The two regular super-exponentials to base exp(1/e) are constructed. An efficient algorithm for the evaluation of these super-exponentials and their inverse functions is suggested and compared to the already published results.
We give a simple uniqueness criterion (and some derived criteria) for holomorphic Abel functions and show that Kneser’s real analytic Abel function of the exponential is subject to this criterion.
We introduce the intuitive method to select an analytic Abel function of an analytic function f at a non-fixpoint. Due to the complexity of this method by involving matrix inversion of increasing size there is little known about its convergence. We show its convergence in the simplest but still complicated case f(x)=bx. We show that the obtained Abel function is, as expected, the logarithm to base b, independent on its development point. As a by-product we obtain a new polynomial approximation sequence for the logarithm to base b.
We give a simple uniqueness criterion (and some derived crite- ria) for holomorphic Abel functions and show that Kneser's real analytic Abel function of the exponential is subject to the criterion. Mathematics Subject Classification (2010). Primary 30D05.
We feature different (mostly underexplored) methods for non-integer iteration that accumulated over the past decades and centuries. We show their relation if known and test them by application to exponentials. We prove that the matrix power iteration, when applied to a fixed point, is the (classic) regular iteration at that fixed point.
We give a simple uniqueness criterion (and some derived crite- ria) for holomorphic Abel functions and show that Kneser's real analytic Abel function of the exponential is subject to the criterion. Mathematics Subject Classification (2010). Primary 30D05.
The holomorphic function h is constructed such that h h z = z!; this function is interpreted as square root of Factorial.
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 boolean AND x we call F super-exponential. h has two real fixed points for b between 1 and e boolean AND(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)boolean AND z. An apparent symmetry of the tetrational to base sqrt(2) disproved.
