expansions in modified Bessel functions

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1: 13.11 Series

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$(n+1)A_{n+1}=(n+b-1)A_{n-1}+(2a-b)A_{n-2},$ $n=2,3,4,\mathrm{\dots }$.

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2: 33.20 Expansions for Small $|ϵ|$

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§33.20(i) Case $ϵ=0$

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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:

pMML, png, TeX

See also:

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

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3: 6.10 Other Series Expansions

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

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4: 13.24 Series

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§13.24(ii) Expansions in Series of Bessel Functions

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5: 10.44 Sums

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§10.44(iii) Neumann-Type Expansions

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6: 33.9 Expansions in Series of Bessel Functions

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§33.9(ii) Bessel Functions and Modified Bessel Functions

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7: 28.23 Expansions in Series of Bessel Functions

§28.23 Expansions in Series of Bessel Functions

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8: 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

… ►For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).

9: 8.7 Series Expansions

§8.7 Series Expansions

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10: 10.72 Mathematical Applications

… ►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). … ►If $f(z)$ has a double zero $z_0$, or more generally $z_0$ is a zero of order $m$, $m=2,3,4,\mathrm{\dots }$, then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. … ►In regions in which the function $f(z)$ has a simple pole at $z=z_0$ and ${(z-z_0)}^{2}g(z)$ is analytic at $z=z_0$ (the case $\lambda =-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm \sqrt{1+4\rho }$, where $\rho$ is the limiting value of ${(z-z_0)}^{2}g(z)$ as $z\to z_0$. …