Define for $z\in \mathbb{C}$, $\alpha\in \mathbb{R}$,
$arg_{\alpha}(z):= Arg(z) + 2n\pi$ where $n\in \mathbb{Z}$ is chosen s.t. $arg_{\alpha}(z)$ lies in $(\alpha, \alpha + 2\pi]$
and $log_{\alpha}(z):= ln|z| + iarg_{\alpha}(z)$
Now fix $\alpha\in \mathbb{R}$. Let $g(z):= log_{\alpha}(z)$ defined on $W:= \mathbb{C}\setminus\{z\in \mathbb{C} \mid z = 0 \text{ or } \arg(z) = \alpha + 2\pi\}$ (so we are taking the plane and removing the branch cut)
and $f(z):= e^z$, defined on the whole plane. I wish to prove that $g'(z)$ exists on $W$ and I would like to compute it's value. Since f is analytic on an open set containing $W$ (namely the whole plane), and $f\circ g = z$ is entire (and I think that $f'(g(z))$ is just $z$ (*), which is nonzero on $W$), and g is continuous on W, I believe this means that g is analytic on $W$ and that (by the so-called "reverse chain rule"):
$g'(z) = \frac{(f\circ g)'(z)}{f'(g(z))} = \frac{1}{z}$. (**)
So far I don't know what my mistake is. I think (**) is wrong, if I recall correctly, this derivative depends on $\alpha$, i.e., it depends on the branch chosen. If I made I mistake, I was thinking it was in (*), but $(e^z)'$ is just $e^z$ on the whole plane so that we should have:
$f'(g(z)) = e^{log_{\alpha}(z) = e^{ln|z| + iarg_{\alpha}(z)}} = |z|e^{arg_{\alpha}(z)} = z$
on W.
Right?
Does anyone know where I am going wrong? F.Y.I., I have done some googling and searching otherwise for material on specifically this matter, when we let $\alpha$ vary through $\mathbb{R}$ (not just principal "$Log(z)$"), to no avail, so an answer to this would be really appreciated.
Best Answer
You did not make a mistake. If $g: D \to \Bbb R$ is any branch of the logarithm on any domain $D \subset \Bbb C\setminus \{ 0 \}$, then $g'(z) = 1/z$.
A “branch of the logarithm” is defined as an inverse function to the exponential function, so this does follow from the chain rule and the fact that $\exp' = \exp$: $$ z = \exp(g(z)) \implies 1 = \exp(g(z)) g'(z) = z g'(z) \implies g'(z) = \frac 1z \, . $$
You can also argue that two branches of the logarithm on the same domain differ by a constant (a multiple of $2 \pi i$), and therefore have the same derivative.