# Local and global logarithms

Plain and simple: Can each entire function $f$ without zeros be written in the form $e^g$ for some entire function $g$? What would be a counter-example?

#### Solutions Collecting From Web of "Local and global logarithms"

Every zero-free holomorphic function $f$ on a simply connected domain $\Omega \subset \mathbb{C}$ has a logarithm.

Since $\Omega$ is simply connected, the logarithmic derivative of $f$ has a primitive $h$, i.e.

$$h'(z) = \frac{f'(z)}{f(z)}.$$

That means the function $f\cdot e^{-h}$ has derivative $\equiv 0$, so is constant. Choosing an appropriate constant, we obtain

$$f(z) = e^{h(z) + c}.$$

Since $\mathbb{C}$ is simply connected, that applies in particular to entire functions.