A nonvanishing holomorphic function on a homologically simply connected domain has a holomorphic logarithm

Given: A homologically simply connected complex domain $\mathrm{\Omega }$ and a holomorphic nowhere-zero $f:\mathrm{\Omega }\to \mathbb{C}$.

Every holomorphic function on a homologically simply connected complex domain has a primitive there (Every holomorphic function on a homologically simply connected domain has a primitive), that is a holomorphic $F$ with $F^{\mathrm{\prime }}$ equal to the function (A primitive of a complex function on an open set).

The complex exponential maps $\mathbb{C}$ onto $\mathbb{C}\setminus \{0\}$ (The complex exponential maps $\mathbb{C}$ onto $\mathbb{C}\setminus \{0\}$).

The complex exponential is entire with ${\exp }^{\mathrm{\prime }}=\exp$ (The complex exponential is entire and its complex derivative is itself).

If $f$ is complex differentiable at $a$ and $g$ at $f(a)$, then $(g\circ f{)}^{\mathrm{\prime }}(a)=g^{\mathrm{\prime }}(f(a))f^{\mathrm{\prime }}(a)$ (The chain rule for complex derivatives).

Linear combinations, products and nonvanishing quotients of functions complex differentiable at a point are complex differentiable there, with the usual formulas (Linearity, product, reciprocal, and quotient rules for complex derivatives).

If $U$ is a complex domain and $g:U\to \mathbb{C}$ is holomorphic with $g^{\mathrm{\prime }}\equiv 0$, then $g$ is constant on $U$ (A holomorphic function with zero derivative on a domain is constant).

$\ker (\exp )=2\pi i\mathbb{Z}$, and $\exp z=\exp w$ exactly when $z-w\in 2\pi i\mathbb{Z}$ ($\ker (\exp )=2\pi i\mathbb{Z}$, and $\exp z=\exp w$ exactly when $z-w\in 2\pi i\mathbb{Z}$).

Every holomorphic function has complex derivatives of all natural orders locally (All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle); a complex differentiable function is continuous (Complex differentiability at a point implies continuity there).

The continuous image of a connected subset is connected (A continuous image of a connected space is connected, and connectedness is a topological property), and a connected subset of $\mathbb{R}$ is order-convex (The connected subspaces of $\mathbb{R}$ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in $\mathbb{R}$").

A complex domain is a nonempty, connected, open subset of $\mathbb{C}$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$), and a homologically simply connected domain is such a domain in which every cycle is null-homologous (Homologically simply connected complex domains).

For $z=a+bi$ with $a,b$ real, $\mathrm{Im}z=b$ and $\mathrm{\mid }z\mathrm{\mid }=\sqrt{a^2+b^2}$ (Real and imaginary parts, complex conjugation, and modulus); the integers form an ordered commutative ring and are discrete in $\mathbb{R}$, so if $m<n$ then $m+\frac{1}{2}$ lies strictly between them and is not an integer (The integers form a commutative ring, The integers form a totally ordered ring, Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$).

For real $x,y$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$ ($\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$).

$\exp (z+w)=\exp z\exp w$ for all complex $z,w$, hence $\exp (-w)\exp (w)=1$ ($\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential).