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Infinitely differentiable generalized logarithmic and exponential functions. (English) Zbl 0742.39005
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 $G(e\sp x)=G(x)+1$, $x\in \bbfR$. The solution $G$ is called a generalized logarithmic function.

39B12Iterative and composite functional equations
33E99Other special functions
30D05Functional equations in the complex domain, iteration and composition of analytic functions
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