Is $e^e$ equal to every point on the circle $z=|e^e|$?

Question

Typically, $e^c$ is taken to be single-valued. However, by the definition of the complex exponential $z^c$, it can be taken to have multiple values. For example, $\sqrt{e} \approx \pm 1.6$. Now, if a complex variable is raised to an integer power, all of the multiple values will be equal by virtue of the behavior of $2n\pi$ mod $2\pi$. Fractional powers will have more than one value, and this leads me to a point of curiosity. Because $e$ is irrational, the only multiple of $e$ that will be equal to an integer is $0e$. Unlike the rational powers of $e$, the multiple values will never stack on top of each other, and intuitively that means that the whole circle of $|z|=|e^e|$ will eventually be reached by some value of the function. Is this true? If not, what are the properties of that set?

Answer 1

Maybe you would like to read this section about complex exponentiation: https://en.wikipedia.org/wiki/Exponentiation#Complex_exponents_with_complex_bases

What you are saying is sort of true, but not really. If anyone ever saw $e^e$, they would always interpret it as a real exponentiation, which has one value $\approx 15.15$. However, if we insist on interpreting one or both of the $e$'s as complex numbers, we can get other values.

A still reasonable interpretation would be $e^e=\exp(e)$, where we think of the input as a complex number. The exponential function $\exp(z)$ is well-defined and unique even for complex inputs, so we get the same real answer. In general $$ e^{x+iy} = e^x\cdot (\cos y + i\sin y) $$

Now, what you are doing is interpreting the base $e$ as a complex number. If we do this, then you are right. We get: $$ e^e := \exp(e\log e) = \exp(e\cdot(1+2k\pi i)) = e^e\cdot e^{e\cdot2k\pi i} , \quad \mathrm{for}\ k\in\mathbb Z $$ The key difference is that $\log$ is now the complex logarithm, which is multivalued. This hits countably infinitely many points on the circle with radius $e^e$, but by no means all of them (since there uncountably many points on the circle). For example, the angle $q\pi$ for a non-zero, rational $q$, will never be attained. However, the points will be dense in the circle, i.e. in any arc on the circle, there will be infinitely many of the points. It is just like how the rational numbers are distributed on the real number line.

But let me just repeat: No one would normally say that $e^e$ has infinitely many values!