§6.2 Definitions and Interrelations

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

§6.2(i) Exponential and Logarithmic Integrals
§6.2(ii) Sine and Cosine Integrals
§6.2(iii) Auxiliary Functions

§6.2(i) Exponential and Logarithmic Integrals

Notes:: See Olver (1997b, pp. 40–41).
Defines:: $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral
Keywords:: complementary exponential integral, exponential integrals, logarithmic integral
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

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

6.2.1 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=\int _{z}^{{\infty}}\frac{e^{{-t}}}{t}dt,$ $z\neq 0$ ,

Defines:: $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral
Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : 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
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where the path does not cross the negative real axis or pass through the origin. 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 $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ assumes its principal value. This is also true of the functions $\mathop{\mathrm{Ci}\/}\nolimits(z)$ and $\mathop{\mathrm{Chi}\/}\nolimits\!\left(z\right)$ defined in §6.2(ii).

6.2.2 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=e^{{-z}}\int _{0}^{{\infty}}\frac{e^{{-t}}}{t+z}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : 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
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6.2.3 $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)=\int _{0}^{z}\frac{1-e^{{-t}}}{t}dt.$

Defines:: $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ : complementary exponential integral
Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/6.2.E3
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$\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ is sometimes called the complementary exponential integral. It is entire.

6.2.4 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)-\mathop{\ln\/}\nolimits z-\EulerConstant.$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ : complementary exponential integral, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $z$ : complex variable
Referenced by:: §6.10(ii), §6.4
Permalink:: http://dlmf.nist.gov/6.2.E4
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In the next three equations $x>0$ .

6.2.5 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)=-\,\pvint _{{-x}}^{{\infty}}\frac{e^{{-t}}}{t}dt=\pvint _{{-\infty}}^{{x}}\frac{e^{{t}}}{t}dt,$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral, $\pvint _{a}^{b}$ : Cauchy principal value and $x$ : real variable
A&S Ref:: 5.1.2
Permalink:: http://dlmf.nist.gov/6.2.E5
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6.2.6 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(-x\right)=-\int _{{x}}^{\infty}\frac{e^{{-t}}}{t}dt=-\mathop{E_{1}\/}\nolimits\!\left(x\right),$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral, $\int$ : integral and $x$ : real variable
Permalink:: http://dlmf.nist.gov/6.2.E6
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6.2.7 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(\pm x\right)=-\mathop{\mathrm{Ein}\/}\nolimits\!\left(\mp x\right)+\mathop{\ln\/}\nolimits x+\EulerConstant.$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ : complementary exponential integral, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral, $\mathop{\ln\/}\nolimits z$ : 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
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( $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ is undefined when $x=0$ , or when $x$ is not real.)

The logarithmic integral is defined by

6.2.8 $\mathop{\mathrm{li}\/}\nolimits\!\left(x\right)=\pvint _{{0}}^{{x}}\frac{dt}{\mathop{\ln\/}\nolimits t}=\mathop{\mathrm{Ei}\/}\nolimits\!\left(\mathop{\ln\/}\nolimits x\right),$ $x>1$ .

Defines:: $\mathop{\mathrm{li}\/}\nolimits\!\left(x\right)$ : logarithmic integral
Symbols:: $dx$ : differential of $x$ , $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\pvint _{a}^{b}$ : 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
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The generalized exponential integral $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)$ , $p\in\Complex$ , is treated in Chapter 8.

§6.2(ii) Sine and Cosine Integrals

Notes:: See Olver (1997b, pp. 41–42).
Keywords:: cosine integrals, sine integrals
Referenced by:: ¶ ‣ §10.15, §10.60(iii), §6.2(i), §8.21(v)
Permalink:: http://dlmf.nist.gov/6.2.ii

6.2.9 $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)=\int _{0}^{z}\frac{\mathop{\sin\/}\nolimits t}{t}dt.$

Defines:: $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : sine integral
Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 5.2.1
Referenced by:: §10.15
Permalink:: http://dlmf.nist.gov/6.2.E9
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$\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ is an odd entire function.

6.2.10 $\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)=-\int _{z}^{\infty}\frac{\mathop{\sin\/}\nolimits t}{t}dt=\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)-\tfrac{1}{2}\pi.$

Defines:: $\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)$ : sine integral
Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : 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
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6.2.11 $\mathop{\mathrm{Ci}\/}\nolimits(z)=-\int _{z}^{\infty}\frac{\mathop{\cos\/}\nolimits t}{t}dt,$

Defines:: $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : 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
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where the path does not cross the negative real axis or pass through the origin. This is the principal value; compare (6.2.1).

6.2.12 $\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)=\int _{0}^{z}\frac{1-\mathop{\cos\/}\nolimits t}{t}dt.$

Defines:: $\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)$ : cosine integral
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral and $z$ : complex variable
A&S Ref:: 5.2.2
Permalink:: http://dlmf.nist.gov/6.2.E12
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$\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)$ is an even entire function.

6.2.13 $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)=-\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)+\mathop{\ln\/}\nolimits z+\EulerConstant.$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)$ : cosine integral, $\mathop{\ln\/}\nolimits z$ : 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
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¶ Values at Infinity

Keywords:: cosine integrals, sine integrals

6.2.14

$\lim _{{x\to\infty}}\mathop{\mathrm{Si}\/}\nolimits\!\left(x\right)=\tfrac{1}{2}\pi,$

$\lim _{{x\to\infty}}\mathop{\mathrm{Ci}\/}\nolimits\!\left(x\right)=0.$

Symbols:: $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : 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
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¶ Hyperbolic Analogs of the Sine and Cosine Integrals

Keywords:: cosine integrals, sine integrals

6.2.15 $\mathop{\mathrm{Shi}\/}\nolimits\!\left(z\right)=\int _{0}^{z}\frac{\mathop{\sinh\/}\nolimits t}{t}dt,$

Defines:: $\mathop{\mathrm{Shi}\/}\nolimits\!\left(z\right)$ : hyperbolic sine integral
Symbols:: $dx$ : differential of $x$ , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral and $z$ : complex variable
A&S Ref:: 5.2.3
Permalink:: http://dlmf.nist.gov/6.2.E15
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6.2.16 $\mathop{\mathrm{Chi}\/}\nolimits\!\left(z\right)=\EulerConstant+\mathop{\ln\/}\nolimits z+\int _{0}^{z}\frac{\mathop{\cosh\/}\nolimits t-1}{t}dt.$

Defines:: $\mathop{\mathrm{Chi}\/}\nolimits\!\left(z\right)$ : hyperbolic cosine integral
Symbols:: $\EulerConstant$ : Euler’s constant, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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
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§6.2(iii) Auxiliary Functions

Keywords:: auxiliary functions for sine and cosine integrals
Permalink:: http://dlmf.nist.gov/6.2.iii

6.2.17 $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)=\phantom{+}\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)\mathop{\sin\/}\nolimits z-\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)\mathop{\cos\/}\nolimits z,$

Defines:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)$ : sine integral, $\mathop{\sin\/}\nolimits z$ : 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
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6.2.18 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)=-\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)\mathop{\cos\/}\nolimits z-\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)\mathop{\sin\/}\nolimits z.$

Defines:: $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)$ : sine integral, $\mathop{\sin\/}\nolimits z$ : 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
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6.2.19 $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}\pi-\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)\mathop{\cos\/}\nolimits z-\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\mathop{\sin\/}\nolimits z,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals, $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : 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
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6.2.20 $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)=\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)\mathop{\sin\/}\nolimits z-\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\mathop{\cos\/}\nolimits z.$