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relations to confluent hypergeometric functions of matrix argument


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1: 35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6(iii) Relations to Bessel Functions of Matrix Argument
2: Software Index
‘✓’ 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. …
  • Open Source Collections and Systems.

    These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

  • 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.

  • Guide to Available Mathematical Software

    A cross index of mathematical software in use at NIST.

  • 3: 33.22 Particle Scattering and Atomic and Molecular Spectra
    Positive-energy functions correspond to processes such as Rutherford scattering and Coulomb excitation of nuclei (Alder et al. (1956)), and atomic photo-ionization and electron-ion collisions (Bethe and Salpeter (1977)). … The solutions to this equation are closely related to the Coulomb functions; see Greiner et al. (1985). … For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, F ( η , ρ ) and G ( η , ρ ) , or f ( ϵ , ; r ) and h ( ϵ , ; r ) , to determine the scattering S -matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function W η , + 1 2 ( 2 ρ ) . … The Coulomb functions given in this chapter are most commonly evaluated for real values of ρ , r , η , ϵ and nonnegative integer values of , but they may be continued analytically to complex arguments and order as indicated in §33.13. …
    4: Bibliography B
  • H. Buchholz (1969) The Confluent Hypergeometric Function with Special Emphasis on Its Applications. Springer-Verlag, New York.
  • W. Bühring (1987b) The behavior at unit argument of the hypergeometric function F 2 3 . SIAM J. Math. Anal. 18 (5), pp. 1227–1234.
  • W. Bühring (1992) Generalized hypergeometric functions at unit argument. Proc. Amer. Math. Soc. 114 (1), pp. 145–153.
  • R. W. Butler and A. T. A. Wood (2002) Laplace approximations for hypergeometric functions with matrix argument. Ann. Statist. 30 (4), pp. 1155–1177.
  • R. W. Butler and A. T. A. Wood (2003) Laplace approximation for Bessel functions of matrix argument. J. Comput. Appl. Math. 155 (2), pp. 359–382.
  • 5: Bibliography D
  • P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou (1999b) Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.
  • K. Dilcher (2002) Bernoulli Numbers and Confluent Hypergeometric Functions. In Number Theory for the Millennium, I (Urbana, IL, 2000), pp. 343–363.
  • C. F. du Toit (1993) Bessel functions J n ( z ) and Y n ( z ) of integer order and complex argument. Comput. Phys. Comm. 78 (1-2), pp. 181–189.
  • T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.
  • T. M. Dunster (2013) Conical functions of purely imaginary order and argument. Proc. Roy. Soc. Edinburgh Sect. A 143 (5), pp. 929–955.
  • 6: Bibliography
  • J. Abad and J. Sesma (1995) Computation of the regular confluent hypergeometric function. The Mathematica Journal 5 (4), pp. 74–76.
  • S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.
  • G. Allasia and R. Besenghi (1991) Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule. Rend. Sem. Mat. Univ. Politec. Torino 49 (3), pp. 315–327.
  • T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.
  • F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.
  • 7: Bibliography M
  • H. Majima, K. Matsumoto, and N. Takayama (2000) Quadratic relations for confluent hypergeometric functions. Tohoku Math. J. (2) 52 (4), pp. 489–513.
  • N. Michel and M. V. Stoitsov (2008) Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions. Comput. Phys. Comm. 178 (7), pp. 535–551.
  • S. C. Milne (1985c) A new symmetry related to 𝑆𝑈 ( n ) for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.
  • T. Morita (2013) A connection formula for the q -confluent hypergeometric function. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 050, 13.
  • R. Morris (1979) The dilogarithm function of a real argument. Math. Comp. 33 (146), pp. 778–787.
  • 8: Errata
  • Equation (35.7.3)

    Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument F 1 2 was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

  • Paragraph Confluent Hypergeometric Functions (in §7.18(iv))

    A note about the multivalued nature of the Kummer confluent hypergeometric function of the second kind U on the right-hand side of (7.18.10) was inserted.

  • Chapter 35 Functions of Matrix Argument

    The generalized hypergeometric function of matrix argument F q p ( a 1 , , a p ; b 1 , , b q ; 𝐓 ) , was linked inadvertently as its single variable counterpart F q p ( a 1 , , a p ; b 1 , , b q ; 𝐓 ) . Furthermore, the Jacobi function of matrix argument P ν ( γ , δ ) ( 𝐓 ) , and the Laguerre function of matrix argument L ν ( γ ) ( 𝐓 ) , were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by P ν ( γ , δ ) ( 𝐓 ) , and L ν ( γ ) ( 𝐓 ) . In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

  • Paragraph Confluent Hypergeometric Functions (in §10.16)

    Confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.

  • Subsection 13.29(v)

    A new Subsection Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.