§6.10 Other Series Expansions

6.10.1		$$E_1\left(z\right)={\mathrm{e}}^{-z}\left(\frac{c_0}{z}+\frac{c_1}{z(z+1)}+\frac{2!c_2}{z(z+1)(z+2)}+\frac{3!c_3}{z(z+1)(z+2)(z+3)}+\mathrm{\cdots }\right),$$
$\mathrm{\Re }z>0$,
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part and $z$: complex variable Permalink: http://dlmf.nist.gov/6.10.E1 Encodings: TeX, pMML, png See also: Annotations for §6.10(i), §6.10 and Ch.6

where

6.10.2		$c_0$	$=1,$
		$c_1$	$=-1,$
		$c_2$	$=\frac{1}{2},$
		$c_3$	$=-\frac{1}{3},$
		$c_4$	$=\frac{1}{6},$
ⓘ 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), §6.10 and Ch.6

and

6.10.3		$$c_k=-\sum _{j=0}^{k-1}\frac{c_j}{k-j},$$
$k\ge 1$.
ⓘ Referenced by: §6.10(i) Permalink: http://dlmf.nist.gov/6.10.E3 Encodings: TeX, pMML, png See also: Annotations for §6.10(i), §6.10 and Ch.6

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).

For the notation see §10.47(ii).

6.10.4		$\mathrm{Si}\left(z\right)$	$=z{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\left({\mathsf{j}}_n\left(\frac{1}{2}z\right)\right)}^2,$
ⓘ Symbols: $\mathrm{Si}\left(z\right)$: sine integral, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer Referenced by: §6.10(ii) Permalink: http://dlmf.nist.gov/6.10.E4 Encodings: TeX, pMML, png See also: Annotations for §6.10(ii), §6.10 and Ch.6
6.10.5		$\mathrm{Cin}\left(z\right)$	$={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{a}_n{\left({\mathsf{j}}_n\left(\frac{1}{2}z\right)\right)}^2,$
ⓘ Symbols: $\mathrm{Cin}\left(z\right)$: cosine integral, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $n$: nonnegative integer and $a_n$: coefficients Permalink: http://dlmf.nist.gov/6.10.E5 Encodings: TeX, pMML, png See also: Annotations for §6.10(ii), §6.10 and Ch.6

6.10.6		$$\mathrm{Ei}\left(x\right)=\gamma +\ln \left|x\right|+\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n(x-a_n){\left({\mathsf{i}}_n^{(1)}\left(\frac{1}{2}x\right)\right)}^2,$$
$x\ne 0$,
ⓘ Symbols: $\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function, ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $x$: real variable, $n$: nonnegative integer and $a_n$: coefficients Referenced by: §6.18(i) Permalink: http://dlmf.nist.gov/6.10.E6 Encodings: TeX, pMML, png See also: Annotations for §6.10(ii), §6.10 and Ch.6

where

6.10.7		$$a_n=(2n+1)\left(1-{(-1)}^n+\psi \left(n+1\right)-\psi \left(1\right)\right),$$
ⓘ Defines: $a_n$: coefficients (locally) Symbols: $\psi \left(z\right)$: 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), §6.10 and Ch.6

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

6.10.8		$$\mathrm{Ein}\left(z\right)=z{\mathrm{e}}^{-z/2}\left({\mathsf{i}}_0^{(1)}\left(\frac{1}{2}z\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{2n+1}{n(n+1)}{\mathsf{i}}_n^{(1)}\left(\frac{1}{2}z\right)\right).$$
ⓘ Symbols: $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{e}$: base of natural logarithm, ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable and $n$: nonnegative integer Referenced by: §6.10(ii) Permalink: http://dlmf.nist.gov/6.10.E8 Encodings: TeX, pMML, png See also: Annotations for §6.10(ii), §6.10 and Ch.6

For (6.10.4)–(6.10.8) and further results see Harris (2000) and Luke (1969b, pp. 56–57). An expansion for $E_1\left(z\right)$ can be obtained by combining (6.2.4) and (6.10.8).