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: 18.40 Methods of Computation

… ►However, for applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. …

10: 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. …