# Monodromy theorem

A sufficient criterion for the single-valuedness of a branch of an analytic function. Let $D$ be a simply-connected domain in the complex space $\mathbf C^n$, $n\geq1$. Now, if an analytic function element $\Sigma(z^0;r)$, with centre $z^0\in D$, can be analytically continued along any path in $D$, then the branch of an analytic function $f(z)$, $z=(z_1,\dots,z_n)$, arising by this analytic continuation is single-valued in $D$. In other words, the branch of the analytic function $f(z)$ defined by the simply-connected domain $D$ and the element $\Sigma(z^0;r)$ with centre $z^0\in D$ must be single-valued. Another equivalent formulation is: If an element $\Sigma(z^0;r)$ can be analytically continued along all paths in an arbitrary domain $D\subset\mathbf C^n$, then the result of this continuation at any point $z^*\in D$ (that is, the element $\Sigma(z^*;r^*)$ with centre $z^*$) is the same for all homotopic paths in $D$ joining $z^0$ to $z^*$.
The monodromy theorem is valid also for analytic functions $f(z)$ defined in domains $D$ on Riemann surfaces or on Riemann domains. See also Complete analytic function.