# Is $\ln{f}$ well-defined?

Let $$f$$ be holomorphic and nonzero on some open set $$\Omega\subseteq \mathbb{C}$$. Then is $$\ln {f}$$ well-defined? Usually, in order for $$\ln {z}$$, we need to choose a branch, e.g., $$(-\infty,0]^c$$, so that the function is well-defined, but how do we know that the image of $$f$$ is in some branch?

My thought process is that if the image of $$f$$ goes all around $$0$$, then we can choose some path $$\gamma$$ such that $$\Gamma = f\circ \gamma$$ circles around 0 and thus

$$\int_\gamma \frac{f'}{f} dz =\int_{\Gamma} \frac{dw}{w} \ne 0\\$$

However, since $$f$$ is holomorphic and nonzero, the most-left integral should be zero.

The problem is that I can't quite show that $$\gamma$$ is $$C^1$$ or at least piecewise $$C^1$$.

• If $\Omega$ is simply connected you can always define a holomorphic logarithm if $f$ doesn't vanish; if $\Omega$ is not simply connected it depends (see $f(z)$=$z$ on the punctured unit disc for a counterexample) but if the image of $f$ is nice (e.g it lies in a simply connected domain that doesn't contain zero), you can also define a holomorphic logarithm – Conrad Feb 17 at 3:47