View source for Monodromy theorem

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A sufficient criterion for the single-valuedness of a [[Branch of an analytic function|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|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|Complete analytic function]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Stoilov,   "The theory of functions of a complex variable" , '''1–2''' , Moscow  (1962)  (In Russian; translated from Rumanian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,   "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Conway,   "Functions of one complex variable" , Springer  (1978)</TD></TR></table>