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11: 10.40 Asymptotic Expansions for Large Argument
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§10.40(ii) Error Bounds for Real Argument and Order
… ►§10.40(iii) Error Bounds for Complex Argument and Order
… ►Bounds for are given by … ►If with bounded and fixed, then …12: 13.7 Asymptotic Expansions for Large Argument
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§13.7(ii) Error Bounds
► … ►Corresponding error bounds for (13.7.2) can be constructed by combining (13.2.41) with (13.7.4)–(13.7.9). … ►Then as with bounded and fixed …13: Bibliography Y
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Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State.
American Journal of Physics 57 (1), pp. 85–86.
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14: 9.7 Asymptotic Expansions
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§9.7(iii) Error Bounds for Real Variables
… ►In (9.7.9)–(9.7.12) the 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 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
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►When the remainder terms are bounded in magnitude by the first neglected terms, and have the same sign as these terms when .
When the remainder terms are bounded in magnitude by times the first neglected terms.
For these and other error bounds see Olver (1997b, pp. 109–112), with and replaced by ; compare (7.11.2).
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►When , and 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 .
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►See Olver (1997b, p. 115) for an expansion of with bounds for the remainder for real and complex values of .
16: 8.11 Asymptotic Approximations and Expansions
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►For bounds on when is real and is complex see Olver (1997b, pp. 109–112).
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►Sharp error bounds and an exponentially-improved extension for (8.11.7) can be found in Nemes (2016).
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►For error bounds and an exponentially-improved extension for this later expansion, see Nemes (2015c).
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►in both cases uniformly with respect to bounded real values of .
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►For sharp error bounds and an exponentially-improved extension, see Nemes (2016).
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17: 12.9 Asymptotic Expansions for Large Variable
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§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
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►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).
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19: 14.31 Other Applications
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►The conical functions 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)).
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