Fourier%E2%80%93Bessel%20expansion
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1: 10.1 Special Notation
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►The main functions treated in this chapter are the Bessel functions , ; Hankel functions , ; modified Bessel functions , ; spherical Bessel functions , , , ; modified spherical Bessel functions , , ; Kelvin functions , , , .
For the spherical Bessel functions and modified spherical Bessel functions the order is a nonnegative integer.
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►Abramowitz and Stegun (1964): , , , , for , , , , respectively, when .
►Jeffreys and Jeffreys (1956): for , for , for .
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►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
2: 35.5 Bessel Functions of Matrix Argument
§35.5 Bessel Functions of Matrix Argument
►§35.5(i) Definitions
… ►§35.5(ii) Properties
… ►§35.5(iii) Asymptotic Approximations
►For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).3: 28.4 Fourier Series
§28.4 Fourier Series
… ►§28.4(ii) Recurrence Relations
… ►§28.4(iii) Normalization
… ►§28.4(v) Change of Sign of
… ►§28.4(vi) Behavior for Small
…4: 1.8 Fourier Series
§1.8 Fourier Series
… ► … ►Uniqueness of Fourier Series
… ►For collections of Fourier-series expansions see Prudnikov et al. (1986a, v. 1, pp. 725–740), Gradshteyn and Ryzhik (2000, pp. 45–49), and Oberhettinger (1973).5: 1.14 Integral Transforms
6: 29.20 Methods of Computation
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►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i).
Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6.
…The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii).
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►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv).
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►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials.
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7: Bibliography D
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Computing spectra of linear operators using the Floquet-Fourier-Hill method.
J. Comput. Phys. 219 (1), pp. 296–321.
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Sur les zéros réels des polynômes de Bernoulli.
Ann. Inst. Fourier (Grenoble) 41 (2), pp. 267–309 (French).
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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters.
J. Number Theory 25 (1), pp. 72–80.
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Error analysis in a uniform asymptotic expansion for the generalised exponential integral.
J. Comput. Appl. Math. 80 (1), pp. 127–161.
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Uniform asymptotic expansions for Charlier polynomials.
J. Approx. Theory 112 (1), pp. 93–133.
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8: Guide to Searching the DLMF
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Table 1: Query Examples
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►For example, for the Bessel function , you can write
Query | Matching records contain |
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"Fourier transform" and series |
both the phrase “Fourier transform” and the word “series”. |
Fourier or series |
at least one of the words “Fourier” or “series”. |
Fourier (transform or series) |
at least one of “Fourier transform” or “Fourier series”. |
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K_n(z)
, BesselK_n(z)
, BesselK(n,z)
, or BesselK[n,z]
.
Note that the first form may match other functions than the Bessel
function, so if you are sure you want Bessel
, you might as well enter one of the other 3 forms.
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9: 27.17 Other Applications
§27.17 Other Applications
►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …10: Bibliography H
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The Laplace transform for expressions that contain a probability function.
Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
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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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Integrals that contain a probability function of complicated arguments.
Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).
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Sums with cylindrical functions that reduce to the probability function and to related functions.
Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).
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Applied and Computational Complex Analysis. Vol. 3: Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions.
Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons Inc.], New York.
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