Properties of Analytic Continuation Along A Curve
Analytic continuation along a curve is essentially unique, in the sense that given two analytic continuations and of along the functions and coincide on Informally, this says that any two analytic continuations of along will end up with the same values in a neighborhood of
If the curve is closed (that is, ), one need not have equal in a neighborhood of For example, if one starts at a point with and the complex logarithm defined in a neighborhood of this point, and one lets be the circle of radius centered at the origin (traveled counterclockwise from ), then by doing an analytic continuation along this curve one will end up with a value of the logarithm at which is plus the original value (see the second illustration on the right).
Read more about this topic: Monodromy Theorem
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