§6.10 Other Series Expansions

Permalink:: http://dlmf.nist.gov/6.10

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

§6.10(i) Inverse Factorial Series

Notes:: See Nielsen (1906a, p. 283). (6.10.3) follows from $1/(1-\mathop{\ln\/}\nolimits\!\left(1-t\right))=\sum _{{k=0}}^{\infty}c_{k}t^{k}$ .
Keywords:: exponential integrals
Referenced by:: §6.18(i)
Permalink:: http://dlmf.nist.gov/6.10.i

6.10.1 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=e^{{-z}}\left(\frac{c_{0}}{z}+\frac{c_{1}}{z(z+1)}+\frac{2!c_{2}}{z(z+1)(z+2)}+\frac{3!c_{3}}{z(z+1)(z+2)(z+3)}+\cdots\right),$ $\realpart{z}>0$ ,

Symbols:: $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $!$ : $n!$ : factorial, $\realpart{}$ : real part and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/6.10.E1
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where

6.10.2

$c_{0}=1,$

$c_{1}=-1,$

$c_{2}=\tfrac{1}{2},$

$c_{3}=-\tfrac{1}{3},$

$c_{4}=\tfrac{1}{6},$

Permalink:: http://dlmf.nist.gov/6.10.E2
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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

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

Keywords:: cosine integrals, exponential integrals, sine integrals
Referenced by:: §10.60(iii), §6.18(i)
Permalink:: http://dlmf.nist.gov/6.10.ii

For the notation see §10.47(ii).

6.10.4 $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)=z\sum _{{n=0}}^{\infty}\left(\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}z\right)\right)^{2},$

Symbols:: $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : sine integral, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : 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

6.10.5 $\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)=\sum _{{n=1}}^{\infty}a_{n}\left(\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}z\right)\right)^{2},$

Symbols:: $\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)$ : cosine integral, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : 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

6.10.6 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)=\EulerConstant+\mathop{\ln\/}\nolimits\left|x\right|+\sum _{{n=0}}^{\infty}(-1)^{n}(x-a_{n})\left(\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(\tfrac{1}{2}x\right)\right)^{2},$ $x\neq 0$ ,

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : 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
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where

6.10.7 $a_{n}=(2n+1)\left(1-(-1)^{n}+\mathop{\psi\/}\nolimits\!\left(n+1\right)-\mathop{\psi\/}\nolimits\!\left(1\right)\right),$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $n$ : nonnegative integer and $a_{n}$ : coefficients
Permalink:: http://dlmf.nist.gov/6.10.E7
Encodings:: TeX, pMML, png

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

6.10.8 $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)=ze^{{-z/2}}\left(\mathop{{\mathsf{i}^{{(1)}}_{{0}}}\/}\nolimits\!\left(\tfrac{1}{2}z\right)+\sum _{{n=1}}^{\infty}\dfrac{2n+1}{n(n+1)}\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(\tfrac{1}{2}z\right)\right).$

Symbols:: $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ : complementary exponential integral, $e$ : base of exponential function, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : 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

For (6.10.4)–(6.10.8) and further results see Harris (2000) and Luke (1969b, pp. 56–57). An expansion for $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ can be obtained by combining (6.2.4) and (6.10.8).

§6.10 Other Series Expansions

Contents

§6.10(i) Inverse Factorial Series

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