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33 Coulomb FunctionsVariables ρ,η

Β§33.5 Limiting Forms for Small ρ, Small |Ξ·|, or Large β„“

Contents
  1. §33.5(i) Small ρ
  2. Β§33.5(ii) Ξ·=0
  3. Β§33.5(iii) Small |Ξ·|
  4. Β§33.5(iv) Large β„“

§33.5(i) Small ρ

As ρ→0 with Ξ· fixed,

33.5.1 Fℓ⁑(Ξ·,ρ) ∼Cℓ⁑(Ξ·)⁒ρℓ+1,
Fℓ′⁑(Ξ·,ρ) ∼(β„“+1)⁒Cℓ⁑(Ξ·)⁒ρℓ.
33.5.2 Gℓ⁑(Ξ·,ρ) βˆΌΟβˆ’β„“(2⁒ℓ+1)⁒Cℓ⁑(Ξ·),
β„“=0,1,2,…,
Gℓ′⁑(Ξ·,ρ) βˆΌβˆ’β„“β’Οβˆ’β„“βˆ’1(2⁒ℓ+1)⁒Cℓ⁑(Ξ·),
β„“=1,2,3,….

Β§33.5(ii) Ξ·=0

33.5.3 Fℓ⁑(0,ρ) =ρ⁒jℓ⁑(ρ),
Gℓ⁑(0,ρ) =βˆ’Οβ’yℓ⁑(ρ).

Equivalently,

33.5.4 Fℓ⁑(0,ρ) =(π⁒ρ/2)1/2⁒Jβ„“+12⁑(ρ),
Gℓ⁑(0,ρ) =βˆ’(π⁒ρ/2)1/2⁒Yβ„“+12⁑(ρ).

For the functions j, y, J, Y see §§10.47(ii), 10.2(ii).

33.5.5 F0⁑(0,ρ) =sin⁑ρ,
G0⁑(0,ρ) =cos⁑ρ,
H0±⁑(0,ρ) =e±i⁒ρ.
33.5.6 Cℓ⁑(0)=2ℓ⁒ℓ!(2⁒ℓ+1)!=1(2⁒ℓ+1)!!.

Β§33.5(iii) Small |Ξ·|

33.5.7 Οƒ0⁑(Ξ·)βˆΌβˆ’Ξ³β’Ξ·,
Ξ·β†’0,

where Ξ³ is Euler’s constant (Β§5.2(ii)).

Β§33.5(iv) Large β„“

As β„“β†’βˆž with Ξ· and ρ (β‰ 0) fixed,

33.5.8 Fℓ⁑(Ξ·,ρ) ∼Cℓ⁑(Ξ·)⁒ρℓ+1,
Gℓ⁑(Ξ·,ρ) βˆΌΟβˆ’β„“(2⁒ℓ+1)⁒Cℓ⁑(Ξ·),
33.5.9 Cℓ⁑(Ξ·)∼eβˆ’Ο€β’Ξ·/2(2⁒ℓ+1)!!∼eβˆ’Ο€β’Ξ·/2⁒eβ„“2⁒(2⁒ℓ)β„“+1.