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Coulomb spheroidal functions


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1: 30.12 Generalized and Coulomb Spheroidal Functions
§30.12 Generalized and Coulomb Spheroidal Functions
Generalized spheroidal wave functions and Coulomb spheroidal functions are solutions of the differential equation …
2: 31.12 Confluent Forms of Heun’s Equation
This has regular singularities at z = 0 and 1 , and an irregular singularity of rank 1 at z = . Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions30.12) are special cases of solutions of the confluent Heun equation. …
3: Bibliography K
  • I. V. Komarov, L. I. Ponomarev, and S. Yu. Slavyanov (1976) Sferoidalnye i kulonovskie sferoidalnye funktsii. Izdat. “Nauka”, Moscow (Russian).
  • 4: Software Index
    Open Source With Book Commercial
    30 Spheroidal Wave Functions
    ‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … In the list below we identify four main sources of software for computing special functions. …
  • Commercial Software.

    Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

  • The following are web-based software repositories with significant holdings in the area of special functions. …
    5: Bibliography
  • M. Abramowitz and P. Rabinowitz (1954) Evaluation of Coulomb wave functions along the transition line. Physical Rev. (2) 96, pp. 77–79.
  • M. Abramowitz (1949) Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.
  • M. Abramowitz (1954) Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.
  • Z. Altaç (1996) Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.
  • G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1992a) Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23 (2), pp. 512–524.
  • 6: Bibliography H
  • 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 (1984) Analytical structure and properties of Coulomb wave functions for real and complex energies. Ann. Physics 155 (2), pp. 461–493.
  • J. Humblet (1985) Bessel function expansions of Coulomb wave functions. J. Math. Phys. 26 (4), pp. 656–659.
  • 7: Bibliography M
  • J. Meixner, F. W. Schäfke, and G. Wolf (1980) Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations: Further Studies. Lecture Notes in Mathematics, Vol. 837, Springer-Verlag, Berlin-New York.
  • N. Michel (2007) Precise Coulomb wave functions for a wide range of complex , η and z . Computer Physics Communications 176 (3), pp. 232–249.
  • J. W. Miles (1975) Asymptotic approximations for prolate spheroidal wave functions. Studies in Appl. Math. 54 (4), pp. 315–349.
  • H. J. W. Müller (1962) Asymptotic expansions of oblate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 211, pp. 33–47.
  • H. J. W. Müller (1963) Asymptotic expansions of prolate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 212, pp. 26–48.
  • 8: Bibliography J
  • D. L. Jagerman (1974) Some properties of the Erlang loss function. Bell System Tech. J. 53, pp. 525–551.
  • D. S. Jones (2001) Asymptotics of the hypergeometric function. Math. Methods Appl. Sci. 24 (6), pp. 369–389.
  • D. S. Jones (2006) Parabolic cylinder functions of large order. J. Comput. Appl. Math. 190 (1-2), pp. 453–469.
  • S. Jorna and C. Springer (1971) Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions p s ¯ n r ( η , h ) and q s ¯ n r ( η , h ) for large h . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.
  • B. R. Judd (1976) Modifications of Coulombic interactions by polarizable atoms. Math. Proc. Cambridge Philos. Soc. 80 (3), pp. 535–539.
  • 9: 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.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • T. A. Beu and R. I. Câmpeanu (1983a) Prolate angular spheroidal wave functions. Comput. Phys. Comm. 30 (2), pp. 187–192.
  • T. A. Beu and R. I. Câmpeanu (1983b) Prolate radial spheroidal wave functions. Comput. Phys. Comm. 30 (2), pp. 177–185.
  • C. J. Bouwkamp (1947) On spheroidal wave functions of order zero. J. Math. Phys. Mass. Inst. Tech. 26, pp. 79–92.
  • 10: Bibliography C
  • B. C. Carlson and G. S. Rushbrooke (1950) On the expansion of a Coulomb potential in spherical harmonics. Proc. Cambridge Philos. Soc. 46, pp. 626–633.
  • J. A. Christley and I. J. Thompson (1994) CRCWFN: Coupled real Coulomb wavefunctions. Comput. Phys. Comm. 79 (1), pp. 143–155.
  • C. W. Clark (1979) Coulomb phase shift. American Journal of Physics 47 (8), pp. 683–684.
  • W. C. Connett, C. Markett, and A. L. Schwartz (1993) Product formulas and convolutions for angular and radial spheroidal wave functions. Trans. Amer. Math. Soc. 338 (2), pp. 695–710.
  • A. R. Curtis (1964a) Coulomb Wave Functions. Roy. Soc. Math. Tables, Vol. 11, Cambridge University Press, Cambridge.