Monodromy Theorem - Monodromy Theorem

Monodromy Theorem

As noticed 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-plane arc 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.

Let be an open disk in the complex plane centered at a point and be a complex-analytic function. Let be another point in the complex plane. If there exists a family of curves with such that and for all the function is continuous, and for each it is possible to do an analytic continuation of along then the analytic continuations of along and will yield the same values at

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.

Let be an open disk in the complex plane centered at a point and be a complex-analytic function. If is an open simply-connected set containing, and it is possible to perform an analytic continuation of on any curve contained in which starts at then admits a direct analytic continuation to meaning that there exists a complex-analytic function whose restriction to is