asymptotic and order symbols
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1: 2.1 Definitions and Elementary Properties
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§2.1(i) Asymptotic and Order Symbols
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2.1.12
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2: 13.9 Zeros
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13.9.16
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3: 18.15 Asymptotic Approximations
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18.15.22
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4: Errata
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Equation (13.9.16)
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Equation (18.15.22)
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Originally was expressed in term of asymptotic symbol . As a consequence of the use of the order symbol on the right-hand side, was replaced by .
Because of the use of the order symbol on the right-hand side, the asymptotic expansion for the generalized Laguerre polynomial was rewritten as an equality.
5: 11.11 Asymptotic Expansions of Anger–Weber Functions
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11.11.11
, ,
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6: Bibliography K
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Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument.
ACM Trans. Math. Software 34 (4), pp. Art. 22, 21.
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Algorithm 912: a module for calculating cylindrical functions of complex order and complex argument.
ACM Trans. Math. Software 37 (4), pp. Art. 47, 25.
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Some completely monotonic functions of positive order.
Math. Comp. 79 (271), pp. 1697–1707.
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An algorithm for solving second order linear homogeneous differential equations.
J. Symbolic Comput. 2 (1), pp. 3–43.
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Asymptotic solution of Maxwell’s equations near caustics.
Izv. Vuz. Radiofiz. 7, pp. 1049–1056.
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7: 11.9 Lommel Functions
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11.9.1
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11.9.4
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§11.9(iii) Asymptotic Expansion
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11.9.9
, .
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►For uniform asymptotic expansions, for large and fixed , of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390).
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8: Bibliography W
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Asymptotic expansions for second-order linear difference equations with a turning point.
Numer. Math. 94 (1), pp. 147–194.
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Asymptotic approximations for certain - and -
symbols.
J. Phys. A 32 (39), pp. 6901–6902.
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Asymptotic expansions for second-order linear difference equations. II.
Stud. Appl. Math. 87 (4), pp. 289–324.
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Asymptotic expansions for second-order linear difference equations.
J. Comput. Appl. Math. 41 (1-2), pp. 65–94.
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Asymptotic solutions of a fourth order differential equation.
Stud. Appl. Math. 118 (2), pp. 133–152.
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9: 14.8 Behavior at Singularities
10: Bibliography T
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The asymptotic expansion of the incomplete gamma functions.
SIAM J. Math. Anal. 10 (4), pp. 757–766.
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Laguerre polynomials: Asymptotics for large degree.
Technical report
Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.
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Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z.
Indagationes Mathematicae.
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Eigenfunction Expansions Associated with Second-Order Differential Equations.
Clarendon Press, Oxford.
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Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades.
J. Appl. Math. Mech. 23, pp. 1549–1565.
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