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1: 4.1 Special Notation

… ►The main functions treated in this chapter are the logarithm $\ln z$, $\mathrm{Ln}z$; the exponential $\exp z$, ${\mathrm{e}}^z$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\arcsin z$, $\mathrm{Arcsin}z$, etc. ; the hyperbolic trigonometric (or just hyperbolic) functions $\sinh z$, $\cosh z$, $\tanh z$, $\mathrm{csch}z$, $\mathrm{sech}z$, $\coth z$; the inverse hyperbolic functions $\mathrm{arcsinh}z$, $\mathrm{Arcsinh}z$, etc. ►Sometimes in the literature the meanings of $\ln$ and $\mathrm{Ln}$ are interchanged; similarly for $\arcsin z$ and $\mathrm{Arcsin}z$, etc. …${\sin }^{-1}z$ for $\arcsin z$ and ${\mathrm{Sin}}^{-1}z$ for $\mathrm{Arcsin}z$.

2: 4.24 Inverse Trigonometric Functions: Further Properties

… ►

4.24.13 $$\mathrm{Arcsin}u\pm \mathrm{Arcsin}v=\mathrm{Arcsin}\left(u{(1-v^2)}^{1/2}\pm v{(1-u^2)}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.32

Permalink:

http://dlmf.nist.gov/4.24.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.14 $$\mathrm{Arccos}u\pm \mathrm{Arccos}v=\mathrm{Arccos}\left(uv\mp {((1-u^2)(1-v^2))}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arccos}z$: general arccosine function

A&S Ref:

4.4.33

Permalink:

http://dlmf.nist.gov/4.24.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.15 $$\mathrm{Arctan}u\pm \mathrm{Arctan}v=\mathrm{Arctan}\left(\frac{u\pm v}{1\mp uv}\right),$$

ⓘ

Symbols:

$\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.34

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.24.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.16 $$\mathrm{Arcsin}u\pm \mathrm{Arccos}v=\mathrm{Arcsin}\left(uv\pm {((1-u^2)(1-v^2))}^{1/2}\right)=\mathrm{Arccos}\left(v{(1-u^2)}^{1/2}\mp u{(1-v^2)}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arccos}z$: general arccosine function and $\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.35

Permalink:

http://dlmf.nist.gov/4.24.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.17 $$\mathrm{Arctan}u\pm \mathrm{Arccot}v=\mathrm{Arctan}\left(\frac{uv\pm 1}{v\mp u}\right)=\mathrm{Arccot}\left(\frac{v\mp u}{uv\pm 1}\right).$$

ⓘ

Symbols:

$\mathrm{Arccot}z$: general arccotangent function and $\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.36

Permalink:

http://dlmf.nist.gov/4.24.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

…

3: 4.8 Identities

… ►

4.8.1 $$\mathrm{Ln}\left(z_1{z}_2\right)=\mathrm{Ln}{z}_1+\mathrm{Ln}{z}_2.$$

ⓘ

Symbols:

$\mathrm{Ln}z$: general logarithm function and $z$: complex variable

A&S Ref:

4.1.6

Referenced by:

§4.8(i)

Permalink:

http://dlmf.nist.gov/4.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

►This is interpreted that every value of $\mathrm{Ln}\left(z_1{z}_2\right)$ is one of the values of $\mathrm{Ln}{z}_1+\mathrm{Ln}{z}_2$, and vice versa. … ►

4.8.3 $$\mathrm{Ln}\frac{z_1}{z_2}=\mathrm{Ln}{z}_1-\mathrm{Ln}{z}_2,$$

ⓘ

Symbols:

$\mathrm{Ln}z$: general logarithm function and $z$: complex variable

A&S Ref:

4.1.8

Permalink:

http://dlmf.nist.gov/4.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

… ►

4.8.5 $$\mathrm{Ln}\left(z^n\right)=n\mathrm{Ln}z,$$ $n\in \mathbb{Z}$,

ⓘ

Symbols:

$\in$: element of, $\mathbb{Z}$: set of all integers, $\mathrm{Ln}z$: general logarithm function, $n$: integer and $z$: complex variable

A&S Ref:

4.1.10

Referenced by:

§4.8(i)

Permalink:

http://dlmf.nist.gov/4.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

… ►

4.8.10 $$\exp \left(\ln z\right)=\exp \left(\mathrm{Ln}z\right)=z.$$

ⓘ

Symbols:

$\exp z$: exponential function, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.2.4

Referenced by:

(25.10.2), §4.8(i)

Permalink:

http://dlmf.nist.gov/4.8.E10

Encodings:

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See also:

Annotations for §4.8(i), §4.8 and Ch.4

…

4: 4.37 Inverse Hyperbolic Functions

… ►

4.37.4 $$\mathrm{Arccsch}z=\mathrm{Arcsinh}\left(1/z\right),$$

ⓘ

Defines:

$\mathrm{Arccsch}z$: general inverse hyperbolic cosecant function

Symbols:

$\mathrm{Arcsinh}z$: general inverse hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E4

Encodings:

pMML, png, TeX

See also:

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

►

4.37.5 $$\mathrm{Arcsech}z=\mathrm{Arccosh}\left(1/z\right),$$

ⓘ

Defines:

$\mathrm{Arcsech}z$: general inverse hyperbolic secant function

Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E5

Encodings:

pMML, png, TeX

See also:

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

►

4.37.6 $$\mathrm{Arccoth}z=\mathrm{Arctanh}\left(1/z\right).$$

ⓘ

Defines:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function

Symbols:

$\mathrm{Arctanh}z$: general inverse hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E6

Encodings:

pMML, png, TeX

See also:

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

… ►Each of the six functions is a multivalued function of $z$. $\mathrm{Arcsinh}z$ and $\mathrm{Arccsch}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. …

5: 5.10 Continued Fractions

… ►

5.10.1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)+z-\left(z-\frac{1}{2}\right)\ln z-\frac{1}{2}\ln \left(2\pi \right)=\frac{a_0}{z+}\frac{a_1}{z+}\frac{a_2}{z+}\frac{a_3}{z+}\frac{a_4}{z+}\frac{a_5}{z+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $z$: complex variable and $a_k$: coefficient

Referenced by:

Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.10.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.10 and Ch.5

…

6: 4.38 Inverse Hyperbolic Functions: Further Properties

… ►

4.38.15 $$\mathrm{Arcsinh}u\pm \mathrm{Arcsinh}v=\mathrm{Arcsinh}\left(u{(1+v^2)}^{1/2}\pm v{(1+u^2)}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arcsinh}z$: general inverse hyperbolic sine function

A&S Ref:

4.6.26

Permalink:

http://dlmf.nist.gov/4.38.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

►

4.38.16 $$\mathrm{Arccosh}u\pm \mathrm{Arccosh}v=\mathrm{Arccosh}\left(uv\pm {((u^2-1)(v^2-1))}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function

A&S Ref:

4.6.27

Permalink:

http://dlmf.nist.gov/4.38.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

►

4.38.17 $$\mathrm{Arctanh}u\pm \mathrm{Arctanh}v=\mathrm{Arctanh}\left(\frac{u\pm v}{1\pm uv}\right),$$

ⓘ

Symbols:

$\mathrm{Arctanh}z$: general inverse hyperbolic tangent function

A&S Ref:

4.6.28

Permalink:

http://dlmf.nist.gov/4.38.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

►

4.38.18 $$\mathrm{Arcsinh}u\pm \mathrm{Arccosh}v=\mathrm{Arcsinh}\left(uv\pm {((1+u^2)(v^2-1))}^{1/2}\right)=\mathrm{Arccosh}\left(v{(1+u^2)}^{1/2}\pm u{(v^2-1)}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function and $\mathrm{Arcsinh}z$: general inverse hyperbolic sine function

A&S Ref:

4.6.29

Permalink:

http://dlmf.nist.gov/4.38.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

►

4.38.19 $$\mathrm{Arctanh}u\pm \mathrm{Arccoth}v=\mathrm{Arctanh}\left(\frac{uv\pm 1}{v\pm u}\right)=\mathrm{Arccoth}\left(\frac{v\pm u}{uv\pm 1}\right).$$

ⓘ

Symbols:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function and $\mathrm{Arctanh}z$: general inverse hyperbolic tangent function

A&S Ref:

4.6.30

Permalink:

http://dlmf.nist.gov/4.38.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

…

7: 5.17 Barnes’ $G$-Function (Double Gamma Function)

… ►

5.17.4 $$\mathrm{Ln}G\left(z+1\right)=\frac{1}{2}z\ln \left(2\pi \right)-\frac{1}{2}z(z+1)+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-{\int }_0^z\mathrm{Ln}\mathrm{\Gamma }\left(t+1\right)dt.$$

ⓘ

Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

►In this equation (and in (5.17.5) below), the $\mathrm{Ln}$’s have their principal values on the positive real axis and are continued via continuity, as in §4.2(i). … ►

5.17.5 $$\mathrm{Ln}G\left(z+1\right)\sim \frac{1}{4}{z}^2+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\ln z-\ln A+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$$

ⓘ

Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $z$: complex variable and $A$: Glaisher’s constant

Referenced by:

§5.17, §5.17, Erratum (V1.0.7) for Equation (5.17.5), Erratum (V1.1.11) for Equation (5.17.5)

Permalink:

http://dlmf.nist.gov/5.17.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.1.11):

For consistency we have replaced $\mathrm{Ln}z$ by $\ln z$.

Errata (effective with 1.0.7):

Originally the term $z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$ was incorrectly stated as $z\mathrm{\Gamma }\left(z+1\right)$. (This error was reported subsequently by Nick Jones on December 11, 2013.)

Reported 2013-08-01 by Gergő Nemes

See also:

Annotations for §5.17 and Ch.5

…

8: 4.2 Definitions

… ►The general logarithm function $\mathrm{Ln}z$ is defined by …This is a multivalued function of $z$ with branch point at $z=0$. … ►Most texts extend the definition of the principal value to include the branch cut … ►In all other cases, $z^a$ is a multivalued function with branch point at $z=0$. … ►This result is also valid when $z^a$ has its principal value, provided that the branch of $\mathrm{Ln}w$ satisfies …

9: 4.7 Derivatives and Differential Equations

… ►

4.7.2 $$\frac{d}{dz}\mathrm{Ln}z=\frac{1}{z},$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{Ln}z$: general logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.7.E2

Encodings:

pMML, png, TeX

See also:

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

… ►

4.7.4 $$\frac{d^n}{{dz}^n}\mathrm{Ln}z={(-1)}^{n-1}(n-1)!z^{-n}.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $!$: factorial (as in $n!$), $\mathrm{Ln}z$: general logarithm function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.7.E4

Encodings:

pMML, png, TeX

See also:

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

… ►

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 $a^z$. …

10: 4.23 Inverse Trigonometric Functions

… ►

4.23.4 $$\mathrm{Arccsc}z=\mathrm{Arcsin}\left(1/z\right),$$

ⓘ

Defines:

$\mathrm{Arccsc}z$: general arccosecant function

Symbols:

$\mathrm{Arcsin}z$: general arcsine function and $z$: complex variable

A&S Ref:

4.4.6 (modified)

Permalink:

http://dlmf.nist.gov/4.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(i), §4.23 and Ch.4

►

4.23.5 $$\mathrm{Arcsec}z=\mathrm{Arccos}\left(1/z\right),$$

ⓘ

Defines:

$\mathrm{Arcsec}z$: general arcsecant function

Symbols:

$\mathrm{Arccos}z$: general arccosine function and $z$: complex variable

A&S Ref:

4.4.7 (modified)

Permalink:

http://dlmf.nist.gov/4.23.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(i), §4.23 and Ch.4

►

4.23.6 $$\mathrm{Arccot}z=\mathrm{Arctan}\left(1/z\right).$$

ⓘ

Defines:

$\mathrm{Arccot}z$: general arccotangent function

Symbols:

$\mathrm{Arctan}z$: general arctangent function and $z$: complex variable

A&S Ref:

4.4.8 (modified)

Permalink:

http://dlmf.nist.gov/4.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(i), §4.23 and Ch.4

… ►Each of the six functions is a multivalued function of $z$. $\mathrm{Arctan}z$ and $\mathrm{Arccot}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. …