Firstly, proof verification on the values of $e^{b \ln a}$:
(Exer 3.51) Part I: Prove $e^{b \ln(a)}$ is single valued $\iff b \in \mathbb Z$. Part II: What if $b \in \mathbb Q$?
I tried:
Part I ($b \in \mathbb Z$)
Pf:
By definition of $a^b$:
$$a^b:=e^{bLn(a)} = e^{b(\ln|a|+iArg(a))} = e^{b\ln|a|}e^{ibArg(a)}$$
By definition of $e^{b\ln(a)}$, $\exists k \in \mathbb Z$ s.t.
$$e^{b\ln(a)} = e^{b(\ln|a|+i\arg(a))} = e^{b\ln|a|}e^{ibArg(a)}e^{2kbi\pi} =: a^be^{2kbi\pi}$$
$\therefore, e^{b\ln(a)}$ is single valued $\iff e^{b\ln(a)} = a^b \iff 1 = e^{2kbi\pi}$
Now $$1 = e^{2kbi\pi} = \cos(2kbi\pi) + i\sin(2kbi\pi) \iff$$
$$\cos(2kbi\pi) = 1 \wedge \sin(2kbi\pi) = 0 \iff bk \in \mathbb Z \iff b \in \mathbb Z$$
$\therefore, e^{b\ln(a)}$ is single valued $\iff e^{b\ln(a)} = a^b \iff b \in \mathbb Z$
QED
Question 1. Where have I gone wrong for Part I, if anywhere?
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Part II ($b \in \mathbb Q$)
Let $b:=\frac pq$, where $p,q$ are coprime positive integers (For any negative, $\cos$ is even and $\sin$ is odd). Now consider $\cos$:
$$\cos(2kbi\pi) := \cos(2k \frac pq i\pi)$$
whose value is different depending where $p$ is in $\{\overline{0},\overline{1}, \dots, \overline{q-1}\}$
$\therefore, e^{b\ln(a)}$ has $q$-values because $p$ has $q$-values $\mod \ q$.
Question 2. Where have I gone wrong for Part II, if anywhere?
Now that that's over with:
Question 3. (Something I noticed on my own in re Exer 3.51 but not asked in Exer 3.51)
What's the connection to the popcorn function aka Thomae's function aka modified Dirichlet $\frac1q 1_{x=\frac pq}$?
In Part II of Exer 3.51, $q$ is extracted from $\frac pq$.
In the popcorn function from elementary real analysis, $\frac1q$ is extracted from $\frac pq$. It seems the popcorn function is computing number of solutions $q$ to a complex equation and then inverting the computation $\frac 1 q$.
Question 4. (Also something I noticed on my own in re Exer 3.51 but not asked in Exer 3.51)
What exactly is the connection to solving the roots of unity?
The roots of unity are the solutions of $z^n=1$, which turn out to be $z=e^{2k\pi\frac{m}{n}}$ whose value is different depending where $m$ is in $\{\overline{0},\overline{1}, \dots, \overline{n-1}\}$. This looks similar to Part II.