exponential integrals

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11: 8.24 Physical Applications

§8.24 Physical Applications

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§8.24(iii) Generalized Exponential Integral

►The function $E_p\left(x\right)$, with $p>0$, appears in theories of transport and radiative equilibrium (Hopf (1934), Kourganoff (1952), Altaç (1996)). ►With more general values of $p$, $E_p\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).

12: 6.11 Relations to Other Functions

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Incomplete Gamma Function

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6.11.1 $$E_1\left(z\right)=\mathrm{\Gamma }(0,z).$$

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Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

5.1.45 (special case of)

Referenced by:

§6.11

Permalink:

http://dlmf.nist.gov/6.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

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Confluent Hypergeometric Function

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6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $z$: complex variable

Referenced by:

§6.11, 5th item, §6.6

Permalink:

http://dlmf.nist.gov/6.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

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13: 6.21 Software

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§6.21(ii) $E_1\left(x\right)$, $\mathrm{Ei}\left(x\right)$, $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $\mathrm{Shi}\left(x\right)$, $\mathrm{Chi}\left(x\right)$, $x\in ℝ$

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§6.21(iii) $E_1\left(z\right)$, $\mathrm{Si}\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Shi}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $z\in ℂ$

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14: 6.6 Power Series

§6.6 Power Series

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

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: 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:

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See also:

Annotations for §6.6 and Ch.6

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6.6.2 $$E_1\left(z\right)=-\gamma -\ln z-\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^n}{n!n}.$$

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Symbols:

$\gamma$: Euler’s constant, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: 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:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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6.6.3 $$E_1\left(z\right)=-\ln z+{\mathrm{e}}^{-z}\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}\psi \left(n+1\right),$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: 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:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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

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Symbols:

$\mathrm{Ein}\left(z\right)$: 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:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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15: 8.26 Tables

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§8.26(iv) Generalized Exponential Integral

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Chiccoli et al. (1988) presents a short table of $E_p\left(x\right)$ for $p=-\frac{9}{2}(1)-\frac{1}{2}$, $0\le x\le 200$ to 14S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Stankiewicz (1968) tabulates $E_n\left(x\right)$ for $n=1(1)10$, $x=0.01(.01)5$ to 7D.

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16: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

§8.20 Asymptotic Expansions of $E_p\left(z\right)$

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§8.20(i) Large $z$

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8.20.1 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z}\left(\sum _{k=0}^{n-1}{(-1)}^k\frac{{\left(p\right)}_k}{z^k}+{(-1)}^n\frac{{\left(p\right)}_n{\mathrm{e}}^z}{z^{n-1}}{E}_{n+p}\left(z\right)\right),$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.51

Permalink:

http://dlmf.nist.gov/8.20.E1

Encodings:

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See also:

Annotations for §8.20(i), §8.20 and Ch.8

… ►For an exponentially-improved asymptotic expansion of $E_p\left(z\right)$ see §2.11(iii). ►

§8.20(ii) Large $p$

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17: 6.7 Integral Representations

§6.7 Integral Representations

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§6.7(i) Exponential Integrals

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6.7.2 $${\mathrm{e}}^x{\int }_0^{\alpha }\frac{{\mathrm{e}}^{-xt}}{1-t}dt=\mathrm{Ei}\left(x\right)-\mathrm{Ei}\left((1-\alpha )x\right),$$ , $x>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral, $\int$: integral and $x$: real variable

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

… ►Many integrals with exponentials and rational functions, for example, integrals of the type $\int {\mathrm{e}}^{z}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be represented in finite form in terms of the function $E_1\left(z\right)$ and elementary functions; see Lebedev (1965, p. 42). …

18: 6.14 Integrals

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§6.14(i) Laplace Transforms

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6.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_1\left(t\right)dt=\frac{1}{a}\ln \left(1+a\right),$$ $\mathrm{\Re }a>-1$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.1.34 (special case of)

Permalink:

http://dlmf.nist.gov/6.14.E1

Encodings:

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See also:

Annotations for §6.14(i), §6.14 and Ch.6

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6.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{Ci}\left(t\right)dt=-\frac{1}{2a}\ln \left(1+a^2\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.2.28

Referenced by:

§6.14(i)

Permalink:

http://dlmf.nist.gov/6.14.E2

Encodings:

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See also:

Annotations for §6.14(i), §6.14 and Ch.6

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§6.14(ii) Other Integrals

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6.14.4 $${\int }_0^{\mathrm{\infty }}{E}_1^2\left(t\right)dt=2\ln 2,$$

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Symbols:

$dx$: differential of $x$ , $E_1\left(z\right)$: exponential integral, $\int$: integral and $\ln z$: principal branch of logarithm function

A&S Ref:

5.1.33

Permalink:

http://dlmf.nist.gov/6.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

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19: 6.10 Other Series Expansions

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§6.10(i) Inverse Factorial Series

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

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6.10.6 $$\mathrm{Ei}\left(x\right)=\gamma +\ln \left|x\right|+\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n(x-a_n){\left({𝗂}_n^{(1)}\left(\frac{1}{2}x\right)\right)}^2,$$ $x\ne 0$,

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function, ${𝗂}_n^{(1)}\left(z\right)$: 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:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

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6.10.8 $$\mathrm{Ein}\left(z\right)=z{\mathrm{e}}^{-z/2}\left({𝗂}_0^{(1)}\left(\frac{1}{2}z\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{2n+1}{n(n+1)}{𝗂}_n^{(1)}\left(\frac{1}{2}z\right)\right).$$

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: 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:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

… ►An expansion for $E_1\left(z\right)$ can be obtained by combining (6.2.4) and (6.10.8).

20: 8.27 Approximations

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DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

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§8.27(ii) Generalized Exponential Integral

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Luke (1975, p. 103) gives Chebyshev-series expansions for $E_1\left(x\right)$ and related functions for $x\ge 5$.

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Luke (1975, p. 106) gives rational and Padé approximations, with remainders, for $E_1\left(z\right)$ and $z^{-1}{\int }_0^z{t}^{-1}(1-{\mathrm{e}}^{-t})dt$ for complex $z$ with $\left|\mathrm{ph}z\right|\le \pi$.

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Verbeeck (1970) gives polynomial and rational approximations for $E_p\left(x\right)=({\mathrm{e}}^{-x}/x)P(z)$, approximately, where $P(z)$ denotes a quotient of polynomials of equal degree in $z=x^{-1}$.