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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: 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). … …
5: 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 …
6: Bibliography L
  • A. Laforgia and M. E. Muldoon (1983) Inequalities and approximations for zeros of Bessel functions of small order. SIAM J. Math. Anal. 14 (2), pp. 383–388.
  • J. S. Lew (1994) On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions. Constr. Approx. 10 (1), pp. 15–30.
  • J. L. López (1999) Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.
  • J. L. López (2000) Asymptotic expansions of symmetric standard elliptic integrals. SIAM J. Math. Anal. 31 (4), pp. 754–775.
  • E. R. Love (1972a) Addendum to: “Changing the order of integration”. J. Austral. Math. Soc. 14, pp. 383–384.
  • 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 P
  • R. B. Paris (1992a) Smoothing of the Stokes phenomenon for high-order differential equations. Proc. Roy. Soc. London Ser. A 436, pp. 165–186.
  • R. B. Paris (2004) Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.
  • R. B. Paris (2005a) A Kummer-type transformation for a F 2 2 hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.
  • A. M. Parkhurst and A. T. James (1974) Zonal Polynomials of Order 1 Through 12 . In Selected Tables in Mathematical Statistics, H. L. Harter and D. B. Owen (Eds.), Vol. 2, pp. 199–388.
  • S. Paszkowski (1991) Evaluation of the Fermi-Dirac integral of half-integer order. Zastos. Mat. 21 (2), pp. 289–301.
  • 9: Bibliography W
  • Z. Wang and R. Wong (2003) Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.
  • R. Wong and T. Lang (1991) On the points of inflection of Bessel functions of positive order. II. Canad. J. Math. 43 (3), pp. 628–651.
  • R. Wong and H. Li (1992a) Asymptotic expansions for second-order linear difference equations. II. Stud. Appl. Math. 87 (4), pp. 289–324.
  • R. Wong and H. Li (1992b) Asymptotic expansions for second-order linear difference equations. J. Comput. Appl. Math. 41 (1-2), pp. 65–94.
  • R. Wong and H. Y. Zhang (2007) Asymptotic solutions of a fourth order differential equation. Stud. Appl. Math. 118 (2), pp. 133–152.
  • 10: 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