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11: 13.8 Asymptotic Approximations for Large Parameters
To obtain approximations for M ( a , b , z ) and U ( a , b , z ) that hold as b , with a > 1 2 - b and z > 0 combine (13.14.4), (13.14.5) with §13.20(i). …
§13.8(iii) Large a
For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i). … For asymptotic approximations to M ( a , b , x ) and U ( a , b , x ) as a - that hold uniformly with respect to x ( 0 , ) and bounded positive values of ( b - 1 ) / | a | , combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii). …
12: Bibliography Z
  • F. A. Zafiropoulos, T. N. Grapsa, O. Ragos, and M. N. Vrahatis (1996) On the Computation of Zeros of Bessel and Bessel-related Functions. In Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.), Utrecht, pp. 409–416.
  • D. Zagier (1989) The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.
  • A. Zarzo, J. S. Dehesa, and R. J. Yañez (1995) Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.
  • J. M. Zhang, X. C. Li, and C. K. Qu (1996) Error bounds for asymptotic solutions of second-order linear difference equations. J. Comput. Appl. Math. 71 (2), pp. 191–212.
  • Q. Zheng (1997) Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.
  • 13: 13.9 Zeros
    §13.9(i) Zeros of M ( a , b , z )
    For fixed a and z in , U ( a , b , z ) has two infinite strings of b -zeros that are asymptotic to the imaginary axis as | b | .
    14: 33.14 Definitions and Basic Properties
    §33.14(i) Coulomb Wave Equation
    §33.14(ii) Regular Solution f ( ϵ , ; r )
    This is a consequence of Kummer’s transformation (§13.2(vii)). …
    §33.14(iii) Irregular Solution h ( ϵ , ; r )
    For nonzero values of ϵ and r the function h ( ϵ , ; r ) is defined by …
    15: 33.2 Definitions and Basic Properties
    §33.2(i) Coulomb Wave Equation
    The function F ( η , ρ ) is recessive (§2.7(iii)) at ρ = 0 , and is defined by …where M κ , μ ( z ) and M ( a , b , z ) are defined in §§13.14(i) and 13.2(i), and …This is a consequence of Kummer’s transformation (§13.2(vii)). … The functions H ± ( η , ρ ) are defined by …
    16: 18.5 Explicit Representations
    Chebyshev
    Related formula: …
    §18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions
    Laguerre
    Hermite
    17: Software Index
  • Research Software.

    This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

  • 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.

  • Software Associated with Books.

    An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

  • 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.

  • 18: 18.34 Bessel Polynomials
    §18.34(i) Definitions and Recurrence Relation
    For the confluent hypergeometric function F 1 1 and the generalized hypergeometric function F 0 2 see §16.2(ii) and §16.2(iv). …
    §18.34(iii) Other Properties
    where primes denote derivatives with respect to x . …
    19: 7.18 Repeated Integrals of the Complementary Error Function
    §7.18(iv) Relations to Other Functions
    Hermite Polynomials
    Confluent Hypergeometric Functions
    Parabolic Cylinder Functions
    Probability Functions
    20: Bibliography M
  • R. S. Maier (2005) On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.
  • T. Masuda (2003) On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46 (1), pp. 121–171.
  • M. Mazzocco (2001b) Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.
  • J. C. P. Miller (1952) On the choice of standard solutions to Weber’s equation. Proc. Cambridge Philos. Soc. 48, pp. 428–435.
  • S. C. Milne (1985c) A new symmetry related to SU ( n ) for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.