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1: Bibliography Q
  • F. Qi (2008) A new lower bound in the second Kershaw’s double inequality. J. Comput. Appl. Math. 214 (2), pp. 610–616.
  • C. K. Qu and R. Wong (1999) “Best possible” upper and lower bounds for the zeros of the Bessel function J ν ( x ) . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.
  • 2: 29.16 Asymptotic Expansions
    Hargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. …
    3: 13.12 Products
    4: 29.9 Stability
    The Lamé equation (29.2.1) with specified values of k , h , ν is called stable if all of its solutions are bounded on ; otherwise the equation is called unstable. …
    5: Bibliography N
  • G. Nemes (2013b) Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.
  • G. Nemes (2014) Error bounds and exponential improvement for the asymptotic expansion of the Barnes G -function. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2172), pp. 20140534, 14.
  • G. Nemes (2015a) Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal. Proc. Roy. Soc. Edinburgh Sect. A 145 (3), pp. 571–596.
  • E. Neuman (2003) Bounds for symmetric elliptic integrals. J. Approx. Theory 122 (2), pp. 249–259.
  • E. Neuman (2013) Inequalities and bounds for the incomplete gamma function. Results Math. 63 (3-4), pp. 1209–1214.
  • 6: 6.12 Asymptotic Expansions
    When | ph z | 1 2 π the remainder is bounded in magnitude by the first neglected term, and has the same sign when ph z = 0 . When 1 2 π | ph z | < π the remainder term is bounded in magnitude by csc ( | ph z | ) times the first neglected term. For these and other error bounds see Olver (1997b, pp. 109–112) with α = 0 . … If the expansion is terminated at the n th term, then the remainder term is bounded by 1 + χ ( n + 1 ) times the next term. … When | ph z | 1 4 π , these remainders are bounded in magnitude by the first neglected terms in (6.12.3) and (6.12.4), respectively, and have the same signs as these terms when ph z = 0 . …
    7: 10.17 Asymptotic Expansions for Large Argument
    §10.17(iii) Error Bounds for Real Argument and Order
    §10.17(iv) Error Bounds for Complex Argument and Order
    Bounds for 𝒱 z , i ( t - ) are given by … Corresponding error bounds for (10.17.3) and (10.17.4) are obtainable by combining (10.17.13) and (10.17.14) with (10.4.4). … For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).
    8: Bibliography O
  • F. W. J. Olver (1964b) Error bounds for asymptotic expansions in turning-point problems. J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.
  • F. W. J. Olver (1967b) Bounds for the solutions of second-order linear difference equations. J. Res. Nat. Bur. Standards Sect. B 71B (4), pp. 161–166.
  • F. W. J. Olver (1974) Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.
  • F. W. J. Olver (1976) Improved error bounds for second-order differential equations with two turning points. J. Res. Nat. Bur. Standards Sect. B 80B (4), pp. 437–440.
  • F. W. J. Olver (1980a) Asymptotic approximations and error bounds. SIAM Rev. 22 (2), pp. 188–203.
  • 9: 18.14 Inequalities
    §18.14(i) Upper Bounds
    Jacobi
    Ultraspherical
    Laguerre
    Hermite
    10: 27.18 Methods of Computation: Primes
    The Sieve of Eratosthenes (Crandall and Pomerance (2005, §3.2)) generates a list of all primes below a given bound. …