I think you get a dense subset of the complex plane very soon, if I'm not mistaken. Specifically, the first i^i will give: exp(-pi^2 (4n + 1) / 4) where n is an arbitrary integer Then i^(i^i) will give: exp(pi i (2m pi i - pi^2 (4n + 1)) / 8) where n,m are arbitrary integers which will yield not only a dense subset of the unit circle, but also infinitely many larger and smaller circles centred on the origin. The next application, i^(i^(i^i)), will give a dense subset of the complex plane immediately.
Sent: Wednesday, November 29, 2017 at 7:18 AM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Complex exponentiation
When z is a nonzero complex number
z = r exp(i*theta)
and alpha is any complex number, then classically:
z^alpha
is defined to be the set of complex values given by
(*) {exp(alpha * log(z))
where log(z) takes exactly all the values of the form
{c + 2K*pi*i | K in Z}
for some complex number c with exp(c) = z, e.g.,
c = ln(r) + i*theta
. Now (*) exp(alpha * (ln(r) + i*theta).
Let
alpha = a + bi
for a, b real. Then we have
z^alpha = {exp((a + bi)*(ln(r) + i*theta))
= {exp(a*ln(r) - b*theta) + i(a*theta + b*ln(r)))
= {exp(a*ln(r) - b*theta_0) i(a*theta_0 + b*ln(r))) * exp(2pi*(-b + a*i))^K
where (say)
-pi < theta_0 <= pi
defines theta_0 uniquely, so the last equation reads:
z^alpha = {u * v^K | K in Z}
for complex numbers u, v with
u = exp(a*ln(r) - b*theta_0) i(a*theta_0 + b*ln(r))),
and
v = exp(2pi*(-b + a*i)).
Now, when people talk about i^i^i^i^.... they usually mean by taking the parentheses from the top down — and using the principal logarithm:
i^z = exp((pi*i/2)*z)
and iterating:
i^...^z = i^(i^(...^(i^z)...))
= exp((pi*i/2)*(exp((pi*i/2)*...*exp((pi*i/2)z)...))
But what if we *didn't* restrict to using the principal logarithm, but instead considered *all* the determinations of
i^previous stuff
each time we exponentiated.
Then at each stage there will be countably many determinations.
Now (as usual) let the *number of exponentiations* approach oo.
QUESTION: What are the limit points of the determinations: --------
Those points z of C that are *limits* of any sequence of points of the form
z_n = something born at exactly n exponentiations,
i.e., such that for every epsilon > 0 there exists N such that there is a point
z_n that is a result of n > N exponentiations with
|z - z_n| < epsilon.
—Dan
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