Extension of tetration to real and complex heights

Abstract

The continuous tetrational function ${^x}r=\tau(r,x)$, the unique solution of equation $\tau(r,x)=r^{\tau(r,x-1)}$ and its differential equation $\tau'(r,x) =q \tau(r,x) \tau'(r,x-1)$, is given explicitly as ${^x}r=\exp_{r}^{\lfloor x \rfloor+1}[\{x\}]_q$, where $x$ is a real variable called height, $r$ is a real constant called base, $\{x\}=x-\lfloor x \rfloor$ is the sawtooth function, $\lfloor x \rfloor$ is the floor function of $x$, and $[\{x\}]_q=(q^{\{x\}}-1)/(q-1)$ is a q-analog of $\{x\}$ with $q=\ln r$, respectively. Though ${^x}r$ is continuous at every point in the real $r-x$ plane, extensions to complex heights and bases have limited domains. The base $r$ can be extended to the complex plane if and only if $x\in \mathbb{Z}$. On the other hand, the height $x$ can be extended to the complex plane at $\Re(x)\notin \mathbb{Z}$. Therefore $r$ and $x$ in ${^x}r$ cannot be complex values simultaneously. Tetrational laws are derived based on the explicit formula of ${^x}r$.