# AuPow

AuPow is specific Abelpower function, id est, the specific Abel function for the Power function.

AuPow can be expressed as elementary function,

\(\mathrm{AdPow}_a(z)=\log_a(\ln(z))\)

Where \(a\) is parameter. Usually, it is assumed that \(a\!>\!0\).

AuPow is inverse function of SuPow, that, in its turn, is superfunction of the power function.

## AdPow and AuPow

Another abelfunction for the power function is called AdPow;

\(\mathrm{AdPow}_a(z)=\log_a(\ln(1/z))\)

AdPow is related with AuPow function with simple relation:

\(\mathrm{AdPow}_a(z)=\mathrm{AuPow}_a(1/z)\)

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.

## Inverse function

For AuPow, the inverse function is SuPow, that is superpower function, id est, superfunction of the power function:

\(\mathrm{AuPow}_a=\mathrm{SuPow}_a^{-1}\)

SuPow satisfies the Transfer equation

\(\mathrm{SuPow}_a(z\!+\!1)=\mathrm{SuPow}_a(z)^a\)

SuPow is elementary function,

\(\mathrm{SuPow}_a(z)= \exp(a^z)\)

## Applications

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.

## References

## Keywords

Abel function AdPow Abel function Elementary function Power function SdPow SuPow Superfunctions SdPow Superfunctions