Monodromy theorem
= Monodromy theorem
{wiki}
The monodromy theorem says that, when <analytic continuation> exists throughout a <domain>, its ambiguity depends on <homotopy>. Let $D\subset\mathbb C$ be a <domain>, $a\in D$, and $g_a$ a <germ of a holomorphic function> at $a$. Assume $g_a$ admits <analytic continuation> along every <continuous path> in $D$ starting at $a$. Two such <paths> ending at $b$ give the same terminal <germ of a holomorphic function> if they are <homotopic> with endpoints fixed in $D$. <A Guide to Complex Variables by Steven G. Krantz>[Krantz, A Guide to Complex Variables, sections 9.3.4–9.3.5, printed pp. 158–159]{ref}
Here a <homotopy> with endpoints fixed is a <continuous map> $H:[0,1]^2\to D$ satisfying
$$
H(0,t)=\gamma_0(t),\quad H(1,t)=\gamma_1(t),\quad H(s,0)=a,\quad H(s,1)=b.
$$
Every intermediate <path> must stay in $D$. Equality means agreement on a <neighbourhood> of $b$, rather than merely equal values at $b$. <A Guide to Complex Variables by Steven G. Krantz>[Sections 9.3.1–9.3.3, printed pp. 157–158: terminal elements and fixed endpoint homotopy]{ref}
The <identity theorem> provides local rigidity; the <monodromy theorem> shows precisely how <topology> controls the remaining global freedom. Deforming a route without crossing a forbidden point cannot change the resulting <germ of a holomorphic function>. Thus a complicated <analytic continuation> calculation can be replaced by one on a simpler <homotopic> route. Distinct <homotopy classes> can still give the same continuation: a <rational function> on its <domain> already gives a single-valued answer along every route, even around a <pole>.