AuPow can be expressed as elementary function,
Where $a$ is parameter. Usually, it is assumed that $a\!>\!0$.
AdPow and AuPow
AdPow is related with AuPow function with simple relation:
For $a\!=\!2$, both functions AdPow and AuPow are shown in Fig.1
For the same $a\!=\!2$, complex map of function AuPow is shown in Fig.2.
SuPow is elementary function,
AuPow appears as example of the Abel function that can be expressed as elementary function. This allows to trace evaluation of superfunction through the regular iteration; the fixed point unity can be used for the calculation.
This example seems to be important for the book Superfunctions.