4 Elementary Functions Logarithm, Exponential, Powers 4.11 Sums 4.13 Lambert

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§4.12 Generalized Logarithms and Exponentials

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Keywords:: exponential function, generalized, generalized exponentials, generalized logarithms, logarithm function
Permalink:: http://dlmf.nist.gov/4.12
See also:: Annotations for Ch.4

A generalized exponential function $\phi(x)$ satisfies the equations

4.12.1	$\displaystyle\phi(x+1)$	$\displaystyle=e^{\phi(x)},$
$-1<x<\infty$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $x$ : real variable and $\phi(z)$ : generalized exponential Permalink: http://dlmf.nist.gov/4.12.E1 Encodings: TeX, pMML, png See also: Annotations for §4.12 and Ch.4
4.12.2	$\displaystyle\phi(0)$	$\displaystyle=0,$
ⓘ Symbols: $\phi(z)$ : generalized exponential Permalink: http://dlmf.nist.gov/4.12.E2 Encodings: TeX, pMML, png See also: Annotations for §4.12 and Ch.4

and is strictly increasing when $0\leq x\leq 1$ . Its inverse $\psi(x)$ is called a generalized logarithm. It, too, is strictly increasing when $0\leq x\leq 1$ , and

4.12.3	$\displaystyle\psi(e^{x})$	$\displaystyle=1+\psi(x),$
$-\infty<x<\infty$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $x$ : real variable and $\psi(x)$ : generalized logarithm Permalink: http://dlmf.nist.gov/4.12.E3 Encodings: TeX, pMML, png See also: Annotations for §4.12 and Ch.4
4.12.4	$\displaystyle\psi(0)$	$\displaystyle=0.$
ⓘ Symbols: $\psi(x)$ : generalized logarithm Permalink: http://dlmf.nist.gov/4.12.E4 Encodings: TeX, pMML, png See also: Annotations for §4.12 and Ch.4

These functions are not unique. The simplest choice is given by

4.12.5		$\phi(x)=\psi(x)=x,$
$0\leq x\leq 1$ .
ⓘ Symbols: $x$ : real variable, $\phi(z)$ : generalized exponential and $\psi(x)$ : generalized logarithm Permalink: http://dlmf.nist.gov/4.12.E5 Encodings: TeX, pMML, png See also: Annotations for §4.12 and Ch.4

Then

4.12.6		$\phi(x)=\ln\left(x+1\right),$
$-1<x<0$ ,
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real variable and $\phi(z)$ : generalized exponential Permalink: http://dlmf.nist.gov/4.12.E6 Encodings: TeX, pMML, png See also: Annotations for §4.12 and Ch.4

and

4.12.7		$\phi(x)=\underbrace{\exp\cdots\exp}_{\left\lfloor x\right\rfloor\text{ times}}% (x-\left\lfloor x\right\rfloor),$
$x>1$ .
ⓘ Symbols: $\exp\NVar{z}$ : exponential function, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $x$ : real variable and $\phi(z)$ : generalized exponential Permalink: http://dlmf.nist.gov/4.12.E7 Encodings: TeX, pMML, png See also: Annotations for §4.12 and Ch.4

Correspondingly,

4.12.8		$\psi(x)=e^{x}-1,$
$-\infty<x<0$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $x$ : real variable and $\psi(x)$ : generalized logarithm Permalink: http://dlmf.nist.gov/4.12.E8 Encodings: TeX, pMML, png See also: Annotations for §4.12 and Ch.4

and

4.12.9		$\psi(x)=\ell+\underbrace{\ln\cdots\ln}_{\ell\text{ times}}x,$
$x>1$ ,
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real variable, $\psi(x)$ : generalized logarithm and $\ell$ : positive integer Permalink: http://dlmf.nist.gov/4.12.E9 Encodings: TeX, pMML, png Notational Change (effective with 1.0.24): The $\ell$ -times repeated application of $\ln$ , previously given as ${\ln}^{(\ell)}$ , has been rewritten using a more conventional notation. See also: Annotations for §4.12 and Ch.4

where $\ell$ is the positive integer determined by the condition

4.12.10		$0\leq\underbrace{\ln\cdots\ln}_{\ell\text{times}}x<1.$
ⓘ Defines: $\ell$ : positive integer (locally) Symbols: $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable Permalink: http://dlmf.nist.gov/4.12.E10 Encodings: TeX, pMML, png Notational Change (effective with 1.0.24): The $\ell$ -times repeated application of $\ln$ , previously given as ${\ln}^{(\ell)}$ , has been rewritten using a more conventional notation. See also: Annotations for §4.12 and Ch.4

Both $\phi(x)$ and $\psi(x)$ are continuously differentiable.

For further information, see Clenshaw et al. (1986). For $C^{\infty}$ generalized logarithms, see Walker (1991). For analytic generalized logarithms, see Kneser (1950).

4.11 Sums 4.13 Lambert

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§4.12 Generalized Logarithms and Exponentials

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