# SdPow

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

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

For given parameter $$a$$,

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

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

## Transfer function

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

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

For this specific transfer function $$T$$, the two real-holomorphix solutions are SuPow and SdPow:

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

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

For the power function both, the superfunctions and the Abel functions can be expressed as elementary functions. For $$a\!=\!2$$, these functions are who functions are shown in Fig.1.

For the same $$a\!=\!2$$, the complex map of function SdPow is shown in FIg.2.