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.

Iteration

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)}.

powertower-yThe 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.

powertower-xConclusion

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  ).


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