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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 \(a\),


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


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



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.



AdPow, Elementary function, Power function, SdPow, SuPow, Superfunction Superpower