6 Exponential, Logarithmic, Sine, and Cosine Integrals Properties 6.9 Continued Fraction 6.11 Relations to Other Functions

§6.10 Other Series Expansions

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Permalink:: http://dlmf.nist.gov/6.10
See also:: Annotations for Ch.6

§6.10(i) Inverse Factorial Series
§6.10(ii) Expansions in Series of Spherical Bessel Functions

§6.10(i) Inverse Factorial Series

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Keywords:: expansion in inverse factorials, exponential integrals
Notes:: See Nielsen (1906a, p. 283). (6.10.3) follows from $1/(1-\ln\left(1-t\right))=\sum_{k=0}^{\infty}c_{k}t^{k}$ .
Referenced by:: §6.18(i)
Permalink:: http://dlmf.nist.gov/6.10.i
See also:: Annotations for §6.10 and Ch.6

6.10.1		$E_{1}\left(z\right)=e^{-z}\left(\frac{c_{0}}{z}+\frac{c_{1}}{z(z+1)}+\frac{2!c% _{2}}{z(z+1)(z+2)}+\frac{3!c_{3}}{z(z+1)(z+2)(z+3)}+\cdots\right),$
$\Re z>0$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $!$ : factorial (as in $n!$ ), $\Re$ : 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	$\displaystyle c_{0}$	$\displaystyle=1,$
	$\displaystyle c_{1}$	$\displaystyle=-1,$
	$\displaystyle c_{2}$	$\displaystyle=\tfrac{1}{2},$
	$\displaystyle c_{3}$	$\displaystyle=-\tfrac{1}{3},$
	$\displaystyle c_{4}$	$\displaystyle=\tfrac{1}{6},$
ⓘ 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		$c_{k}=-\sum_{j=0}^{k-1}\frac{c_{j}}{k-j},$
$k\geq 1$ .
ⓘ 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).

§6.10(ii) Expansions in Series of Spherical Bessel Functions

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Keywords:: cosine integrals, expansion in spherical Bessel functions, expansions in modified spherical Bessel functions, exponential integrals, sine integrals
Referenced by:: §10.60(iii), §6.18(i)
Permalink:: http://dlmf.nist.gov/6.10.ii
See also:: Annotations for §6.10 and Ch.6

For the notation see §10.47(ii).

6.10.4	$\displaystyle\operatorname{Si}\left(z\right)$	$\displaystyle=z\sum_{n=0}^{\infty}\left(\mathsf{j}_{n}\left(\tfrac{1}{2}z% \right)\right)^{2},$
ⓘ Symbols: $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : 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	$\displaystyle\operatorname{Cin}\left(z\right)$	$\displaystyle=\sum_{n=1}^{\infty}a_{n}\left(\mathsf{j}_{n}\left(\tfrac{1}{2}z% \right)\right)^{2},$
ⓘ Symbols: $\operatorname{Cin}\left(\NVar{z}\right)$ : cosine integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : 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		$\operatorname{Ei}\left(x\right)=\gamma+\ln\left\|x\right\|+\sum_{n=0}^{\infty}(-% 1)^{n}(x-a_{n})\left({\mathsf{i}^{(1)}_{n}}\left(\tfrac{1}{2}x\right)\right)^{% 2},$
$x\neq 0$ ,
ⓘ Symbols: $\gamma$ : Euler’s constant, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\ln\NVar{z}$ : principal branch of logarithm function, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : 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: $\psi\left(\NVar{z}\right)$ : 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		$\operatorname{Ein}\left(z\right)=ze^{-z/2}\left({\mathsf{i}^{(1)}_{0}}\left(% \tfrac{1}{2}z\right)+\sum_{n=1}^{\infty}\dfrac{2n+1}{n(n+1)}{\mathsf{i}^{(1)}_% {n}}\left(\tfrac{1}{2}z\right)\right).$
ⓘ Symbols: $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $\mathrm{e}$ : base of natural logarithm, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : 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).

6.9 Continued Fraction 6.11 Relations to Other Functions

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§6.10 Other Series Expansions

Contents

§6.10(i) Inverse Factorial Series

§6.10(ii) Expansions in Series of Spherical Bessel Functions