Proof. Let \(H\) be a homotopy from \(\gamma _0\) to \(\gamma _1\) and write \(\gamma _s\) for the path \(t \mapsto H(t,s)\). Let \(g_s\) denote the terminal element obtained by continuing along \(\gamma _s\), which exists by hypothesis. The heart of the matter is that \(s \mapsto g_s\) is locally constant: if \(s\) and \(s'\) are close enough, the paths \(\gamma _s\) and \(\gamma _{s'}\) run through a common chain of discs, because \(H\) is uniformly continuous on the compact square and the discs used along \(\gamma _s\) have radii bounded below by a positive number. Along that shared chain the two continuations agree step by step, so \(g_s = g_{s'}\).

A locally constant function on the connected set \([0,1]\) is constant, so \(g_0 = g_1\). \(\blacksquare \) □

Proof. In a simply connected domain any two paths from \(a\) to \(b\) are homotopic, so by the theorem the value assigned at \(b\) does not depend on the route taken. Defining \(F(b)\) to be that common value therefore gives a well-defined function on \(D\), and it is analytic near each \(b\) because it coincides there with a function element. \(\blacksquare \)

This is the precise sense in which the earlier examples fail. The punctured plane \(\mathbb {C}\setminus \{0\}\) is not simply connected, and a loop about the origin is not homotopic to a constant path there; so the corollary does not apply and \(\log z\) is free to come back changed. Cut the plane along a ray — say \(\mathbb {C}\setminus (-\infty , 0]\), which is simply connected — and a single-valued branch exists at once. That is what a branch cut is for, and it is why the cut may be placed anywhere so long as it stops the loop from closing. □

Proof. Fix a base point \(a\in D\) and a value \(w_a\) with \(e^{w_a}=a\), possible since \(a\neq 0\). For \(z\in D\) define \[L(z)=w_a+\int _{\gamma }\frac {d\zeta }{\zeta },\] the integral taken along any path \(\gamma \) in \(D\) from \(a\) to \(z\).

\(L\) is well defined

The integrand \(1/\zeta \) is analytic on \(D\), because \(0\notin D\). Since \(D\) is simply connected, Cauchy’s theorem gives \(\int _{\Gamma }\frac {d\zeta }{\zeta }=0\) around every closed contour \(\Gamma \) in \(D\), so two paths from \(a\) to \(z\) give the same value. This is exactly the step that fails on a domain encircling the origin, and it is why \(\log \) is multi-valued there.

\(L\) is analytic with \(L'(z)=1/z\)

An integral of an analytic function taken from a fixed base point is an antiderivative of the integrand, so \(L\) is analytic on \(D\) and \(L'(z)=1/z\).

\(e^{L(z)}=z\)

Consider \(h(z)=ze^{-L(z)}\). Differentiating, \[h'(z)=e^{-L(z)}-z\,L'(z)e^{-L(z)}=e^{-L(z)}\left (1-z\cdot \frac 1z\right )=0 ,\] so \(h\) is constant on the domain \(D\). At \(z=a\), \(h(a)=ae^{-w_a}=1\), hence \(h\equiv 1\) and \(e^{L(z)}=z\) throughout \(D\). So \(L\) is a single-valued analytic branch of \(\log z\).

Any two branches differ by a constant \(2n\pi i\): their difference has zero derivative and takes values in the discrete set of logarithms of \(1\). □