logarithm function

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1: 4.11 Sums

§4.11 Sums

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2: 4.2 Definitions

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§4.2(i) The Logarithm

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

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

… ►Natural logarithms have as base the unique positive number …

3: 4.47 Approximations

§4.47 Approximations

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§4.47(i) Chebyshev-Series Expansions

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4: 4.46 Tables

§4.46 Tables

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5: 5.10 Continued Fractions

§5.10 Continued Fractions

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

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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

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6: 4.1 Special Notation

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$k,m,n$	integers.
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… ►The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. ►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. …

7: 4.10 Integrals

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§4.10(i) Logarithms

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4.10.1 $$\int \frac{dz}{z}=\ln z,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.48

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

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4.10.2 $$\int \ln zdz=z\ln z-z,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.49

Permalink:

http://dlmf.nist.gov/4.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

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4.10.4 $$\int \frac{dz}{z\ln z}=\ln \left(\ln z\right),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.52

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

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4.10.7 $${⨍}_0^x\frac{dt}{\ln t}=\mathrm{li}\left(x\right),$$ $x>1$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

4.1.57

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

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8: 4.8 Identities

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§4.8(i) Logarithms

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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

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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

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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

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4.8.13 $$\ln \left(a^x\right)=x\ln a,$$ $a>0$.

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Symbols:

$\ln z$: principal branch of logarithm function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.8.E13

Encodings:

pMML, png, TeX

See also:

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

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9: 4.3 Graphics

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Figure 4.3.1: $\ln x$ and ${\mathrm{e}}^x$. … Magnify

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§4.3(ii) Complex Arguments: Conformal Maps

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Figure 4.3.3: $\ln \left(x+\mathrm{i}y\right)$ (principal value). … Magnify 3D Help

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10: 5.3 Graphics

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Figure 5.3.2: $\ln \mathrm{\Gamma }\left(x\right)$. … Magnify

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