6 Exponential, Logarithmic, Sine, and Cosine Integrals Properties 6.1 Special Notation 6.3 Graphics

§6.2 Definitions and Interrelations

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Permalink:: http://dlmf.nist.gov/6.2
See also:: Annotations for Ch.6

§6.2(i) Exponential and Logarithmic Integrals

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Defines:: $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral
Keywords:: complementary exponential integral, definition, exponential integrals, interrelations, logarithmic integral, principal value, relation to exponential integrals, relations to other functions
Notes:: See Olver (1997b, pp. 40–41).
Referenced by:: §10.38, §13.18(ii), §13.6(ii), §4.10(i), §6.7(i), §7.5
Permalink:: http://dlmf.nist.gov/6.2.i
See also:: Annotations for §6.2 and Ch.6

The principal value of the exponential integral $E_{1}\left(z\right)$ is defined by

6.2.1		$E_{1}\left(z\right)=\int_{z}^{\infty}\frac{e^{-t}}{t}\,\mathrm{d}t,$
$z\neq 0$ ,
ⓘ Defines: $E_{1}\left(\NVar{z}\right)$ : exponential integral Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $z$ : complex variable A&S Ref: 5.1.1 Referenced by: §6.2(ii), §6.7(iii) Permalink: http://dlmf.nist.gov/6.2.E1 Encodings: TeX, pMML, png See also: Annotations for §6.2(i), §6.2 and Ch.6

where the path does not cross the negative real axis or pass through the origen. As in the case of the logarithm (§4.2(i)) there is a cut along the interval $(-\infty,0]$ and the principal value is two-valued on $(-\infty,0)$ .

Unless indicated otherwise, it is assumed throughout the DLMF that $E_{1}\left(z\right)$ assumes its principal value. This is also true of the functions $\operatorname{Ci}\left(z\right)$ and $\operatorname{Chi}\left(z\right)$ defined in §6.2(ii).

6.2.2		$E_{1}\left(z\right)=e^{-z}\int_{0}^{\infty}\frac{e^{-t}}{t+z}\,\mathrm{d}t,$
$\|\operatorname{ph}z\|<\pi$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\int$ : integral, $\operatorname{ph}$ : phase and $z$ : complex variable A&S Ref: 5.1.1 (in modified form) Referenced by: §2.5(iii), §6.18(i), §6.7(iii) Permalink: http://dlmf.nist.gov/6.2.E2 Encodings: TeX, pMML, png See also: Annotations for §6.2(i), §6.2 and Ch.6

6.2.3		$\operatorname{Ein}\left(z\right)=\int_{0}^{z}\frac{1-e^{-t}}{t}\,\mathrm{d}t.$
ⓘ Defines: $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $z$ : complex variable Permalink: http://dlmf.nist.gov/6.2.E3 Encodings: TeX, pMML, png See also: Annotations for §6.2(i), §6.2 and Ch.6

$\operatorname{Ein}\left(z\right)$ is sometimes called the complementary exponential integral. It is entire.

6.2.4		$E_{1}\left(z\right)=\operatorname{Ein}\left(z\right)-\ln z-\gamma.$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\ln\NVar{z}$ : 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: TeX, pMML, png See also: Annotations for §6.2(i), §6.2 and Ch.6

In the next three equations $x>0$ .

6.2.5		$\operatorname{Ei}\left(x\right)=-\,\pvint_{-x}^{\infty}\frac{e^{-t}}{t}\,% \mathrm{d}t=\pvint_{-\infty}^{x}\frac{e^{t}}{t}\,\mathrm{d}t,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value and $x$ : real variable A&S Ref: 5.1.2 Permalink: http://dlmf.nist.gov/6.2.E5 Encodings: TeX, pMML, png See also: Annotations for §6.2(i), §6.2 and Ch.6

6.2.6		$\operatorname{Ei}\left(-x\right)=-\int_{x}^{\infty}\frac{e^{-t}}{t}\,\mathrm{d% }t=-E_{1}\left(x\right),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\int$ : integral and $x$ : real variable Permalink: http://dlmf.nist.gov/6.2.E6 Encodings: TeX, pMML, png See also: Annotations for §6.2(i), §6.2 and Ch.6

6.2.7		$\operatorname{Ei}\left(\pm x\right)=-\operatorname{Ein}\left(\mp x\right)+\ln x% +\gamma.$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\ln\NVar{z}$ : 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: TeX, pMML, png See also: Annotations for §6.2(i), §6.2 and Ch.6

( $\operatorname{Ei}\left(x\right)$ is undefined when $x=0$ , or when $x$ is not real.)

The logarithmic integral is defined by

6.2.8		$\operatorname{li}\left(x\right)=\pvint_{0}^{x}\frac{\,\mathrm{d}t}{\ln t}=% \operatorname{Ei}\left(\ln x\right),$
$x>1$ .
ⓘ Defines: $\operatorname{li}\left(\NVar{x}\right)$ : logarithmic integral Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value and $x$ : real variable A&S Ref: 5.1.3 Referenced by: §27.12, §6.12(i) Permalink: http://dlmf.nist.gov/6.2.E8 Encodings: TeX, pMML, png See also: Annotations for §6.2(i), §6.2 and Ch.6

The generalized exponential integral $E_{p}\left(z\right)$ , $p\in\mathbb{C}$ , is treated in Chapter 8.

§6.2(ii) Sine and Cosine Integrals

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Keywords:: cosine integrals, definition, principal value, sine integrals
Notes:: See Olver (1997b, pp. 41–42).
Referenced by:: §10.15, §10.60(iii), §6.2(i), §8.21(v)
Permalink:: http://dlmf.nist.gov/6.2.ii
See also:: Annotations for §6.2 and Ch.6

6.2.9		$\operatorname{Si}\left(z\right)=\int_{0}^{z}\frac{\sin t}{t}\,\mathrm{d}t.$
ⓘ Defines: $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 5.2.1 Referenced by: §10.15 Permalink: http://dlmf.nist.gov/6.2.E9 Encodings: TeX, pMML, png See also: Annotations for §6.2(ii), §6.2 and Ch.6

$\operatorname{Si}\left(z\right)$ is an odd entire function.

6.2.10		$\operatorname{si}\left(z\right)=-\int_{z}^{\infty}\frac{\sin t}{t}\,\mathrm{d}% t=\operatorname{Si}\left(z\right)-\tfrac{1}{2}\pi.$
ⓘ Defines: $\operatorname{si}\left(\NVar{z}\right)$ : sine integral Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral and $z$ : complex variable A&S Ref: 5.2.5 (in modified form) 5.2.26 (in modified form) Referenced by: §6.5 Permalink: http://dlmf.nist.gov/6.2.E10 Encodings: TeX, pMML, png See also: Annotations for §6.2(ii), §6.2 and Ch.6

6.2.11		$\operatorname{Ci}(z)=-\int_{z}^{\infty}\frac{\cos t}{t}\,\mathrm{d}t,$
ⓘ Defines: $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $z$ : complex variable A&S Ref: 5.2.27 Referenced by: §10.15 Permalink: http://dlmf.nist.gov/6.2.E11 Encodings: TeX, pMML, png See also: Annotations for §6.2(ii), §6.2 and Ch.6

where the path does not cross the negative real axis or pass through the origen. This is the principal value; compare (6.2.1).

6.2.12		$\operatorname{Cin}\left(z\right)=\int_{0}^{z}\frac{1-\cos t}{t}\,\mathrm{d}t.$
ⓘ Defines: $\operatorname{Cin}\left(\NVar{z}\right)$ : cosine integral Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $z$ : complex variable A&S Ref: 5.2.2 Permalink: http://dlmf.nist.gov/6.2.E12 Encodings: TeX, pMML, png See also: Annotations for §6.2(ii), §6.2 and Ch.6

$\operatorname{Cin}\left(z\right)$ is an even entire function.

6.2.13		$\operatorname{Ci}\left(z\right)=-\operatorname{Cin}\left(z\right)+\ln z+\gamma.$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\operatorname{Cin}\left(\NVar{z}\right)$ : cosine integral, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 5.2.2 (in modified form) Referenced by: §6.4 Permalink: http://dlmf.nist.gov/6.2.E13 Encodings: TeX, pMML, png See also: Annotations for §6.2(ii), §6.2 and Ch.6

Values at Infinity

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Keywords:: cosine integrals, sine integrals, value at infinity
See also:: Annotations for §6.2(ii), §6.2 and Ch.6

6.2.14	$\displaystyle\lim_{x\to\infty}\operatorname{Si}\left(x\right)$	$\displaystyle=\tfrac{1}{2}\pi,$
6.2.14	$\displaystyle\lim_{x\to\infty}\operatorname{Ci}\left(x\right)$	$\displaystyle=0.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral and $x$ : real variable A&S Ref: 5.2.25 (extended form of) Referenced by: §6.16(i) Permalink: http://dlmf.nist.gov/6.2.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §6.2(ii), §6.2(ii), §6.2 and Ch.6

Hyperbolic Analogs of the Sine and Cosine Integrals

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Keywords:: cosine integrals, hyperbolic analog, sine integrals
See also:: Annotations for §6.2(ii), §6.2 and Ch.6

6.2.15	$\displaystyle\operatorname{Shi}\left(z\right)$	$\displaystyle=\int_{0}^{z}\frac{\sinh t}{t}\,\mathrm{d}t,$
ⓘ Defines: $\operatorname{Shi}\left(\NVar{z}\right)$ : hyperbolic sine integral Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral and $z$ : complex variable A&S Ref: 5.2.3 Permalink: http://dlmf.nist.gov/6.2.E15 Encodings: TeX, pMML, png See also: Annotations for §6.2(ii), §6.2(ii), §6.2 and Ch.6
6.2.16	$\displaystyle\operatorname{Chi}\left(z\right)$	$\displaystyle=\gamma+\ln z+\int_{0}^{z}\frac{\cosh t-1}{t}\,\mathrm{d}t.$
ⓘ Defines: $\operatorname{Chi}\left(\NVar{z}\right)$ : hyperbolic cosine integral Symbols: $\gamma$ : Euler’s constant, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 5.2.4 Referenced by: §6.4 Permalink: http://dlmf.nist.gov/6.2.E16 Encodings: TeX, pMML, png See also: Annotations for §6.2(ii), §6.2(ii), §6.2 and Ch.6

§6.2(iii) Auxiliary Functions

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Keywords:: auxiliary functions for sine and cosine integrals, definition
Permalink:: http://dlmf.nist.gov/6.2.iii
See also:: Annotations for §6.2 and Ch.6

6.2.17	$\displaystyle\mathrm{f}\left(z\right)$	$\displaystyle=\phantom{+}\operatorname{Ci}\left(z\right)\sin z-\operatorname{% si}\left(z\right)\cos z,$
ⓘ Defines: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals Symbols: $\cos\NVar{z}$ : cosine function, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\operatorname{si}\left(\NVar{z}\right)$ : sine integral, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 5.2.6 Referenced by: §6.4, §6.5 Permalink: http://dlmf.nist.gov/6.2.E17 Encodings: TeX, pMML, png See also: Annotations for §6.2(iii), §6.2 and Ch.6
6.2.18	$\displaystyle\mathrm{g}\left(z\right)$	$\displaystyle=-\operatorname{Ci}\left(z\right)\cos z-\operatorname{si}\left(z% \right)\sin z.$
ⓘ Defines: $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals Symbols: $\cos\NVar{z}$ : cosine function, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\operatorname{si}\left(\NVar{z}\right)$ : sine integral, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 5.2.7 Referenced by: §6.4, §6.5 Permalink: http://dlmf.nist.gov/6.2.E18 Encodings: TeX, pMML, png See also: Annotations for §6.2(iii), §6.2 and Ch.6
6.2.19	$\displaystyle\operatorname{Si}\left(z\right)$	$\displaystyle=\tfrac{1}{2}\pi-\mathrm{f}\left(z\right)\cos z-\mathrm{g}\left(z% \right)\sin z,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral and $z$ : complex variable A&S Ref: 5.2.8 Referenced by: §6.12(ii) Permalink: http://dlmf.nist.gov/6.2.E19 Encodings: TeX, pMML, png See also: Annotations for §6.2(iii), §6.2 and Ch.6
6.2.20	$\displaystyle\operatorname{Ci}\left(z\right)$	$\displaystyle=\mathrm{f}\left(z\right)\sin z-\mathrm{g}\left(z\right)\cos z.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\sin\NVar{z}$ : sine function, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable A&S Ref: 5.2.9 Referenced by: §6.12(ii) Permalink: http://dlmf.nist.gov/6.2.E20 Encodings: TeX, pMML, png See also: Annotations for §6.2(iii), §6.2 and Ch.6

6.2.21	$\displaystyle\frac{\mathrm{d}\mathrm{f}\left(z\right)}{\mathrm{d}z}$	$\displaystyle=-\mathrm{g}\left(z\right),$
6.2.21	$\displaystyle\frac{\mathrm{d}\mathrm{g}\left(z\right)}{\mathrm{d}z}$	$\displaystyle=\mathrm{f}\left(z\right)-\frac{1}{z}.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable Permalink: http://dlmf.nist.gov/6.2.E21 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §6.2(iii), §6.2 and Ch.6

6.1 Special Notation 6.3 Graphics

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