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6 Exponential, Logarithmic, Sine, and Cosine Integrals Properties 6.5 Further Interrelations 6.7 Integral Representations

§6.6 Power Series

ⓘ

Keywords:: cosine integrals, exponential integrals, power series, sine integrals
Notes:: See Olver (1997b, pp. 40–43). (6.6.3) follows from (6.11.2) and (13.2.9).
Referenced by:: §6.18(i)
Permalink:: http://dlmf.nist.gov/6.6
See also:: Annotations for Ch.6

6.6.1		$\operatorname{Ei}\left(x\right)=\gamma+\ln x+\sum_{n=1}^{\infty}\frac{x^{n}}{n% !\thinspace n},$
$x>0$ .
ⓘ Symbols: $\gamma$ : Euler’s constant, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : 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 See also: Annotations for §6.6 and Ch.6

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 Ch.6

6.6.3		$E_{1}\left(z\right)=-\ln z+e^{-z}\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\psi\left(% n+1\right),$
ⓘ Symbols: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $z$ : complex variable and $n$ : nonnegative integer Referenced by: §2.5(iii), §6.6, §8.19(iv) Permalink: http://dlmf.nist.gov/6.6.E3 Encodings: TeX, pMML, png See also: Annotations for §6.6 and Ch.6

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

6.6.4		$\operatorname{Ein}\left(z\right)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}z^{n}}{n!% \thinspace n},$
ⓘ Symbols: $\operatorname{Ein}\left(\NVar{z}\right)$ : 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 Ch.6

6.6.5		$\operatorname{Si}\left(z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{(2n% +1)!(2n+1)},$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\operatorname{Si}\left(\NVar{z}\right)$ : 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 Ch.6

6.6.6		$\operatorname{Ci}\left(z\right)=\gamma+\ln z+\sum_{n=1}^{\infty}\frac{(-1)^{n}% z^{2n}}{(2n)!(2n)}.$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : 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 See also: Annotations for §6.6 and Ch.6

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

6.5 Further Interrelations 6.7 Integral Representations

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