# Linear fuction Iterates of $$T(z)=A+Bz~$$ at $$~A\!=\!1$$, $$B\!=\!2~$$; $$~y=T^n(x)~$$ versus $$x$$ for various $$n$$

Linear function is function that can be represented in form

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

where $$A$$ and $$B$$ are constants (for example some complex numbers)

## 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 can be expressed through the superfunction and the Abel function in the standard way,

$$(4) ~ ~ ~ \displaystyle T^n(z)=F(n+G(z))$$

Substitution of the Superfunction $$F$$ by (2) and the Abel function $$G$$ by (3) into equation (4) gives aslo linear function

$$(5) ~ ~ ~ \displaystyle T^n(z)= \frac{-A+B^n\Big( A+(-1+B) z\Big)}{-1+B}=A \frac {B^n-1}{B-1} +B^n z$$

with new parameters $$A \frac {B^n-1}{B-1}$$ instead of $$A$$ in (1) and $$B^n$$ instead of $$B$$. With this representation, the number $$n$$ of iteration has no need to be integer. As other holomophic functions, the linear function can be iterated even complex number of times.