Chebyshev series

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2: 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). …

3: 6.20 Approximations

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§6.20(ii) Expansions in Chebyshev Series

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Clenshaw (1962) gives Chebyshev coefficients for $-E_{1}\left(x\right)-\ln|x|$ for $-4\leq x\leq 4$ and $e^{x}E_{1}\left(x\right)$ for $x\geq 4$ (20D).

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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. 321–322) covers $\operatorname{Ein}\left(x\right)$ and $-\operatorname{Ein}\left(-x\right)$ for $0\leq x\leq 8$ (the Chebyshev coefficients are given to 20D); $E_{1}\left(x\right)$ for $x\geq 5$ (20D), and $\operatorname{Ei}\left(x\right)$ for $x\geq 8$ (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

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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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4: 25.20 Approximations

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Cody et al. (1971) gives rational approximations for $\zeta\left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\leq s\leq 5$ , $5\leq s\leq 11$ , $11\leq s\leq 25$ , $25\leq s\leq 55$ . Precision is varied, with a maximum of 20S.

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Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta\left(s+1\right)$ and $\zeta\left(s+k\right)$ , $k=2,3,4,5,8$ , for $0\leq s\leq 1$ (23D).

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

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Antia (1993) gives minimax rational approximations for $\Gamma\left(s+1\right)F_{s}(x)$ , where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals $-\infty<x\leq 2$ and $2\leq x<\infty$ , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$ . For each $s$ there are three sets of approximations, with relative maximum errors $10^{-4},10^{-8},10^{-12}$ .

5: 5.23 Approximations

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§5.23(ii) Expansions in Chebyshev Series

►Luke (1969b) gives the coefficients to 20D for the Chebyshev-series expansions of

\Gamma\left(1+x\right)

1/\Gamma\left(1+x\right)

\Gamma\left(x+3\right)

\ln\Gamma\left(x+3\right)

\psi\left(x+3\right)

, and the first six derivatives of

\psi\left(x+3\right)

for

0\leq x\leq 1

. …Clenshaw (1962) also gives 20D Chebyshev-series coefficients for

\Gamma\left(1+x\right)

and its reciprocal for

0\leq x\leq 1

. …

6: 8.27 Approximations

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Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for $\Gamma\left(a,\omega z\right)$ (by specifying parameters) with $1\leq\omega<\infty$ , and $\gamma\left(a,\omega z\right)$ with $0\leq\omega\leq 1$ ; see also Temme (1994b, §3).

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Luke (1975, p. 103) gives Chebyshev-series expansions for $E_{1}\left(x\right)$ and related functions for $x\geq 5$ .

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7: 9.19 Approximations

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§9.19(ii) Expansions in Chebyshev Series

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MacLeod (1994) supplies Chebyshev-series expansions to cover $\operatorname{Gi}\left(x\right)$ for $0\leq x<\infty$ and $\operatorname{Hi}\left(x\right)$ for $-\infty<x\leq 0$ . The Chebyshev coefficients are given to 20D.

8: 11.15 Approximations

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§11.15(i) Expansions in Chebyshev Series

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Luke (1975, pp. 416–421) gives Chebyshev-series expansions for $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{L}_{n}\left(x\right)$ , $0\leq\left|x\right|\leq 8$ , and $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , $x\geq 8$ , for $n=0,1$ ; $\int_{0}^{x}t^{-m}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}t^{-m}\mathbf{L}_{0}\left(t\right)\,\mathrm{d}t$ , $0\leq\left|x\right|\leq 8$ , $m=0,1$ and $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ , $x\geq 8$ ; the coefficients are to 20D.

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MacLeod (1993) gives Chebyshev-series expansions for $\mathbf{L}_{0}\left(x\right)$ , $\mathbf{L}_{1}\left(x\right)$ , $0\leq x\leq 16$ , and $I_{0}\left(x\right)-\mathbf{L}_{0}\left(x\right)$ , $I_{1}\left(x\right)-\mathbf{L}_{1}\left(x\right)$ , $x\geq 16$ ; the coefficients are to 20D.

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9: 13.31 Approximations

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§13.31(i) Chebyshev-Series Expansions

►Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of

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

and

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

that include the intervals

0\leq x\leq\alpha

and

\alpha\leq x<\infty

, respectively, where

\alpha

is an arbitrary positive constant. …

10: 19.38 Approximations

… ►Cody (1965b) gives Chebyshev-series expansions (§3.11(ii)) with maximum precision 25D. …