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 (from here on called simply 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.

Illustration of analytic continuation along a curve (only a finite number of the disks are shown).
Analytic continuation along a curve of the natural logarithm (the imaginary part of the logarithm is shown only).

Before stating this theorem it is necessary to define analytic continuation along a curve and study its properties.

Analytic continuation along a curve

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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 (a continuous function)   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   (which implies that   and   have a non-empty intersection) and the functions   and   coincide on the intersection  

Properties of analytic continuation along a curve

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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).

Monodromy theorem

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Homotopy with fixed endopoints is necessary for the monodromy theorem to hold.

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-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  

See also

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References

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  • Krantz, Steven G. (1999). Handbook of complex variables. Birkhäuser. ISBN 0-8176-4011-8.
  • Jones, Gareth A.; Singerman, David (1987). Complex functions: an algebraic and geometric viewpoint. Cambridge University Press. ISBN 0-521-31366-X.
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