Linear fuction

Linear function is function that can be represented in form

$(1) ~ ~ ~ T(z)=A+B z$

where $A$ and $B$ are constants.

Abelfunction and Superfunction

Superfunction $F$ for the linear function $T$ by (1) can be written as follows:

$(2) ~ ~ ~ \displaystyle F(z)= A \frac{1-B^z}{1-B}$

The corresponding Abel function can be expressed as follows:

$(3) ~ ~ ~ \displaystyle G(z)= \log_B\Big(1+\frac{B-1}{A}z \big)$

Iterates

Iterate of the linear function is also linear function, although it can be expressed through the superfunction and the Abel function in the standard way,

$(3) ~ ~ ~ \displaystyle T^n(z)=F(n+G(z))$

Keywords

Abel function Holomorphic function, Iterate of function Superfunction