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1: 33.8 Continued Fractions
33.8.1 F F = S + 1 - R + 1 2 T + 1 - R + 2 2 T + 2 - .
If we denote u = F / F and p + i q = H + / H + , then
F = ± ( q - 1 ( u - p ) 2 + q ) - 1 / 2 ,
F = u F ,
G = q - 1 ( u - p ) F ,
2: 33.3 Graphics
§33.3(i) Line Graphs of the Coulomb Radial Functions F ( η , ρ ) and G ( η , ρ )
See accompanying text
Figure 33.3.1: F ( η , ρ ) , G ( η , ρ ) with = 0 , η = - 2 . Magnify
See accompanying text
Figure 33.3.2: F ( η , ρ ) , G ( η , ρ ) with = 0 , η = 0 . Magnify
See accompanying text
Figure 33.3.3: F ( η , ρ ) , G ( η , ρ ) with = 0 , η = 2 . … Magnify
§33.3(ii) Surfaces of the Coulomb Radial Functions F 0 ( η , ρ ) and G 0 ( η , ρ )
3: 25.17 Physical Applications
Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). It has been found possible to perform such regularizations by equating the divergent sums to zeta functions and associated functions (Elizalde (1995)).
4: 33.23 Methods of Computation
The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii ρ and r , respectively, and may be used to compute the regular and irregular solutions. … Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. … This implies decreasing for the regular solutions and increasing for the irregular solutions of §§33.2(iii) and 33.14(iii). … §33.8 supplies continued fractions for F / F and H ± / H ± . Combined with the Wronskians (33.2.12), the values of F , G , and their derivatives can be extracted. …
5: 33.2 Definitions and Basic Properties
§33.2(i) Coulomb Wave Equation
This differential equation has a regular singularity at ρ = 0 with indices + 1 and - , and an irregular singularity of rank 1 at ρ = (§§2.7(i), 2.7(ii)). …
§33.2(ii) Regular Solution F ( η , ρ )
The function F ( η , ρ ) is recessive (§2.7(iii)) at ρ = 0 , and is defined by … F ( η , ρ ) is a real and analytic function of ρ on the open interval 0 < ρ < , and also an analytic function of η when - < η < . …
6: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
F ( η , ρ ) C ( η ) ρ + 1 ,
F ( η , ρ ) ( + 1 ) C ( η ) ρ .
F ( 0 , ρ ) = ρ j ( ρ ) ,
F 0 ( 0 , ρ ) = sin ρ ,
F ( η , ρ ) C ( η ) ρ + 1 ,
7: 33.24 Tables
  • Abramowitz and Stegun (1964, Chapter 14) tabulates F 0 ( η , ρ ) , G 0 ( η , ρ ) , F 0 ( η , ρ ) , and G 0 ( η , ρ ) for η = 0.5 ( .5 ) 20 and ρ = 1 ( 1 ) 20 , 5S; C 0 ( η ) for η = 0 ( .05 ) 3 , 6S.

  • 8: 33.10 Limiting Forms for Large ρ or Large | η |
    F ( η , ρ ) = sin ( θ ( η , ρ ) ) + o ( 1 ) ,
    F ( η , ρ ) ( 2 + 1 ) ! C ( η ) ( 2 η ) + 1 ( 2 η ρ ) 1 / 2 I 2 + 1 ( ( 8 η ρ ) 1 / 2 ) ,
    F 0 ( η , ρ ) e - π η ( π ρ ) 1 / 2 I 1 ( ( 8 η ρ ) 1 / 2 ) ,
    F 0 ( η , ρ ) e - π η ( 2 π η ) 1 / 2 I 0 ( ( 8 η ρ ) 1 / 2 ) ,
    F 0 ( η , ρ ) = ( π ρ ) 1 / 2 J 1 ( ( - 8 η ρ ) 1 / 2 ) + o ( | η | - 1 / 4 ) ,
    9: 33.16 Connection Formulas
    §33.16(i) F and G in Terms of f and h
    33.16.1 F ( η , ρ ) = ( 2 + 1 ) ! C ( η ) ( - 2 η ) + 1 f ( 1 / η 2 , ; - η ρ ) ,
    §33.16(ii) f and h in Terms of F and G when ϵ > 0
    §33.16(iv) s and c in Terms of F and G when ϵ > 0
    10: 33.9 Expansions in Series of Bessel Functions
    33.9.3 F ( η , ρ ) = C ( η ) ( 2 + 1 ) ! ( 2 η ) 2 + 1 ρ - k = 2 + 1 b k t k / 2 I k ( 2 t ) , η > 0 ,
    33.9.4 F ( η , ρ ) = C ( η ) ( 2 + 1 ) ! ( 2 | η | ) 2 + 1 ρ - k = 2 + 1 b k t k / 2 J k ( 2 t ) , η < 0 .