Partial Differential Equation for power-towers - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Partial Differential Equation for power-towers (/showthread.php?tid=417) Partial Differential Equation for power-towers - kobi_78 - 02/01/2010 Hi, I found a pde for power towers of any height. The equation is as following: Define $f_{-1}(a, x) = \log_a{x}$ $f_0(a, x) = x$ $f_{n + 1}(a, x) = a^{f_n(a, x)}$ And $G(x, a) = \frac{1}{a \ln(a)^2 } \sum_{k = -1}^{\infty}{ \frac{1}{\frac{df_{k}}{dx}}} = \frac{1}{a (\ln(a))^2 } \sum_{k = -1}^{\infty}{ \frac{1}{ \prod_{n=1}^{k}{f_{n}(a, x)} \cdot (\ln{a})^k }}$ Where $\prod_{n=1}^{0}{f_{n}(a, x)} = 1$ $\prod_{n=1}^{-1}{f_{n}(a, x)} = \frac{1}{f_0(a, x)}$ Then every $y = f_n(a, x)$ satisfies: $\frac{dy}{da} = G(x, a) \cdot \frac{dy}{dx} - G(y, a)$ I don't understand anything about pdes so I don't know if this says any thing, but I wanted to share it with you guys. I guess we can somehow use this equation to extend natural iteration monomial for larger bases, but I don't really know. Kobi RE: Partial Differential Equation for power-towers - mike3 - 02/02/2010 Hmm. Well if this is the case, then could that mean that there may be a continuum of solutions? If so, could that allow us to define $f_h(a, x)$ for real and complex values of $h$, thereby yielding $\exp^h_a(x)$ for such heights?