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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,


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.

AdPow and AuPow

Another abelfunction for the power function is called AuPow;


AdPow is related with AuPow function with simple relation:


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:


SdPow satisfies the Transfer equation


SdPow is elementary function,

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


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.



Abel function AdPow Abel function Elementary function Power function SdPow Superfunctions SdPow Superfunctions