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11: 5.1 Special Notation

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$j,m,n$	nonnegative integers.
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$x,y$	real variables.
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$a,b,q,s,w$	real or complex variables with .
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12: 5.6 Inequalities

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§5.6(i) Real Variables

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5.6.2 $$\frac{1}{\mathrm{\Gamma }\left(x\right)}+\frac{1}{\mathrm{\Gamma }\left(1/x\right)}\le 2,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(i), §5.6 and Ch.5

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5.6.3 $$\frac{1}{{(\mathrm{\Gamma }\left(x\right))}^2}+\frac{1}{{(\mathrm{\Gamma }\left(1/x\right))}^2}\le 2,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(i), §5.6 and Ch.5

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5.6.6 $$|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|\le |\mathrm{\Gamma }\left(x\right)|,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $x$: real variable and $y$: real variable

A&S Ref:

6.1.26

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

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5.6.9 $$|\mathrm{\Gamma }\left(z\right)|\le {(2\pi )}^{1/2}{|z|}^{x-(1/2)}{\mathrm{e}}^{-\pi |y|/2}\exp \left(\frac{1}{6}{|z|}^{-1}\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $x$: real variable, $y$: real variable and $z$: complex variable

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

13: 12.1 Special Notation

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$x,y$	real variables.
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14: 7.3 Graphics

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§7.3(i) Real Variable

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15: 9.3 Graphics

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§9.3(i) Real Variable

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16: 6.3 Graphics

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§6.3(i) Real Variable

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17: 4.12 Generalized Logarithms and Exponentials

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4.12.1 $$\varphi (x+1)={\mathrm{e}}^{\varphi (x)},$$ ,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $x$: real variable and $\varphi (z)$: generalized exponential

Permalink:

http://dlmf.nist.gov/4.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

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4.12.3 $$\psi ({\mathrm{e}}^x)=1+\psi (x),$$ ,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $x$: real variable and $\psi (x)$: generalized logarithm

Permalink:

http://dlmf.nist.gov/4.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

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4.12.5 $$\varphi (x)=\psi (x)=x,$$ $0\le x\le 1$.

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Symbols:

$x$: real variable, $\varphi (z)$: generalized exponential and $\psi (x)$: generalized logarithm

Permalink:

http://dlmf.nist.gov/4.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

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

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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

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4.12.8 $$\psi (x)={\mathrm{e}}^x-1,$$ ,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $x$: real variable and $\psi (x)$: generalized logarithm

Permalink:

http://dlmf.nist.gov/4.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

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18: 4.5 Inequalities

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4.5.4 $$\ln x\le x-1,$$ $x>0$,

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Symbols:

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

A&S Ref:

4.1.36

Referenced by:

§4.5(i)

Permalink:

http://dlmf.nist.gov/4.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(i), §4.5 and Ch.4

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4.5.8 ,

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Symbols:

$\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

4.2.30

Permalink:

http://dlmf.nist.gov/4.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(ii), §4.5 and Ch.4

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4.5.9 ,

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Symbols:

$\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

4.2.31

Permalink:

http://dlmf.nist.gov/4.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(ii), §4.5 and Ch.4

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4.5.10 $x>-1$,

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Symbols:

$\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

4.2.32

Permalink:

http://dlmf.nist.gov/4.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(ii), §4.5 and Ch.4

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4.5.13 $x>0$, $y>0$,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $x$: real variable and $y$: real variable

A&S Ref:

4.2.36

Referenced by:

§4.5(ii)

Permalink:

http://dlmf.nist.gov/4.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(ii), §4.5 and Ch.4

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19: 10.3 Graphics

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§10.3(i) Real Order and Variable

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§10.3(ii) Real Order, Complex Variable

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§10.3(iii) Imaginary Order, Real Variable

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20: 4.45 Methods of Computation

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§4.45(i) Real Variables

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4.45.1 $$y=x^{2^{-m}}-1.$$

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Symbols:

$m$: integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.45.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.45(i), §4.45(i), §4.45 and Ch.4

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4.45.2 $$\ln x=2^m\ln \left(1+y\right).$$

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Symbols:

$\ln z$: principal branch of logarithm function, $m$: integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.45.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.45(i), §4.45(i), §4.45 and Ch.4

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4.45.3 $$\ln x=\ln \xi +m\ln 10.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/4.45.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.45(i), §4.45(i), §4.45 and Ch.4

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4.45.5 $${\mathrm{e}}^x={10}^m{\mathrm{e}}^y,$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $m$: integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.45.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.45(i), §4.45(i), §4.45 and Ch.4

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