expansions in spherical Bessel functions
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11: 18.15 Asymptotic Approximations
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►These expansions are in terms of Whittaker functions (§13.14).
…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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In Terms of Bessel Functions
… ►For an error bound for the first term in the Airy-function expansions see Olver (1997b, p. 403). …12: 10.49 Explicit Formulas
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§10.49(i) Unmodified Functions
… ►§10.49(ii) Modified Functions
… ► ►§10.49(iii) Rayleigh’s Formulas
… ►§10.49(iv) Sums or Differences of Squares
…13: 18.39 Applications in the Physical Sciences
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►By (1.5.17) the first term in (18.39.21), which is the quantum kinetic energy operator , can be written in spherical coordinates as
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a) Spherical Radial Coulomb Wave Functions Expressed in terms of Laguerre OP’s
… ►c) Spherical Radial Coulomb Wave Functions
… ►See Yamani and Fishman (1975) for for expansions of both the regular and irregular spherical Bessel functions, which are the Pollaczeks with , and Coulomb functions for fixed , Broad and Reinhardt (1976) for a many particle example, and the overview of Alhaidari et al. (2008). …14: 10.75 Tables
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§10.75(ix) Spherical Bessel Functions, Modified Spherical Bessel Functions, and their Derivatives
►Zhang and Jin (1996, pp. 296–305) tabulates , , , , , , , , , 50, 100, , 5, 10, 25, 50, 100, 8S; , , , (Riccati–Bessel functions and their derivatives), , 50, 100, , 5, 10, 25, 50, 100, 8S; real and imaginary parts of , , , , , , , , , 20(10)50, 100, , , 8S. (For the notation replace by , , , , respectively.)
§10.75(x) Zeros and Associated Values of Derivatives of Spherical Bessel Functions
… ►Olver (1960) tabulates , , , , , , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as .
15: Bibliography D
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Complex zeros of linear combinations of spherical Bessel functions and their derivatives.
SIAM J. Math. Anal. 4 (1), pp. 128–133.
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Chebyshev series for the spherical Bessel function
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Comput. Phys. Comm. 18 (1), pp. 73–86.
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Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions.
Proc. Roy. Soc. London Ser. A 440, pp. 37–54.
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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions.
Stud. Appl. Math. 107 (3), pp. 293–323.
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Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions.
SIAM J. Math. Anal. 32 (5), pp. 987–1013.
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16: 10.58 Zeros
§10.58 Zeros
►For the th positive zeros of , , , and are denoted by , , , and , respectively, except that for we count as the first zero of . … ►17: 14.15 Uniform Asymptotic Approximations
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►Here and are the modified Bessel functions (§10.25(ii)).
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►For asymptotic expansions and explicit error bounds, see Dunster (2003b).
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►For the Bessel functions
and see §10.2(ii), and for the
functions associated with and see §2.8(iv).
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►For convergent series expansions see Dunster (2004).
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►See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials as with fixed.
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18: Bibliography H
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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions.
Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).
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Spherical Bessel expansions of sine, cosine, and exponential integrals.
Appl. Numer. Math. 34 (1), pp. 95–98.
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On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order.
Proc. Roy. Soc. London Ser. A 455, pp. 3917–3930.
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Bessel function expansions of Coulomb wave functions.
J. Math. Phys. 26 (4), pp. 656–659.
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Asymptotic expansions of Mathieu functions in wave mechanics.
J. Comput. Phys. 21 (3), pp. 319–325.
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19: Bibliography V
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Expansion of vacuum magnetic fields in toroidal harmonics.
Comput. Phys. Comm. 81 (1-2), pp. 74–90.
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Some novel infinite series of spherical Bessel functions.
Quart. Appl. Math. 42 (3), pp. 321–324.
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Symbolic evaluation of coefficients in Airy-type asymptotic expansions.
J. Math. Anal. Appl. 269 (1), pp. 317–331.
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Expansions in products of Heine-Stieltjes polynomials.
Constr. Approx. 15 (4), pp. 467–480.
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A note on the asymptotic expansion of generalized hypergeometric functions.
Anal. Appl. (Singap.) 12 (1), pp. 107–115.
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20: Bibliography S
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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
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Numerical evaluation of spherical Bessel transforms via fast Fourier transforms.
J. Comput. Phys. 100 (2), pp. 294–296.
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Uniform asymptotic expansions of modified Mathieu functions.
J. Reine Angew. Math. 247, pp. 1–17.
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Asymptotic expansion of Mellin transforms in the complex plane.
Int. J. Pure Appl. Math. 71 (3), pp. 465–480.
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On the calculation of complex zeros of the modified Bessel function of the second kind.
Dokl. Akad. Nauk SSSR 280 (2), pp. 296–299.
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