Monodromy theorem
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.
References
[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) |
Comments
References
[a1] | J.B. Conway, "Functions of one complex variable" , Springer (1978) |
Monodromy theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monodromy_theorem&oldid=36520