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1: 28.12 Definitions and Basic Properties
The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict ν ^ 0 , 1 ; equivalently ν n . …
§28.12(ii) Eigenfunctions me ν ( z , q )
For q = 0 , …
2: 28.2 Definitions and Basic Properties
§28.2(vi) Eigenfunctions
3: 34.11 Higher-Order 3 n j Symbols
§34.11 Higher-Order 3 n j Symbols
4: Bibliography O
  • A. B. Olde Daalhuis and F. W. J. Olver (1995a) Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.
  • A. B. Olde Daalhuis and F. W. J. Olver (1995b) On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.
  • A. B. Olde Daalhuis (1995) Hyperasymptotic solutions of second-order linear differential equations. II. Methods Appl. Anal. 2 (2), pp. 198–211.
  • A. B. Olde Daalhuis (1998a) Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.
  • F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.
  • 5: 10.24 Functions of Imaginary Order
    §10.24 Functions of Imaginary Order
    and J ~ ν ( x ) , Y ~ ν ( x ) are linearly independent solutions of (10.24.1): … In consequence of (10.24.6), when x is large J ~ ν ( x ) and Y ~ ν ( x ) comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). … …
    6: 10.45 Functions of Imaginary Order
    §10.45 Functions of Imaginary Order
    and I ~ ν ( x ) , K ~ ν ( x ) are real and linearly independent solutions of (10.45.1): … The corresponding result for K ~ ν ( x ) is given by …
    7: 10.26 Graphics
    §10.26(i) Real Order and Variable
    §10.26(ii) Real Order, Complex Variable
    §10.26(iii) Imaginary Order, Real Variable
    See accompanying text
    Figure 10.26.7: I ~ 1 / 2 ( x ) , K ~ 1 / 2 ( x ) , 0.01 x 3 . Magnify
    See accompanying text
    Figure 10.26.8: I ~ 1 ( x ) , K ~ 1 ( x ) , 0.01 x 3 . Magnify
    8: Bibliography U
  • Unpublished Mathematical Tables (1944) Mathematics of Computation Unpublished Mathematical Tables Collection.
  • F. Ursell (1984) Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
  • 9: 10.76 Approximations
    Real Variable and Order : Functions
    Real Variable and Order : Zeros
    Real Variable and Order : Integrals
    Complex Variable; Real Order
    Real Variable; Imaginary Order
    10: Frank W. J. Olver
    Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. … Most notably, he served as the Editor-in-Chief and Mathematics Editor of the online NIST Digital Library of Mathematical Functions and its 966-page print companion, the NIST Handbook of Mathematical Functions (Cambridge University Press, 2010). …