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Infinitely differentiable generalized logarithmic and exponential functions. (English) Zbl 0742.39005

By a natural iterative procedure, the author constructs a ${C}^{\infty }$ real function $h$ satisfying the functional equation $h\left({e}^{x}\right)={e}^{h\left(x\right)}-1$, $x\in ℝ$. 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 ${C}^{\infty }$ solutions $G$ of the functional equation $G\left({e}^{x}\right)=G\left(x\right)+1$, $x\in ℝ$.

The solution $G$ is called a generalized logarithmic function.

MSC:
 39B12 Iterative and composite functional equations 33E99 Other special functions 30D05 Functional equations in the complex domain, iteration and composition of analytic functions