In complex analysis, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function along curves starting in the original domain of the function and ending in the larger set. A potential problem of this analytic continuation along a curve strategy is there are usually many curves which end up at the same point in the larger set. The monodromy theorem gives sufficient conditions for analytic continuation to give the same value at a given point regardless of the curve used to get there, so that the resulting extended analytic function is well-defined and single-valued. Before stating this theorem it is necessary to define analytic continuation along a curve and study its properties.
The definition of analytic continuation along a curve is a bit technical, but the basic idea is that one starts with an analytic function defined around a point, and one extends that function along a curve via analytic functions defined on small overlapping disks covering that curve. Formally, consider a curve Let be an analytic function defined on an open disk centered at An analytic continuation of the pair along is a collection of pairs for such that
and
For each is an open disk centered at and is an analytic function.
For each there exists such that for all with one has that and the functions and coincide on the intersection
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, 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, 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.
Monodromy theorem
As noted earlier, two analytic continuations along the same curve yield the same result at the curve's endpoint. However, given two different curves branching out from the same point around which an analytic function is defined, with the curves reconnecting at the end, it is not true in general that the analytic continuations of that function along the two curves will yield the same value at their common endpoint. Indeed, one can consider, as in the previous section, the complex logarithm defined in a neighborhood of a point and the circle centered at the origin and radius Then, it is possible to travel from to in two ways, counterclockwise, on the upper half-planearc of this circle, and clockwise, on the lower half-plane arc. The values of the logarithm at obtained by analytic continuation along these two arcs will differ by If, however, one can continuously deform one of the curves into another while keeping the starting points and ending points fixed, and analytic continuation is possible on each of the intermediate curves, then the analytic continuations along the two curves will yield the same results at their common endpoint. This is called the monodromy theorem and its statement is made precise below. The monodromy theorem makes it possible to extend an analytic function to a larger set via curves connecting a point in the original domain of the function to points in the larger set. The theorem below which states that is also called the monodromy theorem.
