Digital Library of Mathematical Functions
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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.12 Generalized Logarithms and Exponentials

A generalized exponential function ϕ(x) satisfies the equations

4.12.1 ϕ(x+1) =ϕ(x),
-1<x<,
4.12.2 ϕ(0) =0,

and is strictly increasing when 0x1. Its inverse ψ(x) is called a generalized logarithm. It, too, is strictly increasing when 0x1, and

4.12.3 ψ(x) =1+ψ(x),
-<x<,
4.12.4 ψ(0) =0.

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

4.12.5 ϕ(x)=ψ(x)=x,
0x1.

Then

4.12.6 ϕ(x)=ln(x+1),
-1<x<0,

and

4.12.7 ϕ(x)=expexpexp(x-x),
x>1,

where the exponentiations are carried out x times. Correspondingly,

4.12.8 ψ(x)=x-1,
-<x<0,

and

4.12.9 ψ(x)=+ln()x,
x>1,

where ln()x denotes the -th repeated logarithm of x, and is the positive integer determined by the condition

4.12.10 0ln()x<1.

Both ϕ(x) and ψ(x) are continuously differentiable.

For further information, see Clenshaw et al. (1986). For C generalized logarithms, see Walker (1991). For analytic generalized logarithms, see Kneser (1950).