§6.6 Power Series

Notes:: See Olver (1997b, pp. 40–43). (6.6.3) follows from (6.11.2) and (13.2.9).
Keywords:: cosine integrals, exponential integrals, power series, sine integrals
Referenced by:: §6.18(i)
Permalink:: http://dlmf.nist.gov/6.6

6.6.1 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)=\EulerConstant+\mathop{\ln\/}\nolimits x+\sum _{{n=1}}^{\infty}\frac{x^{n}}{n!\thinspace n},$ $x>0$ .

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable and $n$ : nonnegative integer
A&S Ref:: 5.1.10
Referenced by:: §6.5
Permalink:: http://dlmf.nist.gov/6.6.E1
Encodings:: TeX, pMML, png

6.6.2 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=-\EulerConstant-\mathop{\ln\/}\nolimits z-\sum _{{n=1}}^{\infty}\frac{(-1)^{n}z^{n}}{n!\thinspace n}.$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : 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

6.6.3 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=-\mathop{\ln\/}\nolimits z+e^{{-z}}\sum _{{n=0}}^{\infty}\frac{z^{n}}{n!}\mathop{\psi\/}\nolimits\!\left(n+1\right),$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: ¶ ‣ §2.5(iii), §6.6, §8.19(iv)
Permalink:: http://dlmf.nist.gov/6.6.E3
Encodings:: TeX, pMML, png

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

6.6.4 $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)=\sum _{{n=1}}^{\infty}\frac{(-1)^{{n-1}}z^{n}}{n!\thinspace n},$

Symbols:: $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ : complementary exponential integral, $!$ : $n!$ : factorial, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §6.4
Permalink:: http://dlmf.nist.gov/6.6.E4
Encodings:: TeX, pMML, png

6.6.5 $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)=\sum _{{n=0}}^{\infty}\frac{(-1)^{n}z^{{2n+1}}}{(2n+1)!(2n+1)},$

Symbols:: $!$ : $n!$ : factorial, $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : 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

6.6.6 $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)=\EulerConstant+\mathop{\ln\/}\nolimits z+\sum _{{n=1}}^{\infty}\frac{(-1)^{n}z^{{2n}}}{(2n)!(2n)}.$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $z$ : complex variable and $n$ : nonnegative integer
A&S Ref:: 5.2.16
Referenced by:: §6.5
Permalink:: http://dlmf.nist.gov/6.6.E6
Encodings:: TeX, pMML, png

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