expansions in series of

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31: 6.20 Approximations

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§6.20(ii) Expansions in Chebyshev Series

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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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Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$-function (§13.2(i)) from which Chebyshev expansions near infinity for $E_1\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If the scheme can be used in backward direction.

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32: 7.6 Series Expansions

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

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33: 33.9 Expansions in Series of Bessel Functions

§33.9 Expansions in Series of Bessel Functions

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34: 33.20 Expansions for Small $|ϵ|$

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

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35: 28.6 Expansions for Small $q$

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§28.6(i) Eigenvalues

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28.6.19 $$a-{(2n+2)}^2-\frac{q^2}{a-{(2n)}^2-}\frac{q^2}{a-{(2n-2)}^2-}\mathrm{\cdots }\frac{q^2}{a-2^2}=-\frac{q^2}{{(2n+4)}^2-a-}\frac{q^2}{{(2n+6)}^2-a-}\mathrm{\cdots },$$ $a=b_{2n+2}\left(q\right)$.

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Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter, $n$: integer and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.6.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

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36: 25.20 Approximations

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Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover $\zeta \left(s\right)$ for $0\le s\le 1$ (15D), $\zeta \left(s+1\right)$ for $0\le s\le 1$ (20D), and $\ln \xi \left(\frac{1}{2}+\mathrm{i}x\right)$ (§25.4) for $-1\le x\le 1$ (20D). For errata see Piessens and Branders (1972).

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37: 14.18 Sums

… ►For expansions of arbitrary functions in series of Legendre polynomials see §18.18(i), and for expansions of arbitrary functions in series of associated Legendre functions see Schäfke (1961b). …

38: 11.9 Lommel Functions

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§11.9(ii) Expansions in Series of Bessel Functions

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39: 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. … ►For large $x$ and $\left|z\right|$, expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available. …

40: 8.21 Generalized Sine and Cosine Integrals

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Spherical-Bessel-Function Expansions

… ►For (8.21.16), (8.21.17), and further expansions in series of Bessel functions see Luke (1969b, pp. 56–57). …