AdPow

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.

AdPow and AuPow

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.

Keywords

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