Tetrational functions (also called super-exponential functions) are defined by tetration with a constant base. In other words the function f(z) = {}^{z}a is a tetrational function. These functions are originally defined for integer z only, and advanced mathematics is required to find a function that satisfies infinite differentiability and the definition

{}^{z}a = a \U \left( {}^{z-1}a \right)

Power Series

{}^{z}a = \sum_{k=0}^{\infty} t_k(a) (z - z_0)^k

{}^{z}a = \log(z+2) + \sum_{k=0}^{\infty} u_k(a) (z - z_0)^k