# Chebyshev-series expansions

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## 1—10 of 18 matching pages

##### 2: 6.20 Approximations
###### §6.20(ii) Expansions in ChebyshevSeries
• 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).

• 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).

• 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.

• 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.

• ##### 3: 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). …
##### 4: 8.27 Approximations
• 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).

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

• ##### 5: 13.31 Approximations
###### §13.31(i) Chebyshev-SeriesExpansions
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. …
##### 6: 9.19 Approximations
###### §9.19(ii) Expansions in ChebyshevSeries
• 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. The Chebyshev coefficients are given to 20D.

• ##### 7: 25.20 Approximations
• 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).

• 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).

• ##### 8: 19.38 Approximations
Cody (1965b) gives Chebyshev-series expansions3.11(ii)) with maximum precision 25D. …
##### 9: 11.15 Approximations
###### §11.15(i) Expansions in ChebyshevSeries
• 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.

• 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.

• ##### 10: 5.23 Approximations
###### §5.23(ii) Expansions in ChebyshevSeries
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 $0\leq x\leq 1$. …