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11: 20.3 Graphics
See accompanying text
Figure 20.3.2: θ 1 ( π x , q ) , 0 x 2 , q = 0. …Here q Dedekind = e - π y 0 = 0.19 approximately, where y = y 0 corresponds to the maximum value of Dedekind’s eta function η ( i y ) as depicted in Figure 23.16.1. Magnify
12: 33.24 Tables
§33.24 Tables
13: 33.12 Asymptotic Expansions for Large η
§33.12 Asymptotic Expansions for Large η
§33.12(i) Transition Region
Then as η , …
§33.12(ii) Uniform Expansions
14: 33.4 Recurrence Relations and Derivatives
§33.4 Recurrence Relations and Derivatives
Then, with X denoting any of F ( η , ρ ) , G ( η , ρ ) , or H ± ( η , ρ ) , …
15: 33.13 Complex Variable and Parameters
§33.13 Complex Variable and Parameters
The functions F ( η , ρ ) , G ( η , ρ ) , and H ± ( η , ρ ) may be extended to noninteger values of by generalizing ( 2 + 1 ) ! = Γ ( 2 + 2 ) , and supplementing (33.6.5) by a formula derived from (33.2.8) with U ( a , b , z ) expanded via (13.2.42). These functions may also be continued analytically to complex values of ρ , η , and . …
33.13.2 R = ( 2 + 1 ) C ( η ) / C - 1 ( η ) .
16: 33.25 Approximations
§33.25 Approximations
17: 33.1 Special Notation
The main functions treated in this chapter are first the Coulomb radial functions F ( η , ρ ) , G ( η , ρ ) , H ± ( η , ρ ) (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions f ( ϵ , ; r ) , h ( ϵ , ; r ) , s ( ϵ , ; r ) , c ( ϵ , ; r ) (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. …
  • Greene et al. (1979):

    f ( 0 ) ( ϵ , ; r ) = f ( ϵ , ; r ) , f ( ϵ , ; r ) = s ( ϵ , ; r ) , g ( ϵ , ; r ) = c ( ϵ , ; r ) .

  • 18: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
    §33.5(i) Small ρ
    §33.5(ii) η = 0
    33.5.6 C ( 0 ) = 2 ! ( 2 + 1 ) ! = 1 ( 2 + 1 ) !! .
    §33.5(iii) Small | η |
    §33.5(iv) Large
    19: 33.10 Limiting Forms for Large ρ or Large | η |
    §33.10(i) Large ρ
    §33.10(ii) Large Positive η
    §33.10(iii) Large Negative η
    20: 33.3 Graphics
    §33.3 Graphics
    §33.3(i) Line Graphs of the Coulomb Radial Functions F ( η , ρ ) and G ( η , ρ )
    33.3.1 M ( η , ρ ) = ( F 2 ( η , ρ ) + G 2 ( η , ρ ) ) 1 / 2 = | H ± ( η , ρ ) | .
    §33.3(ii) Surfaces of the Coulomb Radial Functions F 0 ( η , ρ ) and G 0 ( η , ρ )
    See accompanying text
    Figure 33.3.8: G 0 ( η , ρ ) , - 2 η 2 , 0 < ρ 5 . Magnify 3D Help