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relations to Anger–Weber functions


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1: 11.10 Anger–Weber Functions
§11.10(vi) Relations to Other Functions
11.10.18 𝐄 ν ( z ) = 1 π ( 1 + cos ( π ν ) ) s 0 , ν ( z ) ν π ( 1 cos ( π ν ) ) s 1 , ν ( z ) .
m 2 = 1 2 n 3 2 .
2: 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.
  • R. Zanovello (1977) Integrali di funzioni di Anger, Weber ed Airy-Hardy. Rend. Sem. Mat. Univ. Padova 58, pp. 275–285 (Italian).
  • R. Zanovello (1995) Numerical analysis of Struve functions with applications to other special functions. Ann. Numer. Math. 2 (1-4), pp. 199–208.
  • Q. Zheng (1997) Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.
  • 3: Bibliography N
  • National Bureau of Standards (1967) Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
  • J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.
  • G. Nemes (2014b) The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.
  • G. Nemes (2014c) The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.
  • Number Theory Web (website)
  • 4: 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.

  • 5: 3.6 Linear Difference Equations
    is satisfied by J n ( 1 ) and Y n ( 1 ) , where J n ( x ) and Y n ( x ) are the Bessel functions of the first kind. …
    Example 2. Weber Function
    The Weber function 𝐄 n ( 1 ) satisfies …Thus the asymptotic behavior of the particular solution 𝐄 n ( 1 ) is intermediate to those of the complementary functions J n ( 1 ) and Y n ( 1 ) ; moreover, the conditions for Olver’s algorithm are satisfied. We apply the algorithm to compute 𝐄 n ( 1 ) to 8S for the range n = 1 , 2 , , 10 , beginning with the value 𝐄 0 ( 1 ) = 0.56865  663 obtained from the Maclaurin series expansion (§11.10(iii)). …