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11: Bibliography O
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Hyperasymptotic solutions of second-order linear differential equations. I.
Methods Appl. Anal. 2 (2), pp. 173–197.
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Hyperasymptotic solutions of second-order linear differential equations. II.
Methods Appl. Anal. 2 (2), pp. 198–211.
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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one.
Proc. Roy. Soc. London Ser. A 454, pp. 1–29.
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Hyperasymptotics for nonlinear ODEs. II. The first Painlevé equation and a second-order Riccati equation.
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2062), pp. 3005–3021.
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A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations.
Proc. Cambridge Philos. Soc. 46 (4), pp. 570–580.
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12: Bibliography D
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A simplified algorithm for the second-order sound fields.
J. Acoust. Soc. Amer. 108 (6), pp. 2759–2764.
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Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point.
SIAM J. Math. Anal. 21 (6), pp. 1594–1618.
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Uniform asymptotic expansions for associated Legendre functions of large order.
Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.
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Conical functions of purely imaginary order and argument.
Proc. Roy. Soc. Edinburgh Sect. A 143 (5), pp. 929–955.
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A survey on orthogonal matrix polynomials satisfying second order differential equations.
J. Comput. Appl. Math. 178 (1-2), pp. 169–190.
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13: 16.21 Differential Equation
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satisfies the differential equation
…This equation is of order
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In consequence of (16.19.1) we may assume, without loss of generality, that .
With the classification of §16.8(i), when the only singularities of (16.21.1) are a regular singularity at and an irregular singularity at .
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►A fundamental set of solutions of (16.21.1) is given by
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14: 10.57 Uniform Asymptotic Expansions for Large Order
15: 10.72 Mathematical Applications
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►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter.
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►In regions in which (10.72.1) has a simple turning point , that is, and are analytic (or with weaker conditions if is a real variable) and is a simple zero of , asymptotic expansions of the solutions for large can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order
(§9.6(i)).
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►If has a double zero , or more generally is a zero of order
, , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order
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…The order of the approximating Bessel functions, or modified Bessel functions, is , except in the case when has a double pole at .
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►Then for large asymptotic approximations of the solutions can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on and ).
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16: 11.10 Anger–Weber Functions
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►The associated Anger–Weber function is defined by
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11.10.6
,
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11.10.7
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11.10.9
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►where
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17: 10.26 Graphics
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