complementary exponential integral

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1: 6.1 Special Notation

… ►The main functions treated in this chapter are the exponential integrals $\mathrm{Ei}\left(x\right)$, $E_1\left(z\right)$, and $\mathrm{Ein}\left(z\right)$; the logarithmic integral $\mathrm{li}\left(x\right)$; the sine integrals $\mathrm{Si}\left(z\right)$ and $\mathrm{si}\left(z\right)$; the cosine integrals $\mathrm{Ci}\left(z\right)$ and $\mathrm{Cin}\left(z\right)$. …

2: 6.20 Approximations

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Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\mathrm{Ein}\left(ax\right)$, $\mathrm{Si}\left(ax\right)$, and $\mathrm{Cin}\left(ax\right)$ for $-1\le x\le 1$, $a\in ℂ$. The coefficients are given in terms of series of Bessel functions.

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Luke (1969b, pp. 321–322) covers $\mathrm{Ein}\left(x\right)$ and $-\mathrm{Ein}\left(-x\right)$ for $0\le x\le 8$ (the Chebyshev coefficients are given to 20D); $E_1\left(x\right)$ for $x\ge 5$ (20D), and $\mathrm{Ei}\left(x\right)$ for $x\ge 8$ (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

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Luke (1969b, pp. 402, 410, and 415–421) gives main diagonal Padé approximations for $\mathrm{Ein}\left(z\right)$, $\mathrm{Si}\left(z\right)$, $\mathrm{Cin}\left(z\right)$ (valid near the origin), and $E_1\left(z\right)$ (valid for large $|z|$); approximate errors are given for a selection of $z$-values.

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Luke (1969b, pp. 411–414) gives rational approximations for $\mathrm{Ein}\left(z\right)$.

3: 6.4 Analytic Continuation

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6.4.1 $$E_1\left(z\right)=\mathrm{Ein}\left(z\right)-\mathrm{Ln}z-\gamma ;$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $E_1\left(z\right)$: exponential integral, $\mathrm{Ln}z$: general logarithm function and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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6.4.3 $$E_1\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\mathrm{Ein}\left(-z\right)-\ln z-\gamma \mp \pi \mathrm{i},$$ $|\mathrm{ph}z|\le \pi$.

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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4: 6.7 Integral Representations

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6.7.7 $${\int }_0^1\frac{{\mathrm{e}}^{-at}\sin \left(bt\right)}{t}dt=\mathrm{\Im }\mathrm{Ein}\left(a+\mathrm{i}b\right),$$ $a,b\in ℝ$,

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line and $\sin z$: sine function

A&S Ref:

5.1.36 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E7

Encodings:

pMML, png, TeX

See also:

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

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6.7.8 $${\int }_0^1\frac{{\mathrm{e}}^{-at}(1-\cos \left(bt\right))}{t}dt=\mathrm{\Re }\mathrm{Ein}\left(a+\mathrm{i}b\right)-\mathrm{Ein}\left(a\right),$$ $a,b\in ℝ$.

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\cos z$: cosine function, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $ℝ$: real line

A&S Ref:

5.1.38 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E8

Encodings:

pMML, png, TeX

See also:

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

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6.7.10 $$\mathrm{Ein}\left(z\right)-\mathrm{Cin}\left(z\right)={\int }_0^{\pi /2}{\mathrm{e}}^{-z\cos t}\sin \left(z\sin t\right)dt,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\cos z$: cosine function, $\mathrm{Cin}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.11 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E10

Encodings:

pMML, png, TeX

See also:

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

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6.7.11 $${\int }_0^1\frac{(1-{\mathrm{e}}^{-at})\cos \left(bt\right)}{t}dt=\mathrm{\Re }\mathrm{Ein}\left(a+\mathrm{i}b\right)-\mathrm{Cin}\left(b\right),$$ $a,b\in ℝ$.

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\cos z$: cosine function, $\mathrm{Cin}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $ℝ$: real line

A&S Ref:

5.2.33 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E11

Encodings:

pMML, png, TeX

See also:

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

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5: 6.2 Definitions and Interrelations

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6.2.3 $$\mathrm{Ein}\left(z\right)={\int }_0^z\frac{1-{\mathrm{e}}^{-t}}{t}dt.$$

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Defines:

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

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable

Permalink:

http://dlmf.nist.gov/6.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

► $\mathrm{Ein}\left(z\right)$ is sometimes called the complementary exponential integral. … ►

6.2.4 $$E_1\left(z\right)=\mathrm{Ein}\left(z\right)-\ln z-\gamma .$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $E_1\left(z\right)$: exponential integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

Referenced by:

§6.10(ii), §6.4

Permalink:

http://dlmf.nist.gov/6.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

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6.2.7 $$\mathrm{Ei}\left(\pm x\right)=-\mathrm{Ein}\left(\mp x\right)+\ln x+\gamma .$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

5.1.40 (in modified form)

Permalink:

http://dlmf.nist.gov/6.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

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6: 6.6 Power Series

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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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7: 6.19 Tables

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Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\mathrm{Si}\left(x\right)$, $-x^{-2}\mathrm{Cin}\left(x\right)$, $x^{-1}\mathrm{Ein}\left(x\right)$, $-x^{-1}\mathrm{Ein}\left(-x\right)$, $x=0(.01)0.5$; $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $\mathrm{Ei}\left(x\right)$, $E_1\left(x\right)$, $x=0.5(.01)2$; $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $x{\mathrm{e}}^{-x}\mathrm{Ei}\left(x\right)$, $x{\mathrm{e}}^x{E}_1\left(x\right)$, $x=2(.1)10$; $x\mathrm{f}\left(x\right)$, $x^2\mathrm{g}\left(x\right)$, $x{\mathrm{e}}^{-x}\mathrm{Ei}\left(x\right)$, $x{\mathrm{e}}^x{E}_1\left(x\right)$, $x^{-1}=0(.005)0.1$; $\mathrm{Si}\left(\pi x\right)$, $\mathrm{Cin}\left(\pi x\right)$, $x=0(.1)10$. Accuracy varies but is within the range 8S–11S.

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8: 6.10 Other Series Expansions

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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:

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See also:

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

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9: 7.18 Repeated Integrals of the Complementary Error Function

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${\mathrm{i}}^{-1}\mathrm{erfc}\left(z\right)={\displaystyle \frac{2}{\sqrt{\pi }}}{\mathrm{e}}^{-z^2},$

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7.18.4 $$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{z^2}\mathrm{erfc}z\right)={(-1)}^n{2}^{n}n!{\mathrm{e}}^{z^2}{\mathrm{i}}^n\mathrm{erfc}\left(z\right),$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$\mathrm{erfc}z$: complementary error function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.9

Permalink:

http://dlmf.nist.gov/7.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iii), §7.18 and Ch.7

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7.18.9 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)={\mathrm{e}}^{-z^2}\left(\frac{1}{2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right)}M(\frac{1}{2}n+\frac{1}{2},\frac{1}{2},z^2)-\frac{z}{2^{n-1}\mathrm{\Gamma }\left(\frac{1}{2}n+\frac{1}{2}\right)}M(\frac{1}{2}n+1,\frac{3}{2},z^2)\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.12

Permalink:

http://dlmf.nist.gov/7.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

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7.18.10 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=\frac{{\mathrm{e}}^{-z^2}}{2^n\sqrt{\pi }}U(\frac{1}{2}n+\frac{1}{2},\frac{1}{2},z^2).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.18(iv), §7.18(iv), Erratum (V1.0.21) for Paragraph Confluent Hypergeometric Functions (in §7.18(iv))

Permalink:

http://dlmf.nist.gov/7.18.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

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7.18.11 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=\frac{{\mathrm{e}}^{-z^2/2}}{\sqrt{2^{n-1}\pi }}U(n+\frac{1}{2},z\sqrt{2}).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.13

Referenced by:

§12.7(ii)

Permalink:

http://dlmf.nist.gov/7.18.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

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10: 19.37 Tables

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Function $\exp \left(-\pi K^{\prime }\left(k\right)/K\left(k\right)\right)(=q(k))$

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