About the Project

Fourier–Bessel expansion

AdvancedHelp

(0.002 seconds)

1—10 of 18 matching pages

1: 10.74 Methods of Computation
FourierBessel Expansion
2: 10.23 Sums
FourierBessel Expansion
3: 28.34 Methods of Computation
  • (b)

    Use of asymptotic expansions and approximations for large q (§§28.8(i), 28.16). See also Zhang and Jin (1996, pp. 482–485).

  • (b)

    Use of asymptotic expansions and approximations for large q (§§28.8(ii)28.8(iv)).

  • (d)

    Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

  • (a)

    Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of q and z .

  • (c)

    Use of asymptotic expansions for large z or large q . See §§28.25 and 28.26.

  • 4: Bibliography O
  • F. Oberhettinger (1973) Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.
  • A. B. Olde Daalhuis (1994) Asymptotic expansions for q -gamma, q -exponential, and q -Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.
  • F. W. J. Olver (1951) A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Proc. Cambridge Philos. Soc. 47, pp. 699–712.
  • F. W. J. Olver (1952) Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.
  • F. W. J. Olver (1954) The asymptotic expansion of Bessel functions of large order. Philos. Trans. Roy. Soc. London. Ser. A. 247, pp. 328–368.
  • 5: Bibliography S
  • O. A. Sharafeddin, H. F. Bowen, D. J. Kouri, and D. K. Hoffman (1992) Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.
  • A. Sidi (2010) A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.
  • D. Slepian and H. O. Pollak (1961) Prolate spheroidal wave functions, Fourier analysis and uncertainty. I. Bell System Tech. J. 40, pp. 43–63.
  • R. S. Strichartz (1994) A Guide to Distribution Theory and Fourier Transforms. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL.
  • S. K. Suslov (2003) An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.
  • 6: Bibliography C
  • S. M. Candel (1981) An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.
  • H. S. Carslaw (1930) Introduction to the Theory of Fourier’s Series and Integrals. 3rd edition, Macmillan, London.
  • I. Cherednik (1995) Macdonald’s evaluation conjectures and difference Fourier transform. Invent. Math. 122 (1), pp. 119–145.
  • H. S. Cohl (2013a) Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.
  • J. P. Coleman and A. J. Monaghan (1983) Chebyshev expansions for the Bessel function J n ( z ) in the complex plane. Math. Comp. 40 (161), pp. 343–366.
  • 7: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
    §28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
    Also, with I n and K n denoting the modified Bessel functions (§10.25(ii)), and again with s = 0 , 1 , 2 , , …
    28.24.10 ε s Ke 2 m ( z , h ) = = 0 A 2 2 m ( h 2 ) A 2 s 2 m ( h 2 ) ( I s ( h e z ) K + s ( h e z ) + I + s ( h e z ) K s ( h e z ) ) ,
    The expansions (28.24.1)–(28.24.13) converge absolutely and uniformly on compact sets of the z -plane. For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).
    8: Bibliography W
  • G. Wolf (2008) On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.
  • R. Wong and J. F. Lin (1978) Asymptotic expansions of Fourier transforms of functions with logarithmic singularities. J. Math. Anal. Appl. 64 (1), pp. 173–180.
  • R. Wong and J.-M. Zhang (1997) Asymptotic expansions of the generalized Bessel polynomials. J. Comput. Appl. Math. 85 (1), pp. 87–112.
  • R. Wong (1973b) On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.
  • E. M. Wright (1935) The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.
  • 9: Bibliography V
  • H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.
  • C. Van Loan (1992) Computational Frameworks for the Fast Fourier Transform. Frontiers in Applied Mathematics, Vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • B. Ph. van Milligen and A. López Fraguas (1994) Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.
  • H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.
  • H. Volkmer (2021) Fourier series representation of Ferrers function P .
  • 10: Bibliography P
  • R. B. Paris (2001a) On the use of Hadamard expansions in hyperasymptotic evaluation. I. Real variables. Proc. Roy. Soc. London Ser. A 457 (2016), pp. 2835–2853.
  • E. Petropoulou (2000) Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.
  • R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.
  • A. Pinkus and S. Zafrany (1997) Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.
  • M. Puoskari (1988) A method for computing Bessel function integrals. J. Comput. Phys. 75 (2), pp. 334–344.