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14 Legendre and Related FunctionsReal Arguments

§14.20 Conical (or Mehler) Functions

Contents
  1. §14.20(i) Definitions and Wronskians
  2. §14.20(ii) Graphics
  3. §14.20(iii) Behavior as x1
  4. §14.20(iv) Integral Representation
  5. §14.20(v) Trigonometric Expansion
  6. §14.20(vi) Generalized Mehler–Fock Transformation
  7. §14.20(vii) Asymptotic Approximations: Large τ, Fixed μ
  8. §14.20(viii) Asymptotic Approximations: Large τ, 0μAτ
  9. §14.20(ix) Asymptotic Approximations: Large μ, 0τAμ
  10. §14.20(x) Zeros and Integrals

§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 (1x2)d2wdx22xdwdx(τ2+14+μ21x2)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 𝖯12+iτμ(x) and 𝖯12+iτμ(x).

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

14.20.2 𝖰^12+iτμ(x)=(eμπi𝖰12+iτμ(x))12πsin(μπ)𝖯12+iτμ(x).

Equivalently,

14.20.3 𝖰^12+iτμ(x)=πeτπsin(μπ)sinh(τπ)2(cosh2(τπ)sin2(μπ))𝖯12+iτμ(x)+π(eτπcos2(μπ)+sinh(τπ))2(cosh2(τπ)sin2(μπ))𝖯12+iτμ(x).

𝖰^12+iτμ(x) exists except when μ=12,32, and τ=0; compare §14.3(i). It is an important companion solution to 𝖯12+iτμ(x) when τ is large; compare §§14.20(vii), 14.20(viii), and 10.25(iii).

14.20.4 𝒲{𝖯12+iτμ(x),𝖯12+iτμ(x)}=2|Γ(μ+12+iτ)|2(1x2).
14.20.5 𝒲{𝖯12+iτμ(x),𝖰^12+iτμ(x)}=π(eτπcos2(μπ)+sinh(τπ))|Γ(μ+12+iτ)|2(cosh2(τπ)sin2(μπ))(1x2),

provided that 𝖰^12+iτμ(x) exists.

Lastly, for the range 1<x<, P12+iτμ(x) is a real-valued solution of (14.20.1); in terms of Q12±iτμ(x) (which are complex-valued in general):

14.20.6 P12+iτμ(x)=ieμπisinh(τπ)|Γ(μ+12+iτ)|2(Q12+iτμ(x)Q12iτμ(x)),
τ0.

§14.20(ii) Graphics

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Figure 14.20.1: 𝖯12+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: 𝖯12+iτ1/2(x), τ=0,1,2,4,8. Magnify
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Figure 14.20.4: 𝖰^12+iτ1/2(x), τ=12,1,2,4. (This function does not exist when τ=0.) Magnify
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Figure 14.20.5: 𝖯12+iτ1(x), τ=0,1,2,4,8. Magnify
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Figure 14.20.6: 𝖰^12+iτ1(x), τ=0,12,1,2,4. Magnify
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Figure 14.20.7: 𝖯12+iτ2(x),τ=0,1,2,4,8. Magnify
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Figure 14.20.8: 𝖰^12+iτ2(x), τ=0,12,1,2,4. Magnify

§14.20(iii) Behavior as x1

The behavior of 𝖯12+iτμ(±x) as x1 is given in §14.8(i). For μ>0 and x1,

14.20.7 𝖰^12+iτμ(x)12Γ(μ)(21x)μ/2,
14.20.8 𝖰^12+iτμ(x)πΓ(μ)(eτπcos2(μπ)+sinh(τπ))2(cosh2(τπ)sin2(μπ))|Γ(μ+12+iτ)|2(21x)μ/2.

§14.20(iv) Integral Representation

When 0<θ<π,

14.20.9 𝖯12+iτ(cosθ)=2π0θcosh(τϕ)2(cosϕcosθ)dϕ.

§14.20(v) Trigonometric Expansion

14.20.10 𝖯12+iτ(cosθ)=1+4τ2+1222sin2(12θ)+(4τ2+12)(4τ2+32)2242sin4(12θ)+,
0θπ.

From (14.20.9) or (14.20.10) it is evident that 𝖯12+iτ(cosθ) is positive for real θ.

§14.20(vi) Generalized Mehler–Fock Transformation

14.20.11 f(τ)=τπsinh(τπ)Γ(12μ+iτ)Γ(12μiτ)1P12+iτμ(x)g(x)dx,

where

14.20.12 g(x)=0P12+iτμ(x)f(τ)dτ.

Special cases:

14.20.13 P12+iτ(x)=cosh(τπ)π1P12+iτ(t)x+tdt,
14.20.14 π0τtanh(τπ)cosh(τπ)P12+iτ(x)P12+iτ(y)dτ=1y+x.

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

For τ and fixed μ,

14.20.15 𝖯12+iτμ(cosθ) =1τμ(θsinθ)1/2Iμ(τθ)(1+O(1/τ)),
14.20.16 𝖰^12+iτμ(cosθ) =1τμ(θsinθ)1/2Kμ(τθ)(1+O(1/τ)),

uniformly for θ(0,πδ], 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 𝖯12+iτμ(x)=σ(μ,τ)(α2+η1+α2x2)1/4Iμ(τη1/2)(1+O(1/τ)),
14.20.18 𝖰^12+iτμ(x)=σ(μ,τ)(α2+η1+α2x2)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(μτarctanα)(μ2+τ2)μ/2.

The variable η is defined implicitly by

14.20.21 (α2+η)1/2+12αlnηαln((α2+η)1/2+α)=arccos(x(1+α2)1/2)+α2ln(1+α2+(α21)x22αx(1+α2x2)1/2(1+α2)(1x2)),

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 𝖯12+iτμ(x)=βexp(μβarctanβ)Γ(μ+1)(1+β2)μ/2eμρ(1+β2x2β2)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((1β2)x2+1+β2+2x(1+β2β2x2)1/21x2)+βarctan(βx1+β2β2x2)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 𝖯12+iτμ(x) is obtainable by replacing eμρ by eμρ on the right-hand side of (14.20.22). Approximations for 𝖯12+iτμ(x) and 𝖰^12+iτμ(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 𝖯12+iτ(x) see Hobson (1931, §237).

For integrals with respect to τ involving 𝖯12+iτ(x), see Prudnikov et al. (1990, pp. 218–228).