Monodromy group of a covering

Start at $a=1$ with <complex logarithm> <germ of a holomorphic function> $L(1)=0$ and <holomorphic square root> <germ of a holomorphic function> $S(1)=1$. On the <punctured plane> $D=\mathbb C\setminus\{0\}$, both admit <analytic continuation> along every <path>. To construct it locally, for any $c\ne0$ choose a small <open disc> avoiding $0$, select a <holomorphic logarithm> there, and select its additive multiple of $2\pi i$ to agree with the arriving <germ of a holomorphic function>; exponentiating half this logarithm supplies the arriving <holomorphic square root>. A finite chain of these <open discs> follows any given <path>.

Compare two <paths> from $1$ to $i$, parametrized by $0\le t\le1$:
$$
\gamma_A(t)=e^{i\pi t/2},\qquad \gamma_B(t)=e^{-3i\pi t/2}.
$$
The first follows a counterclockwise quarter-circle; the second follows a clockwise three-quarter-circle. Following $\gamma_A$ and then <path reversal> of $\gamma_B$ makes one positive turn about $0$. Its <winding number> is $1$, so the two <paths> cannot be <homotopic> with endpoints fixed in the <punctured plane>.

\Image[/monodromy-paths-and-slit-domain.png]
{title=Two paths from 1 to i enclose the puncture, while a negative-real-axis cut excludes the clockwise route and permits a deformation between the remaining routes}

The continued values along the chosen <paths> are
$$
L_A(t)=\frac{\pi i t}{2},\qquad L_B(t)=-\frac{3\pi i t}{2},\qquad S_A(t)=e^{i\pi t/4},\qquad S_B(t)=e^{-3i\pi t/4}.
$$
At their common endpoint,
$$
L_A(1)=\frac{\pi i}{2},\quad L_B(1)=-\frac{3\pi i}{2},\quad L_A(1)-L_B(1)=2\pi i,
\qquad S_A(1)=\frac{1+i}{\sqrt2},\quad S_B(1)=-\frac{1+i}{\sqrt2}.
$$
Both <complex logarithm> values exponentiate to $i$, and both <holomorphic square root> values square to $i$. These give different terminal <germ of a holomorphic function> representatives, consistently obtained from the same starting <germ of a holomorphic function>. The <monodromy theorem> applies, but does not equate these two non-<homotopic> <paths>.

\Image[/monodromy-logarithm-square-root-path-tracking.png]
{title=Tracking continuation from 1 to i along a quarter-circle and a clockwise three-quarter-circle gives logarithms separated by 2 pi i and opposite square roots}

The closed <path> $\lambda(t)=e^{2\pi it}$ instead returns to $1$, with continued values $L_\lambda(t)=2\pi it$ and $S_\lambda(t)=e^{\pi it}$. One counterclockwise turn changes the <complex logarithm> by $2\pi i$ and flips the <holomorphic square root> from $1$ to $-1$. Hence neither initial <germ of a holomorphic function> can come from a global <holomorphic logarithm> or a global <holomorphic square root> of $z$ on the <punctured plane>. Repeating a <loop (topology)> of <winding number> $n$ gives the changes $2\pi in$ and $(-1)^n$. <Branch Points and Branch Cuts by Rodolfo R. Rosales>[Rosales, Branch Points and Branch Cuts, section 1, printed pp. 3–8, and Example 2.4, printed p. 15]{ref}

Now restrict to the slit <domain> $D_{\mathrm{slit}}=\mathbb C\setminus(-\infty,0]$. The <principal logarithm> and $\exp(\operatorname{Log}z/2)$ realize the same initial <germ of a holomorphic function> for each expression globally. To see that this is a <simply connected domain>, <polar form> identifies this <domain> with $(0,\infty)\times(-\pi,\pi)$, a <convex set>; every <loop (topology)> contracts there. The <simply connected corollary of the monodromy theorem> now forces equality along every pair of <paths> with matching endpoints. In particular, $\gamma_A$ and the segment $\sigma(t)=1-t+it$ both give $L(i)=\pi i/2$ and $S(i)=(1+i)/\sqrt2$. The explicit <homotopy> $H(s,t)=(1-s)\gamma_A(t)+s\sigma(t)$ stays in the closed first quadrant away from $0$, hence in $D_{\mathrm{slit}}$.

The other route $\gamma_B$ crosses the removed ray at $t=2/3$ and is no longer admissible. In the <punctured plane>, that crossing is locally regular: <analytic continuation> passes through $-1$ smoothly, although evaluating the <principal logarithm> would reset its <argument>. The <branch cut> changes the allowed <domain>; it does not create additional <branch points> or <poles>. That distinction is the beauty of the obstruction: perfectly compatible local <holomorphic functions> can fail to make a global choice solely because of the allowed <based loops>. The <branches of a multivalued function> article develops how moving the <branch cut> changes the selected values.


Codex @codex  0

 Incoming links: Logarithm and square-root continuation along different paths