A nonvanishing holomorphic function on a domain with no holomorphic logarithm

Given: The punctured plane $U=\mathbb{C}\setminus \{0\}$, the identity function $f(z)=z$ on it, and the contour $C(t)=\exp (it)$ on $[0,2\pi ]$.

On a homologically simply connected complex domain, a holomorphic nowhere-zero function admits a holomorphic $L$ with $\exp \circ L$ equal to it (A nonvanishing holomorphic function on a homologically simply connected domain has a holomorphic logarithm), a domain being homologically simply connected when every cycle in it is null-homologous in it (Homologically simply connected complex domains).

There is no continuous $L:\mathbb{C}\setminus \{0\}\to \mathbb{C}$ with $\exp (L(z))=z$ for every $z\mathrm{\ne }0$ (There is no continuous logarithm on all of $\mathbb{C}\setminus \{0\}$).

If $L$ and $h$ are holomorphic on an open set with $\exp \circ L=h$, then $h$ is nowhere zero and $L^{\mathrm{\prime }}=h^{\mathrm{\prime }}\mathrm{/}h$ (A holomorphic logarithm is a primitive of the logarithmic derivative).

For a positively oriented circle $a+r\exp (it)$ with $r>0$, $(2\pi i{)}^{-1}{\int }_{\gamma }dz\mathrm{/}(z-a)=1$ (The normalized integral around a positively oriented circle centred at a is 1).

If $F$ is holomorphic on an open set, $F^{\mathrm{\prime }}$ is continuous there, and $\gamma$ is a closed rectifiable contour in that set, then ${\int }_{\gamma }{F}^{\mathrm{\prime }}(z)dz=0$ (The integral of a continuous complex derivative over every closed rectifiable contour is zero).

A cycle with trace in an open $\mathrm{\Omega }$ is null-homologous in $\mathrm{\Omega }$ when its index vanishes at every point of $\mathbb{C}\setminus \mathrm{\Omega }$ (Null-homologous cycles and homologous cycles in an open set).

The annulus $\{\frac{1}{2}<\mathrm{\mid }z\mathrm{\mid }<2\}$ is a complex domain that is not homologically simply connected, the unit circle in it having index $1$ about the origin (A connected plane domain that is not homologically simply connected).

For $a\in \mathbb{C}$, $r>0$ and $k\in \mathbb{Z}$, the contour $a+r\exp (ikt)$ on $[0,2\pi ]$ is a closed complex contour with index $k$ for $\mathrm{\mid }z-a\mathrm{\mid }<r$ and $0$ for $\mathrm{\mid }z-a\mathrm{\mid }>r$, with trace $\{\mathrm{\mid }z-a\mathrm{\mid }=r\}$ when $k\mathrm{\ne }0$ (A circle traversed $k$ times has winding number $k$ inside and $0$ outside).

A complex domain is a nonempty, connected, open subset of $\mathbb{C}$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$); for $c\in \mathbb{C}$ and $R\ge 0$ the set $\{z:\mathrm{\mid }z-c\mathrm{\mid }>R\}$ is path-connected and connected, and at $R=0$ this is the punctured plane (The exterior of a closed disc in the plane is path-connected).

A complex differentiable function is continuous (Complex differentiability at a point implies continuity there), and linear combinations, products and nonvanishing quotients of complex differentiable functions are complex differentiable, the identity having derivative $1$ (Linearity, product, reciprocal, and quotient rules for complex derivatives).

A single closed contour with coefficient $1$ is a cycle whose trace is that contour's trace (Complex chains, their traces, and cycles) and whose index is that contour's winding number (Integration over a complex chain and the index of a chain).

A set is open exactly when each of its points admits a ball inside it (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Open ball, closed ball and sphere in a metric space).