# AuPow Fig.1. $$y\!=\!\mathrm{AdPow}_2(x)$$ and $$y\!=\!\mathrm{AuPow}_2(x)$$ Fig.2. $$u\!+\!\mathrm i v\!=\!\mathrm{AuPow}_2(x\!+\!\mathrm i y)$$

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.