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33 Coulomb FunctionsVariables ρ,η

Β§33.9 Expansions in Series of Bessel Functions

Contents
  1. Β§33.9(i) Spherical Bessel Functions
  2. Β§33.9(ii) Bessel Functions and Modified Bessel Functions

Β§33.9(i) Spherical Bessel Functions

33.9.1 Fℓ⁑(Ξ·,ρ)=Οβ’βˆ‘k=0∞ak⁒𝗃ℓ+k⁑(ρ),

where the function 𝗃 is as in Β§10.47(ii), aβˆ’1=0, a0=(2⁒ℓ+1)!!⁒Cℓ⁑(Ξ·), and

33.9.2 k⁒(k+2⁒ℓ+1)2⁒k+2⁒ℓ+1⁒akβˆ’2⁒η⁒akβˆ’1+(kβˆ’2)⁒(k+2β’β„“βˆ’1)2⁒k+2β’β„“βˆ’3⁒akβˆ’2=0,
k=1,2,….

The series (33.9.1) converges for all finite values of η and ρ.

Β§33.9(ii) Bessel Functions and Modified Bessel Functions

In this subsection the functions J, I, and K are as in §§10.2(ii) and 10.25(ii).

With t=2⁒|η|⁒ρ,

33.9.3 Fℓ⁑(Ξ·,ρ)=Cℓ⁑(Ξ·)⁒(2⁒ℓ+1)!(2⁒η)2⁒ℓ+1β’Οβˆ’β„“β’βˆ‘k=2⁒ℓ+1∞bk⁒tk/2⁒Ik⁑(2⁒t),
Ξ·>0,
33.9.4 Fℓ⁑(Ξ·,ρ)=Cℓ⁑(Ξ·)⁒(2⁒ℓ+1)!(2⁒|Ξ·|)2⁒ℓ+1β’Οβˆ’β„“β’βˆ‘k=2⁒ℓ+1∞bk⁒tk/2⁒Jk⁑(2⁒t),
Ξ·<0.

Here b2⁒ℓ=b2⁒ℓ+2=0, b2⁒ℓ+1=1, and

33.9.5 4⁒η2⁒(kβˆ’2⁒ℓ)⁒bk+1+k⁒bkβˆ’1+bkβˆ’2=0,
k=2⁒ℓ+2,2⁒ℓ+3,….

The series (33.9.3) and (33.9.4) converge for all finite positive values of |η| and ρ.

Next, as Ξ·β†’+∞ with ρ (>0) fixed,

33.9.6 Gℓ⁑(Ξ·,ρ)βˆΌΟβˆ’β„“(β„“+12)⁒λℓ⁑(Ξ·)⁒Cℓ⁑(Ξ·)β’βˆ‘k=2⁒ℓ+1∞(βˆ’1)k⁒bk⁒tk/2⁒Kk⁑(2⁒t),

where

33.9.7 λℓ⁑(Ξ·)βˆΌβˆ‘k=2⁒ℓ+1∞(βˆ’1)k⁒(kβˆ’1)!⁒bk.

For other asymptotic expansions of Gℓ⁑(Ξ·,ρ) see FrΓΆberg (1955, Β§8) and Humblet (1985).