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11: 10.40 Asymptotic Expansions for Large Argument
§10.40(ii) Error Bounds for Real Argument and Order
§10.40(iii) Error Bounds for Complex Argument and Order
Bounds for 𝒱 z , ( t ) are given by … If z with | 2 | z | | bounded and m ( 0 ) fixed, then …
12: 13.7 Asymptotic Expansions for Large Argument
§13.7(ii) Error Bounds
See accompanying text
Figure 13.7.1: Regions R 1 , R 2 , R ¯ 2 , R 3 , and R ¯ 3 are the closures of the indicated unshaded regions bounded by the straight lines and circular arcs centered at the origin, with r = | b 2 a | . Magnify
Corresponding error bounds for (13.7.2) can be constructed by combining (13.2.41) with (13.7.4)–(13.7.9). … Then as z with | | z | n | bounded and a , b , m fixed …
13: Bibliography Y
  • K. Yang and M. de Llano (1989) Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.
  • 14: 9.7 Asymptotic Expansions
    §9.7(iii) Error Bounds for Real Variables
    In (9.7.9)–(9.7.12) the n th error term in each infinite series is bounded in magnitude by the first neglected term and has the same sign, provided that the following term in the series is of opposite sign. …
    §9.7(iv) Error Bounds for Complex Variables
    The n th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by … Corresponding bounds for the errors in (9.7.7) to (9.7.14) may be obtained by use of these results and those of §9.2(v) and their differentiated forms. …
    15: 7.12 Asymptotic Expansions
    When | ph z | 1 4 π the remainder terms are bounded in magnitude by the first neglected terms, and have the same sign as these terms when ph z = 0 . When 1 4 π | ph z | < 1 2 π the remainder terms are bounded in magnitude by csc ( 2 | ph z | ) times the first neglected terms. For these and other error bounds see Olver (1997b, pp. 109–112), with α = 1 2 and z replaced by z 2 ; compare (7.11.2). … When | ph z | 1 8 π , R n ( f ) ( z ) and R n ( g ) ( z ) are bounded in magnitude by the first neglected terms in (7.12.2) and (7.12.3), respectively, and have the same signs as these terms when ph z = 0 . … See Olver (1997b, p. 115) for an expansion of G ( z ) with bounds for the remainder for real and complex values of z .
    16: 8.11 Asymptotic Approximations and Expansions
    For bounds on R n ( a , z ) when a is real and z is complex see Olver (1997b, pp. 109–112). … Sharp error bounds and an exponentially-improved extension for (8.11.7) can be found in Nemes (2016). … For error bounds and an exponentially-improved extension for this later expansion, see Nemes (2015c). … in both cases uniformly with respect to bounded real values of y . … For sharp error bounds and an exponentially-improved extension, see Nemes (2016). …
    17: 12.9 Asymptotic Expansions for Large Variable
    §12.9(ii) Bounds and Re-Expansions for the Remainder Terms
    Bounds and re-expansions for the error term in (12.9.1) can be obtained by use of (12.7.14) and §§13.7(ii), 13.7(iii). …
    18: 13.19 Asymptotic Expansions for Large Argument
    Error bounds and exponentially-improved expansions are derivable by combining §§13.7(ii) and 13.7(iii) with (13.14.2) and (13.14.3). …
    19: 14.31 Other Applications
    The conical functions 𝖯 1 2 + i τ m ( x ) appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). …
    20: 18.14 Inequalities
    §18.14(i) Upper Bounds
    Jacobi
    Ultraspherical
    Laguerre
    Hermite