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21: Bibliography D
  • A. M. Din (1981) A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.
  • B. I. Dunlap and B. R. Judd (1975) Novel identities for simple n - j symbols. J. Mathematical Phys. 16, pp. 318–319.
  • T. M. Dunster (1994b) Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane. SIAM J. Math. Anal. 25 (2), pp. 322–353.
  • T. M. Dunster (2001a) Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.
  • T. M. Dunster (2014) Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.
  • 22: Bibliography N
  • G. Nemes (2021) Proofs of two conjectures on the real zeros of the cylinder and Airy functions. SIAM J. Math. Anal. 53 (4), pp. 4328–4349.
  • J. J. Nestor (1984) Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.
  • 23: 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.
  • M. R. Zaghloul (2017) Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the Faddeyeva Function. ACM Trans. Math. Softw. 44 (2), pp. 22:1–22:9.
  • 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.
  • 24: 8.2 Definitions and Basic Properties
    When z 0 , Γ ( a , z ) is an entire function of a , and γ ( a , z ) is meromorphic with simple poles at a = n , n = 0 , 1 , 2 , , with residue ( 1 ) n / n ! . … (8.2.9) also holds when a is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. …
    25: Mathematical Introduction
    complex plane (excluding infinity).
    empty sums zero.
    f ( z ) | C = 0 f ( z ) is continuous at all points of a simple closed contour C in .
    26: 10.74 Methods of Computation
    §10.74(vi) Zeros and Associated Values
    Real Zeros
    Complex Zeros
    Multiple Zeros
    27: Bibliography B
  • P. Baldwin (1985) Zeros of generalized Airy functions. Mathematika 32 (1), pp. 104–117.
  • J. S. Ball (2000) Automatic computation of zeros of Bessel functions and other special functions. SIAM J. Sci. Comput. 21 (4), pp. 1458–1464.
  • R. Barakat and E. Parshall (1996) Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.
  • E. Barouch, B. M. McCoy, and T. T. Wu (1973) Zero-field susceptibility of the two-dimensional Ising model near T c . Phys. Rev. Lett. 31, pp. 1409–1411.
  • W. G. C. Boyd (1990a) Asymptotic Expansions for the Coefficient Functions Associated with Linear Second-order Differential Equations: The Simple Pole Case. In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.), Lecture Notes in Pure and Applied Mathematics, Vol. 124, pp. 53–73.
  • 28: 9.19 Approximations
  • Martín et al. (1992) provides two simple formulas for approximating Ai ( x ) to graphical accuracy, one for < x 0 , the other for 0 x < .

  • Németh (1992, Chapter 8) covers Ai ( x ) , Ai ( x ) , Bi ( x ) , Bi ( x ) , and integrals 0 x Ai ( t ) d t , 0 x Bi ( t ) d t , 0 x 0 v Ai ( t ) d t d v , 0 x 0 v Bi ( t ) d t d v (see also (9.10.20) and (9.10.21)). The Chebyshev coefficients are given to 15D. Chebyshev coefficients are also given for expansions of the second and higher (real) zeros of Ai ( x ) , Ai ( x ) , Bi ( x ) , Bi ( x ) , again to 15D.

  • 29: 28.7 Analytic Continuation of Eigenvalues
    The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). … Therefore w I ( 1 2 π ; a , q ) is irreducible, in the sense that it cannot be decomposed into a product of entire functions that contain its zeros; see Meixner et al. (1980, p. 88). …
    30: 3.3 Interpolation
    3.3.6 R n ( z ) = ω n + 1 ( z ) 2 π i C f ( ζ ) ( ζ z ) ω n + 1 ( ζ ) d ζ ,
    where C is a simple closed contour in D described in the positive rotational sense and enclosing the points z , z 1 , z 2 , , z n . … where ω n + 1 ( ζ ) is given by (3.3.3), and C is a simple closed contour in D described in the positive rotational sense and enclosing z 0 , z 1 , , z n . …
    Example
    To compute the first negative zero a 1 = 2.33810 7410 of the Airy function f ( x ) = Ai ( x ) 9.2). …