A nonvanishing holomorphic function on a disc has a holomorphic logarithm

Given: A disc $D(a,r)=\{z:\mathrm{\mid }z-a\mathrm{\mid }<r\}$ with $r>0$ and a nowhere-zero holomorphic function $h$ on it. It is nonempty since it contains $a$. For each $z$ in it, $\epsilon =r-\mathrm{\mid }z-a\mathrm{\mid }>0$ and $\mathrm{\mid }w-z\mathrm{\mid }<\epsilon$ imply $\mathrm{\mid }w-a\mathrm{\mid }<r$ by the triangle inequality, so the disc is open in $d_{\mathbb{C}}$. For $z,w\in D(a,r)$ and $t\in [0,1]$ the triangle inequality gives $\mathrm{\mid }((1-t)z+tw)-a\mathrm{\mid }\le (1-t)\mathrm{\mid }z-a\mathrm{\mid }+t\mathrm{\mid }w-a\mathrm{\mid }<r$, so every segment between two points of the disc stays in it. By affine-segment and openness transport in The complex coordinate isometry transports topology and affine segments, its coordinate image is a convex open subset of ${\mathbb{R}}^2$ in A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points, hence star-shaped at $\mathrm{\Phi }(a)$ in Star-shaped open subsets of Euclidean space. Each coordinate segment is a continuous path by [L4]; composing it with the coordinate homeomorphism inverse gives a continuous path in the disc. Thus the disc is path connected, hence connected by [L4]; together with nonemptiness and openness this makes it a complex domain in A complex domain is a nonempty connected open subset of $\mathbb{C}$. Also $h^{\mathrm{\prime }}$ is itself holomorphic on the disc, because a holomorphic function has complex derivatives of all orders locally (All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle), so the quotient rule makes $h^{\mathrm{\prime }}\mathrm{/}h$ holomorphic there, $h$ being nowhere zero (Linearity, product, reciprocal, and quotient rules for complex derivatives); and the complex chain rule, the derivative of $\exp$, and the exponential addition law are supplied by The chain rule for complex derivatives, The complex exponential is entire and its complex derivative is itself, and $\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential.

Every holomorphic function on an open set star-shaped with respect to $a$ has a primitive there (Every holomorphic function on a star-shaped domain has a primitive).

A holomorphic function whose derivative vanishes everywhere on a complex domain is constant (A holomorphic function with zero derivative on a domain is constant).

For every nonzero complex number $z$, the solutions of $\exp w=z$ are exactly $\mathrm{Log}z+2\pi ik$ with $k\in \mathbb{Z}$ (All logarithms of $z\mathrm{\ne }0$ are $\mathrm{Log}z+2\pi ik$, $k\in \mathbb{Z}$).

A segment $t\mapsto (1-t)v_0+t{v}_1$ that lies in a subset $A$ is a continuous path in $A$, and a path-connected subset of a topological space is a connected subset (A finite concatenation of straight segments in ${\mathbb{R}}^n$ is a continuous path, Every path-connected space is connected, and every path component lies inside a component, claim 2).