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

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

AdPow can be expressed as elementary function,

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

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

AdPow is inverse function of SdPow, that, in its turn, is superfunction of the power function.

## Contents

Another abelfunction for the power function is called AuPow;

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

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 AdPow is shown in Fig.2.

## Inverse function

For AdPow, the inverse function is SdPow, that is superpower function, id est, superfunction of the power function:

$$\mathrm{AdPow}_a=\mathrm{SdPow}_a^{-1}$$

SdPow satisfies the Transfer equation

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

SdPow is elementary function,

$$\mathrm{SdPow}_a(z)= \exp(-a^z)$$

## Applications

AdPow 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.