logarithmic integral

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1: 6.2 Definitions and Interrelations

… ►

§6.2(i) Exponential and Logarithmic Integrals

… ►The logarithmic integral is defined by ►

6.2.8 $$\mathrm{li}\left(x\right)={⨍}_0^x\frac{dt}{\ln t}=\mathrm{Ei}\left(\ln x\right),$$ $x>1$.

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Defines:

$\mathrm{li}\left(x\right)$: logarithmic integral

Symbols:

$dx$: differential of $x$ , $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function, ${⨍}_a^b$: 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

Encodings:

pMML, png, TeX

See also:

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

… ►

6.2.13 $$\mathrm{Ci}\left(z\right)=-\mathrm{Cin}\left(z\right)+\ln z+\gamma .$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{Cin}\left(z\right)$: cosine integral, $\ln z$: 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:

pMML, png, TeX

See also:

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

… ►

6.2.16 $$\mathrm{Chi}\left(z\right)=\gamma +\ln z+{\int }_0^z\frac{\cosh t-1}{t}dt.$$

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Defines:

$\mathrm{Chi}\left(z\right)$: hyperbolic cosine integral

Symbols:

$\gamma$: Euler’s constant, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\ln z$: 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:

pMML, png, TeX

See also:

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

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2: 6.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the argument. ►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 $\mathrm{Cin}\left(z\right)$. …

3: 6.16 Mathematical Applications

… ►

§6.16(ii) Number-Theoretic Significance of $\mathrm{li}\left(x\right)$

►If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta \left(s\right)$ have real part of $\frac{1}{2}$ (§25.10(i)), then ►

6.16.5 $$\mathrm{li}\left(x\right)-\pi (x)=O\left(\sqrt{x}\ln x\right),$$ $x\to \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $x$: real variable and $\pi (x)$: number of primes $\le x$

A&S Ref:

5.1.50

Permalink:

http://dlmf.nist.gov/6.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(ii), §6.16 and Ch.6

… ►

►►

Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

4: 27.12 Asymptotic Formulas: Primes

… ►

27.12.5 $$\left|\pi \left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp \left(-c{(\ln x)}^{1/2}\right)\right),$$ $x\to \mathrm{\infty }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

Erratum (V1.0.18) for Equation (27.12.5)

Permalink:

http://dlmf.nist.gov/27.12.E5

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.18):

Replaced $\sqrt{\ln x}$ with ${(\ln x)}^{1/2}$ to be consistent with other equations in this section.

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

►For the logarithmic integral $\mathrm{li}\left(x\right)$ see (6.2.8). … ►

27.12.6 $$\left|\pi \left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp \left(-d{(\ln x)}^{3/5}{(\ln \ln x)}^{-1/5}\right)\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number, $d$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.12.E6

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

► $\pi \left(x\right)-\mathrm{li}\left(x\right)$ changes sign infinitely often as $x\to \mathrm{\infty }$; see Littlewood (1914), Bays and Hudson (2000). … ►

27.12.7

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

§27.12

Permalink:

http://dlmf.nist.gov/27.12.E7

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

…

5: 4.10 Integrals

… ►

§4.10(i) Logarithms

… ►

4.10.2 $$\int \ln zdz=z\ln z-z,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.49

Permalink:

http://dlmf.nist.gov/4.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

… ►

4.10.7 $${⨍}_0^x\frac{dt}{\ln t}=\mathrm{li}\left(x\right),$$ $x>1$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

4.1.57

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

… ►For $\mathrm{li}\left(x\right)$ see §6.2(i). … ►Extensive compendia of indefinite and definite integrals of logarithms and exponentials include Apelblat (1983, pp. 16–47), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 107–116), Gröbner and Hofreiter (1950, pp. 52–90), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.3, 1.6, 2.3, 2.6).

6: 6.8 Inequalities

… ►

6.8.1

ⓘ

Symbols:

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

A&S Ref:

5.1.20

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

►

6.8.2

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $x$: real variable

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

►

6.8.3

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $x$: real variable

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

7: 27.21 Tables

… ►Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare $\pi \left(x\right),x/\ln x$, and $\mathrm{li}\left(x\right)$. …

8: 6.3 Graphics

… ►For a graph of $\mathrm{li}\left(x\right)$ see Figure 6.16.2. … ►

►

►

Figure 6.3.3: $|E_1\left(x+\mathrm{i}y\right)|$, $-4\le x\le 4$, $-4\le y\le 4$. …Also, $|E_1\left(z\right)|\to \mathrm{\infty }$ logarithmically as $z\to 0$. Magnify 3D Help

9: 6.14 Integrals

… ►

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:

pMML, png, TeX

See also:

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

►

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

ⓘ

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:

pMML, png, TeX

See also:

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

►

6.14.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{si}\left(t\right)dt=-\frac{1}{a}\arctan a,$$ $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.29

Permalink:

http://dlmf.nist.gov/6.14.E3

Encodings:

pMML, png, TeX

See also:

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

… ►

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

… ►

6.14.7 $${\int }_0^{\mathrm{\infty }}\mathrm{Ci}\left(t\right)\mathrm{si}\left(t\right)dt=\ln 2.$$

ⓘ

Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.32

Referenced by:

§6.14(ii)

Permalink:

http://dlmf.nist.gov/6.14.E7

Encodings:

pMML, png, TeX

See also:

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

…

10: 6.12 Asymptotic Expansions

… ►

§6.12(i) Exponential and Logarithmic Integrals

… ►

6.12.2 $$\mathrm{Ei}\left(x\right)\sim \frac{{\mathrm{e}}^x}{x}\left(1+\frac{1!}{x}+\frac{2!}{x^2}+\frac{3!}{x^3}+\mathrm{\cdots }\right),$$ $x\to +\mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral, $!$: factorial (as in $n!$) and $x$: real variable

Referenced by:

§6.12(i), §6.12(i)

Permalink:

http://dlmf.nist.gov/6.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(i), §6.12 and Ch.6

… ►For the function $\chi$ see §9.7(i). ►The asymptotic expansion of $\mathrm{li}\left(x\right)$ as $x\to \mathrm{\infty }$ is obtainable from (6.2.8) and (6.12.2). … ►

6.12.7 $$R_n^{(\mathrm{f})}(z)={(-1)}^n{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{2n}}{t^2+1}dt,$$

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Defines:

$R_n^{(\mathrm{f})}(z)$: remainder term (locally)

Symbols:

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

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

…