The function defined by an `infinite tower’ of exponents,
is called the Power Tower function. We consider the sequence of successive approximations to this function:
As , the sequence converges for . This result was first proved by Euler. For an earlier post on the power tower, click here.
The Imaginary Power Tower Function
We now examine the case where is replaced by a complex number . Specifically, we look at the case :
We call this the Imaginary Power Tower. Let us start with . Then and . It is quite surprising that is a positive real number, .
The complete approximating sequence is:
If we assume that the sequence converges to a value , then we can write
Writing it is straightforward to show that
This is easily solved numerically to give .
The Spiral Spoor
In Fig. 1 we show the sequence . The points spiral around the limit point , converging towards it. An attempt was made to fit, to the sequence of points, a logarithmic spiral of the form (where is the polar angle relative to the point ). However, this was unsuccessful, as the relationship between successive points is more complex.
In Fig. 2 we show the same sequence of points. The points appear to fall into three distinct sets, each in the form of a spiral. Three logarithmic spirals are superimposed on the plot. The three sets of points lie close to, but not exactly on, the three spiral arcs. It is not clear whether the alignment of points implies a functional relationship of this sort, or whether the pattern is accidental.
To Be Continued: Part II
The Power Tower. Blog post on thatsmaths.com.