expansions in Bessel functions
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21: 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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►In regions in which the function
has a simple pole at and is analytic at (the case
in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large can be constructed in terms of Bessel functions and modified Bessel functions of order , where is the limiting value of as .
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22: 30.9 Asymptotic Approximations and Expansions
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►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986).
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►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995).
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23: 11.10 Anger–Weber Functions
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§11.10(viii) Expansions in Series of Products of Bessel Functions
…24: 10.41 Asymptotic Expansions for Large Order
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25: 18.15 Asymptotic Approximations
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►The latter expansions are in terms of Bessel functions, and are uniform in complex -domains not containing neighborhoods of 1.
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►These expansions are in terms of Bessel functions and modified Bessel functions, respectively.
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26: 10.74 Methods of Computation
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►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument or is sufficiently small in absolute value.
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►In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions.
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27: 10.40 Asymptotic Expansions for Large Argument
§10.40 Asymptotic Expansions for Large Argument
►§10.40(i) Hankel’s Expansions
… ►Products
… ► … ►§10.40(iv) Exponentially-Improved Expansions
…28: Bibliography L
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Expansion of the confluent hypergeometric function in series of Bessel functions.
Math. Tables Aids Comput. 13 (68), pp. 261–271.
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29: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
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►For asymptotic expansions of as
in various sectors of the complex -plane for fixed real values of and fixed real or complex values of , see Wright (1935) when , and Wright (1940b) when .
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