asymptotic behavior for large variable
1—10 of 13 matching pages
§30.11(iii) Asymptotic Behavior…
§15.12(i) Large Variable►For the asymptotic behavior of as with , , fixed, combine (15.2.2) with (15.8.2) or (15.8.8). ►
§15.12(ii) Large… ►As , … ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).
§30.9 Asymptotic Approximations and Expansions►
§30.9(i) Prolate Spheroidal Wave Functions… ►The asymptotic behavior of and as in descending powers of is derived in Meixner (1944). …The behavior of for complex and large is investigated in Hunter and Guerrieri (1982).
§25.11(xii) -Asymptotic Behavior… ►
25.11.42… ►As in the sector , with and fixed, we have the asymptotic expansion … ►Similarly, as in the sector , …
§18.15 Asymptotic Approximations►
§18.15(i) Jacobi… ►For large , fixed , and , Dunster (1999) gives asymptotic expansions of that are uniform in unbounded complex -domains containing . …This reference also supplies asymptotic expansions of for large , fixed , and . … ►The asymptotic behavior of the classical OP’s as 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. …
§19.12 Asymptotic Approximations►With denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of and near the singularity at is given by the following convergent series: … ►For the asymptotic behavior of and as and see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007). … ►Asymptotic approximations for , with different variables, are given in Karp et al. (2007). They are useful primarily when is either small or large compared with 1. …
… ►Then … ►For large , the asymptotic expansion of may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function for that has an inverse transform …with known asymptotic behavior as . …If this integral converges uniformly at each limit for all sufficiently large , then by the Riemann–Lebesgue lemma (§1.8(i)) … ►For integral representations of the and their asymptotic behavior as see Boyd (1995). …
Anharmonic oscillator. II. A study of perturbation theory in large order.
Phys. Rev. D 7, pp. 1620–1636.
Coulomb functions for large charges and small velocities.
Phys. Rev. (2) 97 (2), pp. 542–554.
Asymptotic behavior of the Pollaczek polynomials and their zeros.
Stud. Appl. Math. 96, pp. 307–338.
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.
The behavior at unit argument of the hypergeometric function
SIAM J. Math. Anal. 18 (5), pp. 1227–1234.
10: 8.13 Zeros
… ►When the behavior of the -zeros as functions of can be seen by taking the slice of the surface depicted in Figure 8.3.6. … ►For asymptotic approximations for and as 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 and in the complex -plane for large values of the positive real parameter see Temme (1995a). … ►Approximations to , for large can be found in Kölbig (1970). …