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.

Conclusion

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  ).