### The Power Tower

Look at the function defined by an `infinite tower’ of exponents: $\displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}}$

It would seem that for x>1 this must blow up. But, amazingly, this is not so.

In fact, the function has finite values for positive x up to ${x=\exp(1/e)\approx 1.445}$. We call this function the power tower function.

We can construct the power tower function by an iterative process: $\displaystyle y_1 = x \, \qquad y_{n+1} = x^{y_n} \,.$

This generates the infinite sequence of successive approximations $\displaystyle \left\{y_1, y_2, y_3, \dots \right\} = \left\{ x, x^x, x^{x^x}, \dots \right\}$

We note the convention $\displaystyle x^{x^x} \equiv x^{(x^x)} \qquad\mbox{and \emph{not}}\qquad x^{x^x} = \left(x^x\right)^x = x^{x^2} \,.$

Thus, the tower is constructed downwards; it should really be denoted as $\displaystyle y(x) = {\phantom{ }_{.^{.^{.}}} x^{x^{x}}}$

as each subsequent x is adjoined to the bottom of the tower.

If the sequence converges to y, it follows that $\displaystyle y = x^y$

This gives an explicit expression for x as a function of y: $\displaystyle x = y^{1/y}$

Clearly, this function is well defined for all positive y. If we can invert it, we can get y as a function of x.

Iterative Solution

The logarithm of ${y = x^y}$ gives ${\log y = y \log x}$ or $\displaystyle y = \exp(\xi y)$

where ${\xi = \log x}$. This is in a form suited for iterative solution.

Given a value of ${x}$, and therefore of ${\xi}$, we seek a value ${y}$ such that the graph of ${\exp(\xi y)}$ intersects the diagonal line ${y=y}$. Starting from some value ${y_{(0)}}$, we compute the iterations $\displaystyle y_{(n+1)} = \exp(\xi y_{(n)})$

In the figure below, we sketch the graph of ${\exp(\xi y)}$ for a selection of values of ${\xi}$. For ${\xi<0}$ there is a single root. For ${0<\xi<1/e}$, there are two roots. For ${\xi=1/e}$ there is one double root. Finally, for ${\xi>1/e}$ there are no roots. We plot the function ${x=y^{1/y}}$ in the figure below. It is defined for all positive ${y}$. Its derivative vanishes at ${y=e}$ where it takes its maximum value ${\exp(1/e)}$. The function ${x=y^{1/y}}$ is monotone increasing on the interval ${(0,e)}$ and has an inverse function on this interval. This inverse is the power tower function, plotted in the figure below.

The power tower function is well defined on the domain ${x\in(0,\exp(1/e))}$. This is surprising, as a cursory glance would suggest divergence for ${x>1}$.

It would be interesting to investigate the behaviour of the power tower function for complex values of the argument.

A follow-up post, relating the power tower function to the Lambert W-function.

A brief note with more technical details is here ( PDF  ).