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33 Coulomb FunctionsVariables r,Ο΅

Β§33.19 Power-Series Expansions in r

33.19.1 f⁑(Ο΅,β„“;r)=rβ„“+1β’βˆ‘k=0∞αk⁒rk,

where

33.19.2 Ξ±0 =2β„“+1/(2⁒ℓ+1)!,
Ξ±1 =βˆ’Ξ±0/(β„“+1),
k⁒(k+2⁒ℓ+1)⁒αk+2⁒αkβˆ’1+ϡ⁒αkβˆ’2=0,
k=2,3,….
33.19.3 2⁒π⁒h⁑(Ο΅,β„“;r)=βˆ‘k=02⁒ℓ(2β’β„“βˆ’k)!⁒γkk!⁒(2⁒r)kβˆ’β„“βˆ’βˆ‘k=0∞δk⁒rk+β„“+1βˆ’A⁑(Ο΅,β„“)⁒(2⁒ln⁑|2⁒r/ΞΊ|+β„œβ‘Οˆβ‘(β„“+1+ΞΊ)+β„œβ‘Οˆβ‘(βˆ’β„“+ΞΊ))⁒f⁑(Ο΅,β„“;r),
r≠0.

Here ΞΊ is defined by (33.14.6), A⁑(Ο΅,β„“) is defined by (33.14.11) or (33.14.12), Ξ³0=1, Ξ³1=1, and

33.19.4 Ξ³kβˆ’Ξ³kβˆ’1+14⁒(kβˆ’1)⁒(kβˆ’2β’β„“βˆ’2)⁒ϡ⁒γkβˆ’2=0,
k=2,3,….

Also,

33.19.5 Ξ΄0 =(Ξ²2⁒ℓ+1βˆ’2⁒(ψ⁑(2⁒ℓ+2)+ψ⁑(1))⁒A⁑(Ο΅,β„“))⁒α0,
Ξ΄1 =(Ξ²2⁒ℓ+2βˆ’2⁒(ψ⁑(2⁒ℓ+3)+ψ⁑(2))⁒A⁑(Ο΅,β„“))⁒α1,
33.19.6 k⁒(k+2⁒ℓ+1)⁒δk+2⁒δkβˆ’1+ϡ⁒δkβˆ’2+2⁒(2⁒k+2⁒ℓ+1)⁒A⁑(Ο΅,β„“)⁒αk=0,
k=2,3,…,

with Ξ²0=Ξ²1=0, and

33.19.7 Ξ²kβˆ’Ξ²kβˆ’1+14⁒(kβˆ’1)⁒(kβˆ’2β’β„“βˆ’2)⁒ϡ⁒βkβˆ’2+12⁒(kβˆ’1)⁒ϡ⁒γkβˆ’2=0,
k=2,3,….

The expansions (33.19.1) and (33.19.3) converge for all finite values of r, except r=0 in the case of (33.19.3).