# 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: 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 $x\geq 5$.

• ##### 6: 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). …
##### 7: 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. …
##### 8: 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.

##### 10: 24.8 Series Expansions
###### §24.8(i) Fourier Series
If $n=1,2,\dots$ and $0\leq x\leq 1$, then …
###### §24.8(ii) Other Series
24.8.9 $E_{2n}=(-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{\cosh\left(\tfrac{1}{2}\pi k% \right)}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^{2n}}{e^{2\pi(2k+1)}-1},$ $n=1,2,\dots$.