Monodromy theorem

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A sufficient criterion for the single-valuedness of a branch of an analytic function. Let be a simply-connected domain in the complex space , . Now, if an analytic function element , with centre , can be analytically continued along any path in , then the branch of an analytic function , , arising by this analytic continuation is single-valued in . In other words, the branch of the analytic function defined by the simply-connected domain and the element with centre must be single-valued. Another equivalent formulation is: If an element can be analytically continued along all paths in an arbitrary domain , then the result of this continuation at any point (that is, the element with centre ) is the same for all homotopic paths in joining to .

The monodromy theorem is valid also for analytic functions defined in domains on Riemann surfaces or on Riemann domains. See also Complete analytic function.


[1] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian)
[2] S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian)
[3] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)



[a1] J.B. Conway, "Functions of one complex variable" , Springer (1978)
How to Cite This Entry:
Monodromy theorem. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098