expansions in series of Bessel functions

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21: 6.18 Methods of Computation

… ►For small or moderate values of $x$ and $|z|$, the expansion in power series (§6.6) or in series of spherical Bessel functions (§6.10(ii)) can be used. …

22: 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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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

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23: 6.20 Approximations

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Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\mathrm{Ein}\left(ax\right)$, $\mathrm{Si}\left(ax\right)$, and $\mathrm{Cin}\left(ax\right)$ for $-1\le x\le 1$, $a\in ℂ$. The coefficients are given in terms of series of Bessel functions.

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24: 10.74 Methods of Computation

… ►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 $x$ or $z$ is sufficiently small in absolute value. …

25: 18.18 Sums

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Legendre

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Laguerre

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Hermite

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Ultraspherical

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Hermite

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26: 33.20 Expansions for Small $|ϵ|$

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§33.20(i) Case $ϵ=0$

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§33.20(ii) Power-Series in $ϵ$ for the Regular Solution

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

§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 $2\mathrm{\ell }+2$.

27: 11.13 Methods of Computation

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§11.13(i) Introduction

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§11.13(ii) Series Expansions

►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. For large $|z|$ and/or $|\nu |$ the asymptotic expansions given in §11.6 should be used instead. … ►Other integrals that appear in §11.5(i) have highly oscillatory integrands unless $z$ is small. …

28: 10.31 Power Series

§10.31 Power Series

►For $I_{\nu }\left(z\right)$ see (10.25.2) and (10.27.1). When $\nu$ is not an integer the corresponding expansion for $K_{\nu }\left(z\right)$ is obtained from (10.25.2) and (10.27.4). … ►In particular, ►

10.31.2 $$K_0\left(z\right)=-\left(\ln \left(\frac{1}{2}z\right)+\gamma \right)I_0\left(z\right)+\frac{\frac{1}{4}{z}^2}{{(1!)}^2}+(1+\frac{1}{2})\frac{{(\frac{1}{4}{z}^2)}^2}{{(2!)}^2}+(1+\frac{1}{2}+\frac{1}{3})\frac{{(\frac{1}{4}{z}^2)}^3}{{(3!)}^2}+\mathrm{\cdots }.$$

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Symbols:

$\gamma$: Euler’s constant, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.6.13

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

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29: 10.35 Generating Function and Associated Series

§10.35 Generating Function and Associated Series

►For $z\in ℂ$ and $t\in ℂ\setminus \{0\}$, … ►Jacobi–Anger expansions: for $z,\theta \in ℂ$, … ►

10.35.4 $$1=I_0\left(z\right)-2I_2\left(z\right)+2I_4\left(z\right)-2I_6\left(z\right)+\mathrm{\cdots },$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: complex variable

A&S Ref:

9.6.36

Permalink:

http://dlmf.nist.gov/10.35.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.35 and Ch.10

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10.35.5 $${\mathrm{e}}^{\pm z}=I_0\left(z\right)\pm 2I_1\left(z\right)+2I_2\left(z\right)\pm 2I_3\left(z\right)+\mathrm{\cdots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: complex variable

A&S Ref:

9.6.37,9.6.38

Permalink:

http://dlmf.nist.gov/10.35.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.35 and Ch.10

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30: 10.12 Generating Function and Associated Series

§10.12 Generating Function and Associated Series

►For $z\in ℂ$ and $t\in ℂ\setminus \{0\}$, ►

10.12.1 $${\mathrm{e}}^{\frac{1}{2}z(t-t^{-1})}=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{t}^m{J}_m\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, $m$: integer and $z$: complex variable

A&S Ref:

9.1.41

Referenced by:

§10.12, §10.23(iii), §10.35

Permalink:

http://dlmf.nist.gov/10.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.12 and Ch.10

►Jacobi–Anger expansions: for $z,\theta \in ℂ$, … ►

10.12.4 $$1=J_0\left(z\right)+2J_2\left(z\right)+2J_4\left(z\right)+2J_6\left(z\right)+\mathrm{\cdots },$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind and $z$: complex variable

A&S Ref:

9.1.46

Referenced by:

§10.74(iv)

Permalink:

http://dlmf.nist.gov/10.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.12 and Ch.10

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