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21: Mark J. Ablowitz

… ►ODEs which do not have moveable branch point singularities. …

22: 2.2 Transcendental Equations

… ►

2.2.3 $$t^2-\ln t=y.$$

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $y$: root

Permalink:

http://dlmf.nist.gov/2.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

… ►

2.2.5 $$t^2=y+\ln t=y+\frac{1}{2}\ln y+o\left(1\right),$$

ⓘ

Symbols:

$o\left(x\right)$: order less than, $\ln z$: principal branch of logarithm function and $y$: root

Permalink:

http://dlmf.nist.gov/2.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

… ►

2.2.6 $$t=y^{\frac{1}{2}}\left(1+\frac{1}{4}{y}^{-1}\ln y+o\left(y^{-1}\right)\right),$$ $y\to \mathrm{\infty }$.

ⓘ

Symbols:

$o\left(x\right)$: order less than, $\ln z$: principal branch of logarithm function and $y$: root

Permalink:

http://dlmf.nist.gov/2.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

…

23: 4.12 Generalized Logarithms and Exponentials

… ►

4.12.6 $$\varphi (x)=\ln \left(x+1\right),$$ ,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $x$: real variable and $\varphi (z)$: generalized exponential

Permalink:

http://dlmf.nist.gov/4.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

… ►

4.12.9 $$\psi (x)=\mathrm{\ell }+\underset{\mathrm{\ell }\text{ times}}{\underbrace{\ln \mathrm{\cdots }\ln }}x,$$ $x>1$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $x$: real variable, $\psi (x)$: generalized logarithm and $\mathrm{\ell }$: positive integer

Permalink:

http://dlmf.nist.gov/4.12.E9

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.24):

The $\mathrm{\ell }$-times repeated application of $\ln$, previously given as ${\ln }^{(\mathrm{\ell })}$, has been rewritten using a more conventional notation.

See also:

Annotations for §4.12 and Ch.4

… ►

4.12.10

ⓘ

Defines:

$\mathrm{\ell }$: positive integer (locally)

Symbols:

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

Permalink:

http://dlmf.nist.gov/4.12.E10

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.24):

The $\mathrm{\ell }$-times repeated application of $\ln$, previously given as ${\ln }^{(\mathrm{\ell })}$, has been rewritten using a more conventional notation.

See also:

Annotations for §4.12 and Ch.4

…

24: 14.21 Definitions and Basic Properties

… ► $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ exist for all values of $\nu$, $\mu$, and $z$, except possibly $z=\pm 1$ and $\mathrm{\infty }$, which are branch points (or poles) of the functions, in general. When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). The principal branches of $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are real when $\nu$, $\mu \in ℝ$ and $z\in (1,\mathrm{\infty })$. …

25: 27.11 Asymptotic Formulas: Partial Sums

… ►

27.11.2 $$\sum _{n\le x}d\left(n\right)=x\ln x+(2\gamma -1)x+O\left(\sqrt{x}\right),$$

ⓘ

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

A&S Ref:

24.3.3 III

Referenced by:

§27.11, §27.11, Erratum (V1.1.8) for Section 27.11

Permalink:

http://dlmf.nist.gov/27.11.E2

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

… ►

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

ⓘ

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

… ►

27.11.6 $$\sum _{n\le x}\varphi \left(n\right)=\frac{3}{{\pi }^2}{x}^2+O\left(x\ln x\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.2 III (in slightly different form)

Permalink:

http://dlmf.nist.gov/27.11.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 §27.11 and Ch.27

►

27.11.7 $$\sum _{n\le x}\frac{\varphi \left(n\right)}{n}=\frac{6}{{\pi }^2}x+O\left(\ln x\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.11.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 §27.11 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

…

26: 4.40 Integrals

… ►The results in §§4.40(ii) and 4.40(iv) can be extended to the complex plane by using continuous branches and avoiding singularities. … ►

4.40.3 $$\int \tanh xdx=\ln \left(\cosh x\right).$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.79

Permalink:

http://dlmf.nist.gov/4.40.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

►

4.40.4 $$\int \mathrm{csch}xdx=\ln \left(\tanh \left(\frac{1}{2}x\right)\right),$$ .

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{csch}z$: hyperbolic cosecant function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.80

Permalink:

http://dlmf.nist.gov/4.40.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

… ►

4.40.6 $$\int \coth xdx=\ln \left(\sinh x\right),$$ .

ⓘ

Symbols:

$dx$: differential of $x$ , $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.82

Permalink:

http://dlmf.nist.gov/4.40.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

… ►

4.40.10 $${\int }_0^{\mathrm{\infty }}\frac{\tanh \left(ax\right)-\tanh \left(bx\right)}{x}dx=\ln \left(\frac{a}{b}\right),$$ $a>0$, $b>0$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

…

27: 27.12 Asymptotic Formulas: Primes

… ►

27.12.1 $$\underset{n\to \mathrm{\infty }}{lim}\frac{p_n}{n\ln n}=1,$$

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.12.E1

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 and Ch.27

►

27.12.2 $$p_n>n\ln n,$$ $n=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.12.E2

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 and Ch.27

… ►

27.12.4 $$\pi \left(x\right)\sim \sum _{k=1}^{\mathrm{\infty }}\frac{(k-1)!x}{{(\ln x)}^k}.$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number, $k$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.12.E4

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 and Ch.27

… ►

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

… ►

27.12.7

ⓘ

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

…

28: 32.11 Asymptotic Approximations for Real Variables

… ►

32.11.8 $$d^2=-{\pi }^{-1}\ln \left(1-k^2\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $k$: real and $d$: constant

Permalink:

http://dlmf.nist.gov/32.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(ii), §32.11 and Ch.32

… ►where $\mathrm{\Gamma }$ is the gamma function (§5.2(i)), and the branch of the $\mathrm{ph}$ function is immaterial. … ►

32.11.17 $$d^2={\pi }^{-1}\ln \left(1+k^2\right),$$ $\mathrm{sign}\left(k\right)={(-1)}^n$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{sign}x$: sign of, $n$: integer, $k$: real and $d$: constant

Permalink:

http://dlmf.nist.gov/32.11.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(iii), §32.11 and Ch.32

… ►

32.11.20 $$\psi (x)=\frac{2}{3}\sqrt{2}{x}^{3/2}-\frac{3}{2}{\rho }^2\ln x.$$

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $x$: real, $\rho$: real constant and $\psi (z)$: function

Permalink:

http://dlmf.nist.gov/32.11.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(iii), §32.11 and Ch.32

… ►and the branch of the $\mathrm{ph}$ function is immaterial. …

29: 4.23 Inverse Trigonometric Functions

… ►The function ${(1-t^2)}^{1/2}$ assumes its principal value when $t\in (-1,1)$; elsewhere on the integration paths the branch is determined by continuity. …$\mathrm{Arctan}z$ and $\mathrm{Arccot}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. … ►The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the $z$-plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. …The principal branches are denoted by $\arcsin z$, $\arccos z$, $\arctan z$, respectively. … ►

…

Figure 4.23.1: $z$-plane. …

…

30: 4.4 Special Values and Limits

… ►

4.4.1 $$\ln 1=0,$$

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function

A&S Ref:

4.1.12

Permalink:

http://dlmf.nist.gov/4.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(i), §4.4 and Ch.4

►

4.4.2 $$\ln \left(-1\pm \mathrm{i}0\right)=\pm \pi \mathrm{i},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\ln z$: principal branch of logarithm function

A&S Ref:

4.1.14 (modified)

Permalink:

http://dlmf.nist.gov/4.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(i), §4.4 and Ch.4

►

4.4.3 $$\ln \left(\pm \mathrm{i}\right)=\pm \frac{1}{2}\pi \mathrm{i}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\ln z$: principal branch of logarithm function

A&S Ref:

4.1.15

Permalink:

http://dlmf.nist.gov/4.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(i), §4.4 and Ch.4

… ►

4.4.13 $$\underset{x\to \mathrm{\infty }}{lim}{x}^{-a}\ln x=0,$$ $\mathrm{\Re }a>0$,

ⓘ

Symbols:

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

A&S Ref:

4.1.30

Permalink:

http://dlmf.nist.gov/4.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(iii), §4.4 and Ch.4

►

4.4.14 $$\underset{x\to 0}{lim}{x}^a\ln x=0,$$ $\mathrm{\Re }a>0$,

ⓘ

Symbols:

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

A&S Ref:

4.1.31

Permalink:

http://dlmf.nist.gov/4.4.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(iii), §4.4 and Ch.4

…