# SuPow

Fig.1. Quadratic function (black curve) and two its superfunctions
Fig.2. $$u\!+\!\mathrm i v \!=\! \mathrm{SuPow}_2(x\!+\!\mathrm i y)$$

SuPow is specific superpower function, id est, the superfunction of the power function $$~z\mapsto z^a\!=\!\exp(\ln(z) \,a)~$$

For given parameter $$a$$,

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

Usually, it is assumed, that $$a\!>\!1$$.

## Transfer function

Function $$F\!=\!\mathrm{SuPow}_a$$ is superfunction for the specific power function $$T(z)\!=\!z^a$$. The superfunction satisfies the transfer equation

$$T(F(z))=F(z\!+\!1)$$

another solution is denoted as SdPow;

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

For $$a\!=\!2$$ graphics $$y=T(x)$$, $$y=\mathrm{SuPow}_2(x)$$ and $$y=\mathrm{SdPow}_2(x)$$ are shown in Fig.1.

For $$a\!=\!2$$, complex map of function $$\mathrm{SuPow}_a$$ is shown in Fig.2.

## SuPow and SdPow

Function $$\mathrm{SuPow}_a$$ is periodic, as well as function $$\mathrm{SdPow}_a$$

The period $$P=2\pi\mathrm i / \ln(a)$$ is the same for both these functions.

For $$a\!>\!1$$, the period is pure imaginary. A little bit more than one period fits the range of map in Fig.2.

Such a periodicity is typical for a superfunction of a transfer function with a real fixed point. Some of thiese function are listed in the Table of superfunctions.

Functions SuPow and SdPow are related with expression

$$\mathrm{SuPow}_a(z)=\mathrm{SdPow}_a(z\!+\!P/2)$$

Similar relations hold also for many other pairs of periodic superfunctions of some transfer function with two fixed points

However, namely the superpower functions have also the specific relation,

$$\displaystyle \mathrm{SuPow}_a(z)=\frac{1}{\mathrm{SdPow}_a(z)}$$

## Inverse function

$$u\!+\!\mathrm i v=\mathrm{AuPow}_2(x\!+\!\mathrm i y)$$

The inverse function AuPow$$=\mathrm{SuPow}^{-1}$$ is also elementary function;

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

AuPow is the Abel function for the same transfer function $$T(z)=z^a$$ and satisfies the Abel equation

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

Complex map of function $$\mathrm{AuPow}_2$$ is shown in figure at right.

For $$a\!>\!1$$, function $$\mathrm{AuPow}_2$$ has cut line along the real axis from $$1$$ to $$-\infty$$.