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1: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
2: 10.72 Mathematical Applications
Simple Turning Points
In regions in which (10.72.1) has a simple turning point z 0 , that is, f ( z ) and g ( z ) are analytic (or with weaker conditions if z = x is a real variable) and z 0 is a simple zero of f ( z ) , asymptotic expansions of the solutions w for large u can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order 1 3 9.6(i)). … In regions in which the function f ( z ) has a simple pole at z = z 0 and ( z z 0 ) 2 g ( z ) is analytic at z = z 0 (the case λ = 1 in §10.72(i)), asymptotic expansions of the solutions w of (10.72.1) for large u can be constructed in terms of Bessel functions and modified Bessel functions of order ± 1 + 4 ρ , where ρ is the limiting value of ( z z 0 ) 2 g ( z ) as z z 0 . … In (10.72.1) assume f ( z ) = f ( z , α ) and g ( z ) = g ( z , α ) depend continuously on a real parameter α , f ( z , α ) has a simple zero z = z 0 ( α ) and a double pole z = 0 , except for a critical value α = a , where z 0 ( a ) = 0 . …
3: 13.27 Mathematical Applications
For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i). …
4: 27.18 Methods of Computation: Primes
Two simple algorithms for proving primality require a knowledge of all or part of the factorization of n 1 , n + 1 , or both; see Crandall and Pomerance (2005, §§4.1–4.2). …
5: 5.2 Definitions
It is a meromorphic function with no zeros, and with simple poles of residue ( 1 ) n / n ! at z = n . 1 / Γ ( z ) is entire, with simple zeros at z = n . … ψ ( z ) is meromorphic with simple poles of residue 1 at z = n . …
6: 31.6 Path-Multiplicative Solutions
This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the z -plane that encircles s 1 and s 2 once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor e 2 ν π i . …
7: 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.
  • 8: 3.8 Nonlinear Equations
    §3.8 Nonlinear Equations
    If f ( z 0 ) = 0 and f ( z 0 ) 0 , then z 0 is a simple zero of f . … If ζ is a simple zero, then the iteration converges locally and quadratically. … If the wanted zero ξ is simple, then the method converges locally with order of convergence p = 1 2 ( 1 + 5 ) = 1.618 . … Then the sensitivity of a simple zero z to changes in α is given by …
    9: 8.27 Approximations
  • DiDonato (1978) gives a simple approximation for the function F ( p , x ) = x p e x 2 / 2 x e t 2 / 2 t p d t (which is related to the incomplete gamma function by a change of variables) for real p and large positive x . This takes the form F ( p , x ) = 4 x / h ( p , x ) , approximately, where h ( p , x ) = 3 ( x 2 p ) + ( x 2 p ) 2 + 8 ( x 2 + p ) and is shown to produce an absolute error O ( x 7 ) as x .

  • 10: 11.13 Methods of Computation
    For simple and effective approximations to 𝐇 0 ( z ) and 𝐇 1 ( z ) see Aarts and Janssen (2016). …