# §6.6 Power Series

 6.6.1 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)=\EulerConstant+\mathop{\ln\/}% \nolimits x+\sum_{n=1}^{\infty}\frac{x^{n}}{n!\thinspace n},$ $x>0$.
 6.6.2 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=-\EulerConstant-\mathop{\ln\/}% \nolimits z-\sum_{n=1}^{\infty}\frac{(-1)^{n}z^{n}}{n!\thinspace n}.$
 6.6.3 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=-\mathop{\ln\/}\nolimits z+e^{-z}% \sum_{n=0}^{\infty}\frac{z^{n}}{n!}\mathop{\psi\/}\nolimits\!\left(n+1\right),$

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

 6.6.4 $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)=\sum_{n=1}^{\infty}\frac{(-1)% ^{n-1}z^{n}}{n!\thinspace n},$
 6.6.5 $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)=\sum_{n=0}^{\infty}\frac{(-1)^% {n}z^{2n+1}}{(2n+1)!(2n+1)},$
 6.6.6 $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)=\EulerConstant+\mathop{\ln\/}% \nolimits z+\sum_{n=1}^{\infty}\frac{(-1)^{n}z^{2n}}{(2n)!(2n)}.$

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