# §6.6 Power Series

 6.6.1 $\mathrm{Ei}\left(x\right)=\gamma+\ln x+\sum_{n=1}^{\infty}\frac{x^{n}}{n!% \thinspace n},$ $x>0$.
 6.6.2 $E_{1}\left(z\right)=-\gamma-\ln z-\sum_{n=1}^{\infty}\frac{(-1)^{n}z^{n}}{n!% \thinspace n}.$ ⓘ Symbols: $\gamma$: Euler’s constant, $E_{1}\left(\NVar{z}\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln\NVar{z}$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer A&S Ref: 5.1.11 Referenced by: §6.4, §6.5 Permalink: http://dlmf.nist.gov/6.6.E2 Encodings: TeX, pMML, png See also: Annotations for 6.6 and 6
 6.6.3 $E_{1}\left(z\right)=-\ln z+e^{-z}\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\psi\left(% n+1\right),$

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

 6.6.4 $\mathrm{Ein}\left(z\right)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}z^{n}}{n!% \thinspace n},$ ⓘ Symbols: $\mathrm{Ein}\left(\NVar{z}\right)$: complementary exponential integral, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer Referenced by: §6.4 Permalink: http://dlmf.nist.gov/6.6.E4 Encodings: TeX, pMML, png See also: Annotations for 6.6 and 6
 6.6.5 $\mathrm{Si}\left(z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{(2n+1)!(2% n+1)},$ ⓘ Symbols: $!$: factorial (as in $n!$), $\mathrm{Si}\left(\NVar{z}\right)$: sine integral, $z$: complex variable and $n$: nonnegative integer A&S Ref: 5.2.14 Referenced by: §6.5 Permalink: http://dlmf.nist.gov/6.6.E5 Encodings: TeX, pMML, png See also: Annotations for 6.6 and 6
 6.6.6 $\mathrm{Ci}\left(z\right)=\gamma+\ln z+\sum_{n=1}^{\infty}\frac{(-1)^{n}z^{2n}% }{(2n)!(2n)}.$

The series in this section converge for all finite values of $x$ and $|z|$.