# §6.10 Other Series Expansions

## §6.10(i) Inverse Factorial Series

 6.10.1 $E_{1}\left(z\right)=e^{-z}\left(\frac{c_{0}}{z}+\frac{c_{1}}{z(z+1)}+\frac{2!c% _{2}}{z(z+1)(z+2)}+\frac{3!c_{3}}{z(z+1)(z+2)(z+3)}+\cdots\right),$ $\Re z>0$,

where

 6.10.2 $\displaystyle c_{0}$ $\displaystyle=1,$ $\displaystyle c_{1}$ $\displaystyle=-1,$ $\displaystyle c_{2}$ $\displaystyle=\tfrac{1}{2},$ $\displaystyle c_{3}$ $\displaystyle=-\tfrac{1}{3},$ $\displaystyle c_{4}$ $\displaystyle=\tfrac{1}{6},$ ⓘ Permalink: http://dlmf.nist.gov/6.10.E2 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §6.10(i), §6.10 and Ch.6

and

 6.10.3 $c_{k}=-\sum_{j=0}^{k-1}\frac{c_{j}}{k-j},$ $k\geq 1$. ⓘ Referenced by: §6.10(i) Permalink: http://dlmf.nist.gov/6.10.E3 Encodings: TeX, pMML, png See also: Annotations for §6.10(i), §6.10 and Ch.6

For a more general result (incomplete gamma function), and also for a result for the logarithmic integral, see Nielsen (1906a, p. 283: Formula (3) is incorrect).

## §6.10(ii) Expansions in Series of Spherical Bessel Functions

For the notation see §10.47(ii).

 6.10.4 $\displaystyle\operatorname{Si}\left(z\right)$ $\displaystyle=z\sum_{n=0}^{\infty}\left(\mathsf{j}_{n}\left(\tfrac{1}{2}z% \right)\right)^{2},$ ⓘ Symbols: $\operatorname{Si}\left(\NVar{z}\right)$: sine integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer Referenced by: §6.10(ii) Permalink: http://dlmf.nist.gov/6.10.E4 Encodings: TeX, pMML, png See also: Annotations for §6.10(ii), §6.10 and Ch.6 6.10.5 $\displaystyle\operatorname{Cin}\left(z\right)$ $\displaystyle=\sum_{n=1}^{\infty}a_{n}\left(\mathsf{j}_{n}\left(\tfrac{1}{2}z% \right)\right)^{2},$
 6.10.6 $\operatorname{Ei}\left(x\right)=\gamma+\ln\left|x\right|+\sum_{n=0}^{\infty}(-% 1)^{n}(x-a_{n})\left({\mathsf{i}^{(1)}_{n}}\left(\tfrac{1}{2}x\right)\right)^{% 2},$ $x\neq 0$,

where

 6.10.7 $a_{n}=(2n+1)\left(1-(-1)^{n}+\psi\left(n+1\right)-\psi\left(1\right)\right),$ ⓘ Defines: $a_{n}$: coefficients (locally) Symbols: $\psi\left(\NVar{z}\right)$: psi (or digamma) function and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/6.10.E7 Encodings: TeX, pMML, png See also: Annotations for §6.10(ii), §6.10 and Ch.6

and $\psi$ denotes the logarithmic derivative of the gamma function (§5.2(i)).

 6.10.8 $\operatorname{Ein}\left(z\right)=ze^{-z/2}\left({\mathsf{i}^{(1)}_{0}}\left(% \tfrac{1}{2}z\right)+\sum_{n=1}^{\infty}\dfrac{2n+1}{n(n+1)}{\mathsf{i}^{(1)}_% {n}}\left(\tfrac{1}{2}z\right)\right).$

For (6.10.4)–(6.10.8) and further results see Harris (2000) and Luke (1969b, pp. 56–57). An expansion for $E_{1}\left(z\right)$ can be obtained by combining (6.2.4) and (6.10.8).