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asymptotic behavior for large variable

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1: 28.31 Equations of Whittaker–Hill and Ince
Asymptotic Behavior
2: 30.11 Radial Spheroidal Wave Functions
§30.11(iii) Asymptotic Behavior
3: 15.12 Asymptotic Approximations
§15.12(i) Large Variable
For the asymptotic behavior of F ( a , b ; c ; z ) as z with a , b , c fixed, combine (15.2.2) with (15.8.2) or (15.8.8).
§15.12(ii) Large c
As λ , … For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).
4: 30.9 Asymptotic Approximations and Expansions
§30.9 Asymptotic Approximations and Expansions
§30.9(i) Prolate Spheroidal Wave Functions
The asymptotic behavior of λ n m ( γ 2 ) and a n , k m ( γ 2 ) as n in descending powers of 2 n + 1 is derived in Meixner (1944). …The behavior of λ n m ( γ 2 ) for complex γ 2 and large | λ n m ( γ 2 ) | is investigated in Hunter and Guerrieri (1982).
5: 25.11 Hurwitz Zeta Function
§25.11(xii) a -Asymptotic Behavior
25.11.41 ζ ( s , a + 1 ) = ζ ( s ) - s ζ ( s + 1 ) a + O ( a 2 ) .
25.11.42 ζ ( s , α + i β ) 0 ,
As a in the sector | ph a | π - δ ( < π ) , with s ( 1 ) and δ fixed, we have the asymptotic expansion … Similarly, as a in the sector | ph a | 1 2 π - δ ( < 1 2 π ) , …
6: 18.15 Asymptotic Approximations
§18.15 Asymptotic Approximations
§18.15(i) Jacobi
For large β , fixed α , and 0 n / β c , Dunster (1999) gives asymptotic expansions of P n ( α , β ) ( z ) that are uniform in unbounded complex z -domains containing z = ± 1 . …This reference also supplies asymptotic expansions of P n ( α , β ) ( z ) for large n , fixed α , and 0 β / n c . … The asymptotic behavior of the classical OP’s as x ± with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. …
7: 19.12 Asymptotic Approximations
§19.12 Asymptotic Approximations
With ψ ( x ) denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of K ( k ) and E ( k ) near the singularity at k = 1 is given by the following convergent series: … For the asymptotic behavior of F ( ϕ , k ) and E ( ϕ , k ) as ϕ 1 2 π - and k 1 - see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007). … Asymptotic approximations for Π ( ϕ , α 2 , k ) , with different variables, are given in Karp et al. (2007). They are useful primarily when ( 1 - k ) / ( 1 - sin ϕ ) is either small or large compared with 1. …
8: 2.4 Contour Integrals
Then … For large t , the asymptotic expansion of q ( t ) may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function F ( z ) for Q ( z ) that has an inverse transform …with known asymptotic behavior as t + . …If this integral converges uniformly at each limit for all sufficiently large t , then by the Riemann–Lebesgue lemma (§1.8(i)) … For integral representations of the b 2 s and their asymptotic behavior as s see Boyd (1995). …
9: Bibliography B
  • C. M. Bender and T. T. Wu (1973) Anharmonic oscillator. II. A study of perturbation theory in large order. Phys. Rev. D 7, pp. 1620–1636.
  • L. C. Biedenharn, R. L. Gluckstern, M. H. Hull, and G. Breit (1955) Coulomb functions for large charges and small velocities. Phys. Rev. (2) 97 (2), pp. 542–554.
  • R. Bo and R. Wong (1996) Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.
  • A. A. Bogush and V. S. Otchik (1997) Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.
  • W. Bühring (1987b) The behavior at unit argument of the hypergeometric function F 2 3 . SIAM J. Math. Anal. 18 (5), pp. 1227–1234.
  • 10: 8.13 Zeros
    When - 5 a 4 the behavior of the x -zeros as functions of a can be seen by taking the slice γ * ( a , x ) = 0 of the surface depicted in Figure 8.3.6. … For asymptotic approximations for x + ( a ) and x - ( a ) as a - see Tricomi (1950b), with corrections by Kölbig (1972b). For more accurate asymptotic approximations see Thompson (2012). … For information on the distribution and computation of zeros of γ ( a , λ a ) and Γ ( a , λ a ) in the complex λ -plane for large values of the positive real parameter a see Temme (1995a). … Approximations to a n * , x n * for large n can be found in Kölbig (1970). …