# §6.10 Other Series Expansions

## §6.10(i) Inverse Factorial Series

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

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)

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)

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 $\displaystyle\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ $\displaystyle=z\sum_{n=0}^{\infty}\left(\mathop{\mathsf{j}_{n}\/}\nolimits\!% \left(\tfrac{1}{2}z\right)\right)^{2},$ 6.10.5 $\displaystyle\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)$ $\displaystyle=\sum_{n=1}^{\infty}a_{n}\left(\mathop{\mathsf{j}_{n}\/}\nolimits% \!\left(\tfrac{1}{2}z\right)\right)^{2},$
 6.10.6 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)=\gamma+\mathop{\ln\/}\nolimits% \left|x\right|+\sum_{n=0}^{\infty}(-1)^{n}(x-a_{n})\left(\mathop{{\mathsf{i}^{% (1)}_{n}}\/}\nolimits\!\left(\tfrac{1}{2}x\right)\right)^{2},$ $x\neq 0$,

where

 6.10.7 $a_{n}=(2n+1)\left(1-(-1)^{n}+\mathop{\psi\/}\nolimits\!\left(n+1\right)-% \mathop{\psi\/}\nolimits\!\left(1\right)\right),$ Defines: $a_{n}$: coefficients (locally) Symbols: $\mathop{\psi\/}\nolimits\!\left(\NVar{z}\right)$: 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)

and $\mathop{\psi\/}\nolimits$ denotes the logarithmic derivative of the gamma function (§5.2(i)).

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

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