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11: 4.2 Definitions

… ►In all other cases, $z^a$ is a multivalued function with branch point at $z=0$. …

12: 25.2 Definition and Expansions

… ►

25.2.5 $${\gamma }_n=\underset{m\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^m\frac{{(\ln k)}^n}{k}-\frac{{(\ln m)}^{n+1}}{n+1}\right).$$

ⓘ

Defines:

${\gamma }_n$: Stieltjes constants

Symbols:

$\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

definition, Stieltjes constants

Sources:

Hardy (1912, p. 215); Ivić (1985, (1.12), p. 4)

Referenced by:

§25.2(ii), §25.6(ii), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

…

13: 15.4 Special Cases

… ►

§15.4(i) Elementary Functions

►The following results hold for principal branches when , and by analytic continuation elsewhere. … ►

15.4.1 $$F(1,1;2;z)=-z^{-1}\ln \left(1-z\right),$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Referenced by:

§15.4(i)

Permalink:

http://dlmf.nist.gov/15.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(i), §15.4 and Ch.15

… ►

§15.4(ii) Argument Unity

… ►

§15.4(iii) Other Arguments

…

14: 4.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►

$k,m,n$	integers.
…

…

15: 13.14 Definitions and Basic Properties

… ►Standard solutions are: … ►In general $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are many-valued functions of $z$ with branch points at $z=0$ and $z=\mathrm{\infty }$. The principal branches correspond to the principal branches of the functions $z^{\frac{1}{2}+\mu }$ and $U(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z)$ on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i). … ►In all other cases …

16: 4.45 Methods of Computation

… ►For other values of $x$ set $x={10}^m\xi$, where $1/10\le \xi \le 10$ and $m\in \mathbb{Z}$. … ►The other trigonometric functions can be found from the definitions (4.14.4)–(4.14.7). … ►

Other Methods

… ►For other methods see Miel (1981). … ►Initial approximations are obtainable, for example, from the power series (4.13.6) (with $t\ge 0$) when $x$ is close to $-1/\mathrm{e}$, from the asymptotic expansion (4.13.10) when $x$ is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of $x$. …

17: 4.6 Power Series

… ►

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

►

4.6.4 $$\ln z=2\left(\left(\frac{z-1}{z+1}\right)+\frac{1}{3}{\left(\frac{z-1}{z+1}\right)}^3+\frac{1}{5}{\left(\frac{z-1}{z+1}\right)}^5+\mathrm{\cdots }\right),$$ $\mathrm{\Re }z\ge 0$, $z\ne 0$,

ⓘ

Symbols:

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

A&S Ref:

4.1.27

Permalink:

http://dlmf.nist.gov/4.6.E4

Encodings:

pMML, png, TeX

See also:

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

►

4.6.5 $$\ln \left(\frac{z+1}{z-1}\right)=2\left(\frac{1}{z}+\frac{1}{3z^3}+\frac{1}{5z^5}+\mathrm{\cdots }\right),$$ $|z|\ge 1$, $z\ne \pm 1$,

ⓘ

Symbols:

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

A&S Ref:

4.1.28

Permalink:

http://dlmf.nist.gov/4.6.E5

Encodings:

pMML, png, TeX

See also:

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

…

18: 10.16 Relations to Other Functions

§10.16 Relations to Other Functions

►

Elementary Functions

… ►

Parabolic Cylinder Functions

… ►

Confluent Hypergeometric Functions

… ►

Generalized Hypergeometric Functions

…

19: 27.12 Asymptotic Formulas: Primes

… ►

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

ⓘ

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

…

20: 6.2 Definitions and Interrelations

… ►

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

ⓘ

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

… ►

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

ⓘ

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

… ►The logarithmic integral is defined by ►

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

ⓘ

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

ⓘ

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

…