applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA

Infinitely Iterated Exponentiation
of a Complex Number

Infinite Exponentiation

An infinite exponentiation is something which is raised to a power which is something raised to a power ad infinitum>
Suppose

G = a^{aa…} which can be represented as
G = a^{G}

However a little manipulation turns it into a seemingly trivial problem. The manipulation is to take the G-th root of both sides
giving

G^{1/G} = a

Now if we want a value of a that gives G as a solution we need only take the G-th root of G and we have the answer. For example,
for G=2, a=2^{½}=√2. Thus

√2^{√2√2…} = 2

A previous study worked out the analysis when G is a real number
It was found that there is convergence if and only if a<e^{1/e} where e=2.721828….

When a and G are complex number the equation a=G^{1/G} is actually two equations.
Let G=R·e^{iΦ} and a=r·^{iθ}.
Then