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11: 4.5 Inequalities

… ►

§4.5(i) Logarithms

►

4.5.1 $x>-1$, $x\ne 0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.1.33

Referenced by:

§4.5(i), §4.5(ii)

Permalink:

http://dlmf.nist.gov/4.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(i), §4.5 and Ch.4

►

4.5.2 , $x\ne 0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.1.34

Referenced by:

§4.5(i), §4.5(ii)

Permalink:

http://dlmf.nist.gov/4.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(i), §4.5 and Ch.4

… ►

4.5.4 $$\ln x\le x-1,$$ $x>0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.1.36

Referenced by:

§4.5(i)

Permalink:

http://dlmf.nist.gov/4.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(i), §4.5 and Ch.4

… ►For more inequalities involving the logarithm function see Mitrinović (1964, pp. 75–77), Mitrinović (1970, pp. 272–276), and Bullen (1998, pp. 159–160). …

12: 27.11 Asymptotic Formulas: Partial Sums

… ►

27.11.3 $$\sum _{n\le x}\frac{d\left(n\right)}{n}=\frac{1}{2}{(\ln x)}^2+2\gamma \ln x+O\left(1\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $d_k\left(n\right)$: divisor function, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E3

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 §27.11 and Ch.27

… ►

27.11.8 $$\sum _{p\le x}\frac{1}{p}=\ln \ln x+A+O\left(\frac{1}{\ln x}\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\ln z$: principal branch of logarithm function, $p,p_1,\mathrm{\dots }$: prime numbers, $x$: real number and $A$: constant

Permalink:

http://dlmf.nist.gov/27.11.E8

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 §27.11 and Ch.27

… ►

27.11.10 $$\sum _{p\le x}\frac{\ln p}{p}=\ln x+O\left(1\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\ln z$: principal branch of logarithm function, $p,p_1,\mathrm{\dots }$: prime numbers and $x$: real number

Permalink:

http://dlmf.nist.gov/27.11.E10

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 §27.11 and Ch.27

►

27.11.11 $$\sum _{\begin{array}{c}p\le x\\ p\equiv h(modk)\end{array}}\frac{\ln p}{p}=\frac{1}{\varphi \left(k\right)}\ln x+O\left(1\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\equiv$: modular equivalence, $\ln z$: principal branch of logarithm function, $k$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $x$: real number

Referenced by:

§27.11, §27.11

Permalink:

http://dlmf.nist.gov/27.11.E11

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 §27.11 and Ch.27

… ►The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if $\left(h,k\right)=1$, then the number of primes $p\le x$ with $p\equiv h(modk)$ is asymptotic to $x/(\varphi \left(k\right)\ln x)$ as $x\to \mathrm{\infty }$.

13: 6.15 Sums

… ►

6.15.1 $$\sum _{n=1}^{\mathrm{\infty }}\mathrm{Ci}\left(\pi n\right)=\frac{1}{2}(\ln 2-\gamma ),$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

►

6.15.2 $$\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{si}\left(\pi n\right)}{n}=\frac{1}{2}\pi (\ln \pi -1),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

►

6.15.3 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\mathrm{Ci}\left(2\pi n\right)=1-\ln 2-\frac{1}{2}\gamma ,$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

►

6.15.4 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{\mathrm{si}\left(2\pi n\right)}{n}=\pi (\frac{3}{2}\ln 2-1).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

…

14: 27 Functions of Number Theory

…

15: 6 Exponential, Logarithmic, Sine, and
Cosine Integrals

Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals

…

16: 4.6 Power Series

… ►

§4.6(i) Logarithms

►

4.6.1 $$\ln \left(1+z\right)=z-\frac{1}{2}{z}^2+\frac{1}{3}{z}^3-\mathrm{\cdots },$$ $|z|\le 1$, $z\ne -1$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.24

Referenced by:

§4.45(i), §4.6(i), (8.15.2)

Permalink:

http://dlmf.nist.gov/4.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

►

4.6.2 $$\ln z=\left(\frac{z-1}{z}\right)+\frac{1}{2}{\left(\frac{z-1}{z}\right)}^2+\frac{1}{3}{\left(\frac{z-1}{z}\right)}^3+\mathrm{\cdots },$$ $\mathrm{\Re }z\ge \frac{1}{2}$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.1.25

Permalink:

http://dlmf.nist.gov/4.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

►

4.6.3 $$\ln z=(z-1)-\frac{1}{2}{(z-1)}^2+\frac{1}{3}{(z-1)}^3-\mathrm{\cdots },$$ $|z-1|\le 1$, $z\ne 0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.26

Permalink:

http://dlmf.nist.gov/4.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

… ►

4.6.6 $$\ln \left(z+a\right)=\ln a+2\left(\left(\frac{z}{2a+z}\right)+\frac{1}{3}{\left(\frac{z}{2a+z}\right)}^3+\frac{1}{5}{\left(\frac{z}{2a+z}\right)}^5+\mathrm{\cdots }\right),$$ $a>0$, $\mathrm{\Re }z\ge -a$, $z\ne -a$.

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.1.29

Permalink:

http://dlmf.nist.gov/4.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

…

17: 4.7 Derivatives and Differential Equations

… ►

§4.7(i) Logarithms

►

4.7.1 $$\frac{d}{dz}\ln z=\frac{1}{z},$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.46

Permalink:

http://dlmf.nist.gov/4.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(i), §4.7 and Ch.4

… ►For a nonvanishing analytic function $f(z)$, the general solution of the differential equation … ►

4.7.6 $$w(z)=\mathrm{Ln}\left(f(z)\right)+\text{ constant}.$$

ⓘ

Defines:

$w$: solution (locally)

Symbols:

$\mathrm{Ln}z$: general logarithm function, $z$: complex variable and $f(z)$: non-vanishing analytic function

Permalink:

http://dlmf.nist.gov/4.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(i), §4.7 and Ch.4

… ►When $a^z$ is a general power, $\ln a$ is replaced by the branch of $\mathrm{Ln}a$ used in constructing $a^z$. …

18: 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 $\mathrm{Cin}\left(z\right)$. …

19: 4.47 Approximations

§4.47 Approximations

►

§4.47(i) Chebyshev-Series Expansions

►Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arctan$, $\mathrm{arcsinh}$. … ►Hart et al. (1968) give $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arccos$, $\arctan$, $\sinh$, $\cosh$, $\tanh$, $\mathrm{arcsinh}$, $\mathrm{arccosh}$. … ►Luke (1975, Chapter 3) supplies real and complex approximations for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\arctan$, $\mathrm{arcsinh}$. …

20: 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

ⓘ

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