expansions in Bessel functions

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1: 33.20 Expansions for Small $|ϵ|$

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§33.20(i) Case $ϵ=0$

… ►where $A(ϵ,\mathrm{\ell })$ is given by (33.14.11), (33.14.12), and ►

33.20.8 $${𝖧}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(2r)}^{(p+1)/2}{C}_{k,p}{Y}_{2\mathrm{\ell }+1+p}\left(\sqrt{8r}\right),$$ $r>0$,

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $C_{k,p}$: coefficients and ${𝖧}_k(\mathrm{\ell };r)$: coefficient

Permalink:

http://dlmf.nist.gov/33.20.E8

Encodings:

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See also:

Annotations for §33.20(iii), §33.20 and Ch.33

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§33.20(iv) Uniform Asymptotic Expansions

… ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

2: 13.24 Series

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§13.24(ii) Expansions in Series of Bessel Functions

… ►Additional expansions in terms of Bessel functions are given in Luke (1959). …

3: 30.10 Series and Integrals

… ►For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).

4: 12.18 Methods of Computation

… ►These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …

5: 6.10 Other Series Expansions

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§6.10(ii) Expansions in Series of Spherical Bessel Functions

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6: 10.66 Expansions in Series of Bessel Functions

§10.66 Expansions in Series of Bessel Functions

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7: 33.9 Expansions in Series of Bessel Functions

§33.9 Expansions in Series of Bessel Functions

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§33.9(i) Spherical Bessel Functions

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§33.9(ii) Bessel Functions and Modified Bessel Functions

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8: 10.23 Sums

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Partial Fractions

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§10.23(iii) Series Expansions of Arbitrary Functions

… ►For other types of expansions of arbitrary functions in series of Bessel functions, see Watson (1944, Chapters 17–19) and Erdélyi et al. (1953b, §§ 7.10.2–7.10.4). … … ►For collections of sums of series involving Bessel or Hankel functions see Erdélyi et al. (1953b, §7.15), Gradshteyn and Ryzhik (2000, §§8.51–8.53), Hansen (1975), Luke (1969b, §9.4), Prudnikov et al. (1986b, pp. 651–691 and 697–700), and Wheelon (1968, pp. 48–51).

9: 7.6 Series Expansions

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§7.6(ii) Expansions in Series of Spherical Bessel Functions

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10: 8.7 Series Expansions

§8.7 Series Expansions

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8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$, .

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Proof sketch:

Integrate (18.12.13) from $z=0$ to $z=1$ and for the integral on the left-hand side use the substitution $z=\frac{t}{1+t}$. The resulting integral is (8.6.5). The constraint follows from (18.15.14).

Referenced by:

Erratum (V1.1.8) for Equation (8.7.6)

Permalink:

http://dlmf.nist.gov/8.7.E6

Encodings:

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Addition (effective with 1.1.8):

The constraint was updated to include .

Suggested 2022-10-14 by Walter Gautschi

See also:

Annotations for §8.7 and Ch.8

►For an expansion for $\gamma (a,\mathrm{i}x)$ in series of Bessel functions $J_n\left(x\right)$ that converges rapidly when $a>0$ and $x$ ($\ge 0$) is small or moderate in magnitude see Barakat (1961).