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

Abstract:

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$\hat {\phantom {x}}$x we call F super-exponential. h has two real fixed points for b between 1 and e$\hat {\phantom {x}}$(1/e). Exemplary we choose the base b=sqrt(2) and portray 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 half-plane. All four super-exponentials are periodic along the imaginary axis. Only the strictly increasing super-exponential at 2 can satisfy F(0)=1 and can hence be called tetrational.

We develop numerical algorithms for the precise evaluation of these functions and their inverses in the complex plane. We graph the two corresponding different half-iterates of h(z)=sqrt(2)$\hat {\phantom {x}}$z. An apparent symmetry of the tetrational to base sqrt(2) disproved.