Portrait of the four regular super-exponentials to base sqrt(2)

We introduce the concept of regular super-functions at a fixed point. It is derived from the concept of regular iteration. A super-function F of h is a solution of F(z+1)=h(F(z)). We provide a condition for F being entire, we also give two uniqueness criteria for regular super-functions. In the particular case h(x)=b^x we call F super-exponential. h has two real fixed points for b between 1 and e^(1/e). Exemplary we choose the base b=sqrt(2) and portrait the four classes of real regular super-exponentials in the complex plane. There are two at fixed point 2 and two at fixed point 4. Each class is given by the translations along the x-axis of a suitable representative. Both super-exponentials at fixed point 4 -- one strictly increasing and one strictly decreasing -- are entire. Both super-exponentials at fixed point 2 -one strictly increasing and one strictly decreasing -- are holomorphic on a right halfplane. All four super-exponentials are periodic along the imaginary axis. Only the s...

• Type: Journal
• Year: 2010
• Where: MOC
• Authors: Dmitrii Kouznetsov, Henryk Trappmann
• Keywords: Fixed Point, MOC 2010, Regular, Regular Super-functions, Security Privacy