principal branches (or values)

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11: 6.14 Integrals

… ►

6.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_1\left(t\right)dt=\frac{1}{a}\ln \left(1+a\right),$$ $\mathrm{\Re }a>-1$,

ⓘ

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

… ►

6.14.4 $${\int }_0^{\mathrm{\infty }}{E}_1^2\left(t\right)dt=2\ln 2,$$

ⓘ

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

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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

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12: 10.2 Definitions

… ►

§10.2(ii) Standard Solutions

… ►The principal branch corresponds to the principal branches of $J_{\pm \nu }\left(z\right)$ in (10.2.3) and (10.2.4), with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$. … ►The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$. … ► … ►

Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.

► ►►

Pair	Interval or Region
…

►

13: 4.3 Graphics

… ►Figure 4.3.2 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the negative real axis, where $w={\mathrm{e}}^z$ and $z=\ln w$ (principal value). … ►

►

►

Figure 4.3.3: $\ln \left(x+\mathrm{i}y\right)$ (principal value). There is a branch cut along the negative real axis. Magnify 3D Help

…

14: 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

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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

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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

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15: 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

…

16: 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

…

17: 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

…

18: 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

…

19: 4.40 Integrals

… ►

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

… ►

4.40.13 $$\int \mathrm{arctanh}xdx=x\mathrm{arctanh}x+\frac{1}{2}\ln \left(1-x^2\right),$$ ,

ⓘ

Symbols:

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

A&S Ref:

4.6.45 (modified)

Permalink:

http://dlmf.nist.gov/4.40.E13

Encodings:

pMML, png, TeX

See also:

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

…

20: 4.37 Inverse Hyperbolic Functions

… ►The principal values (or principal branches) of the inverse $\sinh$, $\cosh$, and $\tanh$ are obtained by introducing cuts in the $z$-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. …The principal branches are denoted by $\mathrm{arcsinh}$, $\mathrm{arccosh}$, $\mathrm{arctanh}$ respectively. … ►

4.37.19 $$\mathrm{arccosh}z=\ln \left(\pm {(z^2-1)}^{1/2}+z\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },1)$,

ⓘ

Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction and $z$: complex variable

Referenced by:

§4.37(iv), §4.37(iii), §4.37(iv)

Permalink:

http://dlmf.nist.gov/4.37.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

… ►

4.37.20 $$\mathrm{arccosh}\left(\mathrm{i}y\right)=\pm \frac{1}{2}\pi \mathrm{i}+\ln \left({(y^2+1)}^{1/2}\pm y\right),$$ $y\gtrless 0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function and $y$: real variable

Referenced by:

§4.37(iv)

Permalink:

http://dlmf.nist.gov/4.37.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

… ►

4.37.22 $$\mathrm{arccosh}x=\pm \ln \left(\mathrm{i}{(1-x^2)}^{1/2}+x\right),$$ $x\in (-1,1]$,

ⓘ

Symbols:

$\in$: element of, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.37.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

…