Did we mention already the tangent? It has this nice addition theorem:
\( \tan(x+y)=\frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)} \)
which brings us the superfunction:
\( \sigma(z)=\tan(z\cdot\arctan( c)) \)
\( \sigma(z+1)=\frac{\sigma(z)+c}{1-c\sigma(z)}=f(\sigma(z)) \)
for the function
\( f(z)=\frac{z+c}{1-cz} \)
\( f \) is another particular case of a linear fraction (where the regular iteration at both fixed points coincide).
The two (non-real) fixed points are:
\( \frac{z+c}{1-cz}=z \), \( z+c=z-cz^2 \)
\( z=\pm i \) for \( c\neq 0 \)
\( \tan(x+y)=\frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)} \)
which brings us the superfunction:
\( \sigma(z)=\tan(z\cdot\arctan( c)) \)
\( \sigma(z+1)=\frac{\sigma(z)+c}{1-c\sigma(z)}=f(\sigma(z)) \)
for the function
\( f(z)=\frac{z+c}{1-cz} \)
\( f \) is another particular case of a linear fraction (where the regular iteration at both fixed points coincide).
The two (non-real) fixed points are:
\( \frac{z+c}{1-cz}=z \), \( z+c=z-cz^2 \)
\( z=\pm i \) for \( c\neq 0 \)