The Imaginary Power Tower: Part I

The function defined by an `infinite tower’ of exponents,

$\displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}}$

is called the Power Tower function. We consider the sequence of successive approximations to this function:

$\displaystyle y_0 = 1 \qquad y_1 = x \qquad \dots \qquad y_{n+1} = x^{y_n} \,.$

As ${n\rightarrow\infty}$ , the sequence ${\{y_n\}}$ converges for ${e^{-e}<x<e^{1/e}}$ . 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 ${x}$ is replaced by a complex number ${z}$ . Specifically, we look at the case ${z=i}$ :

$\displaystyle q = {i^{i^{i^{.^{.^{.}}}}}} \,.$

We call this the Imaginary Power Tower. Let us start with ${q_0 = 1}$ . Then ${q_{1} = i^{1} = i}$ and ${q_{2} = i^{i} = e^{-\pi/2}}$ . It is quite surprising that ${i^{i}}$ is a positive real number, ${\exp(-\pi/2)}$ .

The complete approximating sequence is:

$\displaystyle q_0 = 1 \qquad q_{1} = i^{1} = i \qquad q_{2} = i^{i} = e^{-\pi/2} \qquad \dots \qquad q_{n+1} = i^{q_n}$

If we assume that the sequence ${\{q_n\}}$ converges to a value ${Q}$ , then we can write

$\displaystyle Q = i^{Q}$

Writing ${Q = \varrho\exp(i\vartheta)}$ it is straightforward to show that

$\displaystyle \vartheta \tan\vartheta = \log\left[\frac{\pi}{2}\frac{\cos\vartheta}{\vartheta}\right] \qquad\mbox {and}\qquad \varrho = \frac{2}{\pi}\frac{\vartheta}{\cos\vartheta}$

This is easily solved numerically to give ${Q=(0.438283, 0.360592)}$ .

The Spiral Spoor

ImaginaryPowerTower-Spiral1 In Fig. 1 we show the sequence ${\{q_n\}}$ . The points spiral around the limit point ${Q}$ , converging towards it. An attempt was made to fit, to the sequence of points, a logarithmic spiral of the form ${z=Q+ a\exp(b\varphi)}$ (where ${\varphi}$ is the polar angle relative to the point ${Q}$ ). However, this was unsuccessful, as the relationship between successive points is more complex.

ImaginaryPowerTower-Spiral3 In Fig. 2 we show the same sequence of points. The points ${q_n}$ 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

Sources

${\bullet}$ The Power Tower. Blog post on thatsmaths.com.

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