Neumann-type expansions

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2: 16.22 Asymptotic Expansions

§16.22 Asymptotic Expansions

►Asymptotic expansions of

{G^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right)

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).

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: 29.16 Asymptotic Expansions

§29.16 Asymptotic Expansions

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6: 6.20 Approximations

… ►

§6.20(ii) Expansions in Chebyshev Series

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

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Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\operatorname{Ein}\left(ax\right)$ , $\operatorname{Si}\left(ax\right)$ , and $\operatorname{Cin}\left(ax\right)$ for $-1\leq x\leq 1$ , $a\in\mathbb{C}$ . 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 $|\operatorname{ph}z|<\pi$ the scheme can be used in backward direction.

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§6.20(iii) Padé-Type and Rational Expansions

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7: 20.8 Watson’s Expansions

§20.8 Watson’s Expansions

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

8: 12.16 Mathematical Applications

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

9: 12.20 Approximations

§12.20 Approximations

►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions

U\left(a,b,x\right)

and

M\left(a,b,x\right)

(§13.2(i)) whose regions of validity include intervals with endpoints

x=\infty

and

x=0

, respectively. As special cases of these results a Chebyshev-series expansion for

U\left(a,x\right)

valid when

\lambda\leq x<\infty

follows from (12.7.14), and Chebyshev-series expansions for

U\left(a,x\right)

and

V\left(a,x\right)

valid when

0\leq x\leq\lambda

follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). …

10: 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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