generalized logarithms

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1: 4.44 Other Applications

§4.44 Other Applications

… ►For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). …

2: 4.12 Generalized Logarithms and Exponentials

§4.12 Generalized Logarithms and Exponentials

►A generalized exponential function $\varphi (x)$ satisfies the equations …Its inverse $\psi (x)$ is called a generalized logarithm. … ►For $C^{\mathrm{\infty }}$ generalized logarithms, see Walker (1991). For analytic generalized logarithms, see Kneser (1950).

3: 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

…

4: 4.2 Definitions

… ►

§4.2(i) The Logarithm

►The general logarithm function $\mathrm{Ln}z$ is defined by ►

4.2.1 $$\mathrm{Ln}z={\int }_1^z\frac{dt}{t},$$ $z\ne 0$,

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Defines:

$\mathrm{Ln}z$: general logarithm function

Symbols:

$dx$: differential of $x$ , $\int$: integral and $z$: complex variable

A&S Ref:

4.1.4

Referenced by:

§4.2(i)

Permalink:

http://dlmf.nist.gov/4.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(i), §4.2 and Ch.4

… ►With this definition the general logarithm is given by … ►

§4.2(ii) Logarithms to a General Base $a$

…

5: 4.8 Identities

… ►

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

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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

… ►

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

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

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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

… ►

4.8.8 $$\mathrm{Ln}\left(\exp z\right)=z+2k\pi \mathrm{i},$$ $k\in \mathbb{Z}$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\mathrm{Ln}z$: general logarithm function, $k$: integer and $z$: complex variable

A&S Ref:

4.2.2

Permalink:

http://dlmf.nist.gov/4.8.E8

Encodings:

pMML, png, TeX

See also:

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

… ►

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

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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:

pMML, png, TeX

See also:

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

…

6: 4.7 Derivatives and Differential Equations

… ►

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

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

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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

►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 $a^z$. …

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

… ►

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: 5.9 Integral Representations

… ►

5.9.10 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+2{\int }_0^{\mathrm{\infty }}\frac{\arctan \left(t/z\right)}{{\mathrm{e}}^{2\pi t}-1}dt,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

6.1.50

Referenced by:

(5.9.10_1), §5.9(i), §5.9(ii), 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/5.9.E10

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 §5.9(i), §5.9(i), §5.9 and Ch.5

… ►

5.9.10_1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)-\frac{z}{\pi }{\int }_0^{\mathrm{\infty }}\frac{\ln \left(1-{\mathrm{e}}^{-2\pi t}\right)}{t^2+z^2}dt,$$

ⓘ

Symbols:

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

Proof sketch:

Can be obtained from (5.9.10) via integration by parts.

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E10_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

►

5.9.10_2 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}\left(\frac{1}{{\mathrm{e}}^t-1}-\frac{1}{t}+\frac{1}{2}\right)\frac{dt}{t},$$

ⓘ

Symbols:

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

Source:

Whittaker and Watson (1927, §12.31)

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E10_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

… ►

5.9.11 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)=-\gamma z-\frac{1}{2\pi \mathrm{i}}{\int }_{-c-\mathrm{\infty }\mathrm{i}}^{-c+\mathrm{\infty }\mathrm{i}}\frac{\pi z^{-s}}{s\sin \left(\pi s\right)}\zeta \left(-s\right)ds,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $s$: real or complex variable and $z$: complex variable

Referenced by:

§5.9(i), §5.9(ii), 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/5.9.E11

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 $\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+1\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

…

9: 6.4 Analytic Continuation

… ►

6.4.1 $$E_1\left(z\right)=\mathrm{Ein}\left(z\right)-\mathrm{Ln}z-\gamma ;$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $E_1\left(z\right)$: exponential integral, $\mathrm{Ln}z$: general logarithm function and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

…

10: 3.1 Arithmetics and Error Measures

… ►To eliminate overflow or underflow in finite-precision arithmetic numbers are represented by using generalized logarithms ${\ln }_{\mathrm{\ell }}(x)$ given by …