Fourier?Bessel expansion

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1: 16.22 Asymptotic Expansions

§16.22 Asymptotic Expansions

►Asymptotic expansions of $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for large $z$ are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer $G$-functions with large parameters see Fields (1973, 1983).

2: 4.47 Approximations

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§4.47(i) Chebyshev-Series Expansions

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3: 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. …

4: 29.16 Asymptotic Expansions

§29.16 Asymptotic Expansions

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5: 6.20 Approximations

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§6.20(ii) Expansions in Chebyshev Series

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Luke and Wimp (1963) covers $\mathrm{Ei}\left(x\right)$ for $x\le -4$ (20D), and $\mathrm{Si}\left(x\right)$ and $\mathrm{Ci}\left(x\right)$ for $x\ge 4$ (20D).

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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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§6.20(iii) Padé-Type and Rational Expansions

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6: 20.8 Watson’s Expansions

§20.8 Watson’s Expansions

… ►This reference and Bellman (1961, pp. 46–47) include other expansions of this type.

7: 12.16 Mathematical Applications

… ► … ►Integral transforms and sampling expansions are considered in Jerri (1982).

8: 12.20 Approximations

§12.20 Approximations

►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $U(a,b,x)$ and $M(a,b,x)$ (§13.2(i)) whose regions of validity include intervals with endpoints $x=\mathrm{\infty }$ and $x=0$, respectively. As special cases of these results a Chebyshev-series expansion for $U(a,x)$ valid when follows from (12.7.14), and Chebyshev-series expansions for $U(a,x)$ and $V(a,x)$ valid when $0\le x\le \lambda$ follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). …

9: 6.12 Asymptotic Expansions

§6.12 Asymptotic Expansions

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§6.12(i) Exponential and Logarithmic Integrals

… ►For the function $\chi$ see §9.7(i). … ►

§6.12(ii) Sine and Cosine Integrals

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10: 28.16 Asymptotic Expansions for Large $q$

§28.16 Asymptotic Expansions for Large $q$

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28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

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See also:

Annotations for §28.16 and Ch.28

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