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21: 33.9 Expansions in Series of Bessel Functions
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§33.9(i) Spherical Bessel Functions
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§33.9(ii) Bessel Functions and Modified Bessel Functions
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33.9.3 F β„“ ⁑ ( Ξ· , ρ ) = C β„“ ⁑ ( Ξ· ) ⁒ ( 2 ⁒ β„“ + 1 ) ! ( 2 ⁒ Ξ· ) 2 ⁒ β„“ + 1 ⁒ ρ β„“ ⁒ k = 2 ⁒ β„“ + 1 b k ⁒ t k / 2 ⁒ I k ⁑ ( 2 ⁒ t ) , Ξ· > 0 ,
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33.9.4 F β„“ ⁑ ( Ξ· , ρ ) = C β„“ ⁑ ( Ξ· ) ⁒ ( 2 ⁒ β„“ + 1 ) ! ( 2 ⁒ | Ξ· | ) 2 ⁒ β„“ + 1 ⁒ ρ β„“ ⁒ k = 2 ⁒ β„“ + 1 b k ⁒ t k / 2 ⁒ J k ⁑ ( 2 ⁒ t ) , Ξ· < 0 .
22: 33.22 Particle Scattering and Atomic and Molecular Spectra
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𝗄 Scaling
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Z Scaling
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i ⁒ 𝗄 Scaling
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§33.22(iii) Conversions Between Variables
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23: 29.2 Differential Equations
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29.2.9 d 2 w d η 2 + ( g ν ⁒ ( ν + 1 ) ⁒ ⁑ ( η ) ) ⁒ w = 0 ,
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24: 33.23 Methods of Computation
§33.23 Methods of Computation
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§33.23(vii) WKBJ Approximations
25: 33.8 Continued Fractions
§33.8 Continued Fractions
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33.8.1 F β„“ F β„“ = S β„“ + 1 R β„“ + 1 2 T β„“ + 1 R β„“ + 2 2 T β„“ + 2 β‹― .
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33.8.2 H β„“ ± H β„“ ± = c ± i ρ ⁒ a ⁒ b 2 ⁒ ( ρ Ξ· ± i ) + ( a + 1 ) ⁒ ( b + 1 ) 2 ⁒ ( ρ Ξ· ± 2 ⁒ i ) + β‹― ,
26: 33.6 Power-Series Expansions in ρ
§33.6 Power-Series Expansions in ρ
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33.6.1 F β„“ ⁑ ( Ξ· , ρ ) = C β„“ ⁑ ( Ξ· ) ⁒ k = β„“ + 1 A k β„“ ⁒ ( Ξ· ) ⁒ ρ k ,
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33.6.2 F β„“ ⁑ ( Ξ· , ρ ) = C β„“ ⁑ ( Ξ· ) ⁒ k = β„“ + 1 k ⁒ A k β„“ ⁒ ( Ξ· ) ⁒ ρ k 1 ,
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33.6.4 A k β„“ ⁒ ( Ξ· ) = ( i ) k β„“ 1 ( k β„“ 1 ) ! ⁒ F 1 2 ⁑ ( β„“ + 1 k , β„“ + 1 i ⁒ Ξ· ; 2 ⁒ β„“ + 2 ; 2 ) .
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33.6.5 H β„“ ± ⁑ ( Ξ· , ρ ) = e ± i ⁒ ΞΈ β„“ ⁑ ( Ξ· , ρ ) ( 2 ⁒ β„“ + 1 ) ! ⁒ Ξ“ ⁑ ( β„“ ± i ⁒ Ξ· ) ⁒ ( k = 0 ( a ) k ( 2 ⁒ β„“ + 2 ) k ⁒ k ! ⁒ ( βˆ“ 2 ⁒ i ⁒ ρ ) a + k ⁒ ( ln ⁑ ( βˆ“ 2 ⁒ i ⁒ ρ ) + ψ ⁑ ( a + k ) ψ ⁑ ( 1 + k ) ψ ⁑ ( 2 ⁒ β„“ + 2 + k ) ) k = 1 2 ⁒ β„“ + 1 ( 2 ⁒ β„“ + 1 ) ! ⁒ ( k 1 ) ! ( 2 ⁒ β„“ + 1 k ) ! ⁒ ( 1 a ) k ⁒ ( βˆ“ 2 ⁒ i ⁒ ρ ) a k ) ,
27: 33.7 Integral Representations
§33.7 Integral Representations
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33.7.1 F β„“ ⁑ ( Ξ· , ρ ) = ρ β„“ + 1 ⁒ 2 β„“ ⁒ e i ⁒ ρ ( Ο€ ⁒ Ξ· / 2 ) | Ξ“ ⁑ ( β„“ + 1 + i ⁒ Ξ· ) | ⁒ 0 1 e 2 ⁒ i ⁒ ρ ⁒ t ⁒ t β„“ + i ⁒ Ξ· ⁒ ( 1 t ) β„“ i ⁒ Ξ· ⁒ d t ,
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33.7.2 H β„“ ⁑ ( Ξ· , ρ ) = e i ⁒ ρ ⁒ ρ β„“ ( 2 ⁒ β„“ + 1 ) ! ⁒ C β„“ ⁑ ( Ξ· ) ⁒ 0 e t ⁒ t β„“ i ⁒ Ξ· ⁒ ( t + 2 ⁒ i ⁒ ρ ) β„“ + i ⁒ Ξ· ⁒ d t ,
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33.7.3 H β„“ ⁑ ( Ξ· , ρ ) = i ⁒ e Ο€ ⁒ Ξ· ⁒ ρ β„“ + 1 ( 2 ⁒ β„“ + 1 ) ! ⁒ C β„“ ⁑ ( Ξ· ) ⁒ 0 ( exp ⁑ ( i ⁒ ( ρ ⁒ tanh ⁑ t 2 ⁒ Ξ· ⁒ t ) ) ( cosh ⁑ t ) 2 ⁒ β„“ + 2 + i ⁒ ( 1 + t 2 ) β„“ ⁒ exp ⁑ ( ρ ⁒ t + 2 ⁒ Ξ· ⁒ arctan ⁑ t ) ) ⁒ d t ,
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33.7.4 H β„“ + ⁑ ( Ξ· , ρ ) = i ⁒ e Ο€ ⁒ Ξ· ⁒ ρ β„“ + 1 ( 2 ⁒ β„“ + 1 ) ! ⁒ C β„“ ⁑ ( Ξ· ) ⁒ 1 i ⁒ e i ⁒ ρ ⁒ t ⁒ ( 1 t ) β„“ i ⁒ Ξ· ⁒ ( 1 + t ) β„“ + i ⁒ Ξ· ⁒ d t .
28: Bibliography M
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  • I. G. Macdonald (1972) Affine root systems and Dedekind’s Ξ· -function. Invent. Math. 15 (2), pp. 91–143.
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  • N. Michel (2007) Precise Coulomb wave functions for a wide range of complex β„“ , Ξ· and z . Computer Physics Communications 176 (3), pp. 232–249.
  • 29: 33.16 Connection Formulas
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    §33.16(i) F β„“ and G β„“ in Terms of f and h
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    33.16.1 F β„“ ⁑ ( Ξ· , ρ ) = ( 2 ⁒ β„“ + 1 ) ! ⁒ C β„“ ⁑ ( Ξ· ) ( 2 ⁒ Ξ· ) β„“ + 1 ⁒ f ⁑ ( 1 / Ξ· 2 , β„“ ; Ξ· ⁒ ρ ) ,
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    33.16.2 G β„“ ⁑ ( Ξ· , ρ ) = Ο€ ⁒ ( 2 ⁒ Ξ· ) β„“ ( 2 ⁒ β„“ + 1 ) ! ⁒ C β„“ ⁑ ( Ξ· ) ⁒ h ⁑ ( 1 / Ξ· 2 , β„“ ; Ξ· ⁒ ρ ) ,
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    §33.16(ii) f and h in Terms of F β„“ and G β„“ when Ο΅ > 0
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    §33.16(iv) s and c in Terms of F β„“ and G β„“ when Ο΅ > 0
    30: 8.18 Asymptotic Expansions of I x ⁑ ( a , b )
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    8.18.9 I x ⁑ ( a , b ) 1 2 ⁒ erfc ⁑ ( Ξ· ⁒ b / 2 ) + 1 2 ⁒ Ο€ ⁒ ( a + b ) ⁒ ( x x 0 ) a ⁒ ( 1 x 1 x 0 ) b ⁒ k = 0 ( 1 ) k ⁒ c k ⁑ ( Ξ· ) ( a + b ) k ,
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    8.18.10 1 2 ⁒ η 2 = x 0 ⁒ ln ⁑ ( x x 0 ) + ( 1 x 0 ) ⁒ ln ⁑ ( 1 x 1 x 0 ) ,