A nonvanishing holomorphic function on such a domain has holomorphic roots of every positive order

Given: A homologically simply connected complex domain $\mathrm{\Omega }$, a holomorphic nowhere-zero $f:\mathrm{\Omega }\to \mathbb{C}$, and a natural $m\ge 1$.

On a homologically simply connected complex domain, a holomorphic nowhere-zero $f$ admits a holomorphic $L$ with $\exp \circ L=f$, and any two such differ by a constant in $2\pi i\mathbb{Z}$ (A nonvanishing holomorphic function on a homologically simply connected domain has a holomorphic logarithm, Homologically simply connected complex domains).

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

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

The composite of functions complex differentiable at the relevant points is complex differentiable, with $(g\circ f{)}^{\mathrm{\prime }}(a)=g^{\mathrm{\prime }}(f(a))f^{\mathrm{\prime }}(a)$ (The chain rule for complex derivatives); constant multiples of complex differentiable functions are complex differentiable (Linearity, product, reciprocal, and quotient rules for complex derivatives).

For a natural $m\ge 1$, the $m$th roots of unity are exactly the numbers $\exp (2\pi ik\mathrm{/}m)$ for natural $k$ with $0\le k<m$ (The $n$-th roots of a complex number and the $n$ distinct roots of unity for every $n\ge 1$).

Natural powers satisfy $z^0=1$ and $z^{j+1}=z^{j}z$ (Integer powers in the complex field).

If a property holds at $0$ and passes from $j$ to $j+1$, it holds for every natural number (The principle of mathematical induction).

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 $e^{i\pi }+1=0$).

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