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Coulomb functions: variables r,ϵ

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11: 33.1 Special Notation
  • Greene et al. (1979):

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

  • 12: 33.19 Power-Series Expansions in r
    §33.19 Power-Series Expansions in r
    33.19.1 f ( ϵ , ; r ) = r + 1 k = 0 α k r k ,
    33.19.3 2 π h ( ϵ , ; r ) = k = 0 2 ( 2 k ) ! γ k k ! ( 2 r ) k k = 0 δ k r k + + 1 A ( ϵ , ) ( 2 ln | 2 r / κ | + ψ ( + 1 + κ ) + ψ ( + κ ) ) f ( ϵ , ; r ) , r 0 .
    13: 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 ) ,
    33.4.2 R X 1 T X + R + 1 X + 1 = 0 , 1 ,
    33.4.3 X = R X 1 S X , 1 ,
    33.4.4 X = S + 1 X R + 1 X + 1 , 0 .
    14: 33.8 Continued Fractions
    §33.8 Continued Fractions
    33.8.1 F F = S + 1 R + 1 2 T + 1 R + 2 2 T + 2 .
    For R , S , and T see (33.4.1).
    33.8.2 H ± H ± = c ± i ρ a b 2 ( ρ η ± i ) + ( a + 1 ) ( b + 1 ) 2 ( ρ η ± 2 i ) + ,
    If we denote u = F / F and p + i q = H + / H + , then …
    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 . The quantities C ( η ) , σ ( η ) , and R , given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that …
    33.13.2 R = ( 2 + 1 ) C ( η ) / C 1 ( η ) .
    16: Bibliography D
  • G. Delic (1979b) Chebyshev series for the spherical Bessel function j l ( r ) . Comput. Phys. Comm. 18 (1), pp. 73–86.
  • A. R. DiDonato and A. H. Morris (1986) Computation of the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 12 (4), pp. 377–393.
  • A. R. DiDonato and A. H. Morris (1992) Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.
  • C. Dunkl, M. Ismail, and R. Wong (Eds.) (2000) Special Functions. World Scientific Publishing Co., Inc., River Edge, NJ.
  • A. Dzieciol, S. Yngve, and P. O. Fröman (1999) Coulomb wave functions with complex values of the variable and the parameters. J. Math. Phys. 40 (12), pp. 6145–6166.
  • 17: Bibliography H
  • R. L. Hall, N. Saad, and K. D. Sen (2010) Soft-core Coulomb potentials and Heun’s differential equation. J. Math. Phys. 51 (2), pp. Art. ID 022107, 19 pages.
  • M. Hiyama and H. Nakamura (1997) Two-center Coulomb functions. Comput. Phys. Comm. 103 (2-3), pp. 209–216.
  • L. E. Hoisington and G. Breit (1938) Calculation of Coulomb wave functions for high energies. Phys. Rev. 54 (8), pp. 627–628.
  • M. H. Hull and G. Breit (1959) Coulomb Wave Functions. In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.), pp. 408–465.
  • J. Humblet (1985) Bessel function expansions of Coulomb wave functions. J. Math. Phys. 26 (4), pp. 656–659.
  • 18: Bibliography B
  • A. R. Barnett (1981a) An algorithm for regular and irregular Coulomb and Bessel functions of real order to machine accuracy. Comput. Phys. Comm. 21 (3), pp. 297–314.
  • A. R. Barnett (1981b) KLEIN: Coulomb functions for real λ and positive energy to high accuracy. Comput. Phys. Comm. 24 (2), pp. 141–159.
  • A. R. Barnett (1982) COULFG: Coulomb and Bessel functions and their derivatives, for real arguments, by Steed’s method. Comput. Phys. Comm. 27, pp. 147–166.
  • A. R. Barnett (1996) The Calculation of Spherical Bessel Functions and Coulomb Functions. In Computational Atomic Physics: Electron and Positron Collisions with Atoms and Ions, K. Bartschat and J. Hinze (Eds.), pp. 181–202.
  • L. C. Biedenharn, R. L. Gluckstern, M. H. Hull, and G. Breit (1955) Coulomb functions for large charges and small velocities. Phys. Rev. (2) 97 (2), pp. 542–554.
  • 19: 18.39 Applications in the Physical Sciences
    c) Spherical Radial Coulomb Wave Functions
    The radial Coulomb wave functions R n , l ( r ) , solutions of …
    d) Radial Coulomb Wave Functions Expressed in Terms of the Associated Coulomb–Laguerre OP’s
    Discretized and Continuum Expansions of Scattering Eigenfunctions in terms of Pollaczek Polynomials: J-matrix Theory
    The Coulomb–Pollaczek polynomials provide alternate representations of the positive energy Coulomb wave functions of Chapter 33. …
    20: Bibliography C
  • B. C. Carlson (1985) The hypergeometric function and the R -function near their branch points. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.
  • B. C. Carlson (2006b) Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric R -functions. Math. Comp. 75 (255), pp. 1309–1318.
  • C. W. Clark (1979) Coulomb phase shift. American Journal of Physics 47 (8), pp. 683–684.
  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.
  • A. R. Curtis (1964a) Coulomb Wave Functions. Roy. Soc. Math. Tables, Vol. 11, Cambridge University Press, Cambridge.