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Weber parabolic cylinder functions


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1: 12.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: U ( a , z ) , V ( a , z ) , U ¯ ( a , z ) , and W ( a , z ) . …
2: Bibliography O
  • F. W. J. Olver (1959) Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. J. Res. Nat. Bur. Standards Sect. B 63B, pp. 131–169.
  • 3: Bibliography F
  • L. Fox (1960) Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments. National Physical Laboratory Mathematical Tables, Vol. 4. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.
  • 4: Bibliography V
  • R. S. Varma (1941) An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
  • 5: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2011b) Fast and accurate computation of the Weber parabolic cylinder function W ( a , x ) . IMA J. Numer. Anal. 31 (3), pp. 1194–1216.
  • 6: Bibliography M
  • J. C. P. Miller (Ed.) (1955) Tables of Weber Parabolic Cylinder Functions. Her Majesty’s Stationery Office, London.
  • 7: Bibliography S
  • H. Shanker (1940a) On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
  • H. Shanker (1940b) On certain integrals and expansions involving Weber’s parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 158–166.
  • 8: Software Index
    Open Source With Book Commercial
    12 Parabolic Cylinder 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. …
    9: 32.10 Special Function Solutions
    §32.10 Special Function Solutions
    §32.10(iv) Fourth Painlevé Equation
    P IV  has solutions expressible in terms of parabolic cylinder functions12.2) iff either …When a + 1 2 is zero or a negative integer the U parabolic cylinder functions reduce to Hermite polynomials (§18.3) times an exponential function; thus …
    10: 10.16 Relations to Other Functions
    Elementary Functions
    Airy Functions
    Parabolic Cylinder Functions
    Confluent Hypergeometric Functions
    Generalized Hypergeometric Functions