ-μ(x)=πe-τπsin(μπ)sinh(τπ)2(cosh2(τπ)-sin2(μπ))P-12+iτ-μ(x)+π(e-τπcos2(μπ)+sinh(τπ))2(cosh2(τπ)-sin2(μπ))P-12+iτ-μ(-x).

Q^-12+iτ-μ(x) exists except when μ=12,32, and τ=0; compare §14.3(i). It is an important companion solution to P-12+iτ-μ(x) when τ is large; compare §§14.20(vii), 14.20(viii), and 10.25(iii).

14.20.4 𝒲{P-12+iτ-μ(x),P-12+iτ-μ(-x)}=2|Γ(μ+12+iτ)|2(1-x2).
14.20.5 𝒲{P-12+iτ-μ(x),Q^-12+iτ-μ(x)}=π(e-τπcos2(μπ)+sinh(τπ))|Γ(μ+12+iτ)|2(cosh2(τπ)-sin2(μπ))(1-x2),

provided that Q^-12+iτ-μ(x) exists.

Lastly, for the range 1<x<, P-12+iτ-μ(x) is a real-valued solution of (14.20.1); in terms of Q-12±iτμ(x) (which are complex-valued in general):

14.20.6 P-12+iτ-μ(x)=ie-μπisinh(τπ)|Γ(μ+12+iτ)|2(Q-12+iτμ(x)-Q-12-iτμ(x)),

§14.20(ii) Graphics

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Figure 14.20.1: P-12+iτ0(x), τ=0,1,2,4,8. Magnify
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Figure 14.20.2: Q^-12+iτ0(x), τ=0,12,1,2,4. Magnify
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Figure 14.20.3: P-12+iτ-1/2(x), τ=0,1,2,4,8. Magnify
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Figure 14.20.4: Q^-12+iτ-1/2(x), τ=12,1,2,4. (This function does not exist when τ=0.) Magnify
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Figure 14.20.5: P-12+iτ-1(x), τ=0,1,2,4,8. Magnify
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Figure 14.20.6: Q^-12+iτ-1(x), τ=0,12,1,2,4. Magnify
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Figure 14.20.7: P-12+iτ-2(x),τ=0,1,2,4,8. Magnify
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Figure 14.20.8: Q^-12+iτ-2(x), τ=0,12,1,2,4. Magnify

§14.20(iii) Behavior as x1

The behavior of P-12+iτ-μ(±x) as x1- is given in §14.8(i). For μ>0 and x1-,

14.20.7 Q^-12+iτμ(x)12Γ(μ)(21-x)μ/2,
14.20.8 Q^-12+iτ-μ(x)πΓ(μ)(e-τπcos2(μπ)+sinh(τπ))2(cosh2(τπ)-sin2(μπ))|Γ(μ+12+iτ)|2(21-x)μ/2.

§14.20(iv) Integral Representation

When 0<θ<π,

14.20.9 P-12+iτ(cosθ)=2π0θcosh(τϕ)2(cosϕ-cosθ)dϕ.

§14.20(v) Trigonometric Expansion

14.20.10 P-12+iτ(cosθ)=1+4τ2+1222sin2(12θ)+(4τ2+12)(4τ2+32)2242sin4(12θ)+,

From (14.20.9) or (14.20.10) it is evident that P-12+iτ(cosθ) is positive for real θ.

§14.20(vi) Generalized Mehler–Fock Transformation

14.20.11 f(τ)=τπsinh(τπ)Γ(12-μ+iτ)Γ(12-μ-iτ)1P-12+iτμ(x)g(x)dx,


14.20.12 g(x)=0P-12+iτμ(x)f(τ)dτ.

Special cases:

14.20.13 P-12+iτ(x)=cosh(τπ)π1P-12+iτ(t)x+tdt,
14.20.14 π0τtanh(τπ)cosh(τπ)P-12+iτ(x)P-12+iτ(y)dτ=1y+x.

§14.20(vii) Asymptotic Approximations: Large τ, Fixed μ

For τ and fixed μ,

14.20.15 P-12+iτ-μ(cosθ) =1τμ(θsinθ)1/2Iμ(τθ)(1+O(1/τ)),
14.20.16 Q^-12+iτ-μ(cosθ) =1τμ(θsinθ)1/2Kμ(τθ)(1+O(1/τ)),

uniformly for θ(0,π-δ], where I and K are the modified Bessel functions (§10.25(ii)) and δ is an arbitrary constant such that 0<δ<π. For asymptotic expansions and explicit error bounds, see Olver (1997b, pp. 473–474). See also Žurina and Karmazina (1966).

§14.20(viii) Asymptotic Approximations: Large τ, 0μAτ

In this subsection and §14.20(ix), A and δ denote arbitrary constants such that A>0 and 0<δ<2.

As τ,

14.20.17 P-12+iτ-μ(x)=σ(μ,τ)(α2+η1+α2-x2)1/4Iμ(τη1/2)(1+O(1/τ)),
14.20.18 Q^-12+iτ-μ(x)=σ(μ,τ)(α2+η1+α2-x2)1/4Kμ(τη1/2)(1+O(1/τ)),

uniformly for x[-1+δ,1) and μ[0,Aτ]. Here

14.20.19 α=μ/τ,
14.20.20 σ(μ,τ)=exp(μ-τarctanα)(μ2+τ2)μ/2.

The variable η is defined implicitly by

14.20.21 (α2+η)1/2+12αlnη-αln((α2+η)1/2+α)=arccos(x(1+α2)1/2)+α2ln(1+α2+(α2-1)x2-2αx(1+α2-x2)1/2(1+α2)(1-x2)),

where the inverse trigonometric functions take their principal values. The interval -1<x<1 is mapped one-to-one to the interval 0<η<, with the points x=-1 and x=1 corresponding to η= and η=0, respectively.

§14.20(ix) Asymptotic Approximations: Large μ, 0τAμ

As μ,

14.20.22 P-12+iτ-μ(x)=βexp(μβarctanβ)Γ(μ+1)(1+β2)μ/2e-μρ(1+β2-x2β2)1/4(1+O(1μ)),

uniformly for x(-1,1) and τ[0,Aμ]. Here

14.20.23 β=τ/μ,

and the variable ρ is defined by

14.20.24 ρ=12ln((1-β2)x2+1+β2+2x(1+β2-β2x2)1/21-x2)+βarctan(βx1+β2-β2x2)-12ln(1+β2),

with the inverse tangent taking its principal value. The interval -1<x<1 is mapped one-to-one to the interval -<ρ<, with the points x=-1, x=0, and x=1 corresponding to ρ=-, ρ=0, and ρ=, respectively.

With the same conditions, the corresponding approximation for P-12+iτ-μ(-x) is obtainable by replacing e-μρ by eμρ on the right-hand side of (14.20.22). Approximations for P-12+iτμ(x) and Q^-12+iτ-μ(x) can then be achieved via (14.9.7) and (14.20.3).

For extensions to complex arguments (including the range 1<x<), asymptotic expansions, and explicit error bounds, see Dunster (1991). For the case of purely imaginary order and argument see Dunster (2013).

§14.20(x) Zeros and Integrals

For zeros of P-12+iτ(x) see Hobson (1931, §237).

For integrals with respect to τ involving P-12+iτ(x), see Prudnikov et al. (1990, pp. 218–228).