For an analytic function $f$ that does not vanish on a simply connected region, we may define its logarithm to be the function:
$$\log f=g(z):=\int_{y}\frac{f'}{f}dz+c_0.$$
Where $\gamma$ is some path starting at an arbitrary point in the region, and ending at $z$; while $c_0$ satisfies $e^{c_0}=f(z_0)$.
I believe that this logarithm should satisfy under certain conditions that: $$\log f=\log |f|+iarg(f).$$
Am I right, or this is too difficult in general?
Best Answer
The function $g$ satisfies $g' = \frac{f'}{f}$ in the given domain, so that $$ (f e^{-g} )' = f' e^{-g} - f g' e^{-g} = 0 \\ \implies f e^{-g} = \text{const} = f(z_0) e^{-g(z_0)} = f(z_0) e^{-c_0} = 1 \, . $$ Therefore $e^g = f$, i.e. $g$ is “a holomorphic logarithm” of $f$ in the domain. In particular $$ f(z) = e^{g(z)} = e^{\operatorname{Re} g(z)} e^{ i \operatorname{Im}g(z)} $$ which implies that $$ |f(z)| = e^{\operatorname{Re}g(z)} \implies \operatorname{Re}g(z) = \log |f(z)| $$ and that $ \operatorname{Im}g(z)$ is an argument of $f(z)$. So $$ g(z) = \log |f(z)| + i \operatorname{arg}f(z) $$ in the sense that $\operatorname{arg}f(z)$ is a continuous function which is an argument of $f(z)$ for each $z$.