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Full Version: Partial Differential Equation for power-towers
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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
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?