View source for Monodromy theorem

A sufficient criterion for the single-valuedness of a [[Branch of an analytic function|branch of an analytic function]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m0646901.png" /> be a simply-connected domain in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m0646902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m0646903.png" />. Now, if an analytic function element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m0646904.png" />, with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m0646905.png" />, can be analytically continued along any path in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m0646906.png" />, then the branch of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m0646907.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m0646908.png" />, arising by this [[Analytic continuation|analytic continuation]] is single-valued in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m0646909.png" />. In other words, the branch of the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469010.png" /> defined by the simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469011.png" /> and the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469012.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469013.png" /> must be single-valued. Another equivalent formulation is: If an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469014.png" /> can be analytically continued along all paths in an arbitrary domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469015.png" />, then the result of this continuation at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469016.png" /> (that is, the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469017.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469018.png" />) is the same for all homotopic paths in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469019.png" /> joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469020.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469021.png" />.

The monodromy theorem is valid also for analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469022.png" /> defined in domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064690/m06469023.png" /> 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>