§4.12 Generalized Logarithms and Exponentials

Keywords:: exponential function, generalized exponentials, generalized logarithms, logarithm function
Permalink:: http://dlmf.nist.gov/4.12

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

4.12.1 $\phi(x+1)=e^{{\phi(x)}},$ $-1<x<\infty$ ,

Symbols:: $e$ : base of exponential function, $x$ : real variable and $\phi(z)$ : generalized exponential
Permalink:: http://dlmf.nist.gov/4.12.E1
Encodings:: TeX, pMML, png

4.12.2 $\phi(0)=0,$

Symbols:: $\phi(z)$ : generalized exponential
Permalink:: http://dlmf.nist.gov/4.12.E2
Encodings:: TeX, pMML, png

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 $\psi(e^{x})=1+\psi(x),$ $-\infty<x<\infty$ ,

Symbols:: $e$ : base of exponential function, $x$ : real variable and $\psi(x)$ : generalized logarithm
Permalink:: http://dlmf.nist.gov/4.12.E3
Encodings:: TeX, pMML, png

4.12.4 $\psi(0)=0.$

Symbols:: $\psi(x)$ : generalized logarithm
Permalink:: http://dlmf.nist.gov/4.12.E4
Encodings:: TeX, pMML, png

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

Then

4.12.6 $\phi(x)=\mathop{\ln\/}\nolimits\!\left(x+1\right),$ $-1<x<0$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : 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

and

4.12.7 $\phi(x)=\mathop{\exp\/}\nolimits\mathop{\exp\/}\nolimits\cdots\mathop{\exp\/}\nolimits\!\left(x-\left\lfloor x\right\rfloor\right),$ $x>1$ ,

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $\left\lfloor x\right\rfloor$ : floor of $x$ , $x$ : real variable and $\phi(z)$ : generalized exponential
Permalink:: http://dlmf.nist.gov/4.12.E7
Encodings:: TeX, pMML, png

where the exponentiations are carried out $\left\lfloor x\right\rfloor$ times. Correspondingly,

4.12.8 $\psi(x)=e^{x}-1,$ $-\infty<x<0$ ,

Symbols:: $e$ : base of exponential function, $x$ : real variable and $\psi(x)$ : generalized logarithm
Permalink:: http://dlmf.nist.gov/4.12.E8
Encodings:: TeX, pMML, png

and

4.12.9 $\psi(x)=\ell+{\mathop{\ln\/}\nolimits^{{(\ell)}}}x,$ $x>1$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : 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

where ${\mathop{\ln\/}\nolimits^{{(\ell)}}}x$ denotes the $\ell$ -th repeated logarithm of $x$ , and $\ell$ is the positive integer determined by the condition

4.12.10 $0\leq{\mathop{\ln\/}\nolimits^{{(\ell)}}}x<1.$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable and $\ell$ : positive integer
Permalink:: http://dlmf.nist.gov/4.12.E10
Encodings:: TeX, pMML, png

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

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