# series expansions

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

• 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: 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: 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. …
##### 6: 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 $x\geq 5$.

• ##### 7: 13.24 Series
###### §13.24(i) Expansions in Series of Whittaker Functions
For expansions of arbitrary functions in series of $M_{\kappa,\mu}\left(z\right)$ functions see Schäfke (1961b).
###### §13.24(ii) Expansions in Series of Bessel Functions
For other series expansions see Prudnikov et al. (1990, §6.6). …
##### 8: 8.25 Methods of Computation
###### §8.25(i) SeriesExpansions
Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. …
##### 9: 11.15 Approximations
###### §11.15(i) Expansions in Chebyshev Series
• 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.