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Zbl 0742.39005
Walker, Peter
Infinitely differentiable generalized logarithmic and exponential functions. (English)
[J] Math. Comput. 57, No.196, 723-733 (1991). ISSN 0025-5718; ISSN 1088-6842/e

By a natural iterative procedure, the author constructs a $C\sp \infty$ real function $h$ satisfying the functional equation $h(e\sp x)=e\sp{h(x)}-1$, $x\in \bbfR$. Using this $h$ and some of his earlier results [Bull. Austr. Math. Soc. 38, No. 3, 351-356 (1988; Zbl 0643.39002) and J. Math. Anal. Appl. 155, No. 1, 93-110 (1991; Zbl 0716.39006)] he finds $C\sp \infty$ solutions $G$ of the functional equation $G(e\sp x)=G(x)+1$, $x\in \bbfR$.\par The solution $G$ is called a generalized logarithmic function.
[J.Matkowski (Bielsko-Biała)]

MSC 2000:: *39B12 Iteraterative functional equations
33E99 Special functions
30D05 Functional equations in the complex domain

Keywords: generalized exponential functions; infinitely differentiable solutions; iterative procedure; generalized logarithmic function

Citations: Zbl 0643.39002; Zbl 0716.39006

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