Look at the function defined by an `infinite tower’ of exponents:
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 . We call this function the power tower function.
We can construct the power tower function by an iterative process:
This generates the infinite sequence of successive approximations
We note the convention
Thus, the tower is constructed downwards; it should really be denoted as
as each subsequent x is adjoined to the bottom of the tower.
If the sequence converges to y, it follows that
This gives an explicit expression for x as a function of 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.
The logarithm of gives or
where . This is in a form suited for iterative solution.
Given a value of , and therefore of , we seek a value such that the graph of intersects the diagonal line . Starting from some value , we compute the iterations
In the figure below, we sketch the graph of for a selection of values of . For there is a single root. For , there are two roots. For there is one double root. Finally, for there are no roots.
We plot the function in the figure below. It is defined for all positive . Its derivative vanishes at where it takes its maximum value .
The power tower function is well defined on the domain . This is surprising, as a cursory glance would suggest divergence for .
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 ).