Monodromy theorem
Encyclopedia
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, the monodromy theorem is an important result about analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
of a complex-analytic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
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.
Before stating this theorem it is necessary to define analytic continuation along a curve and study its properties.
Analytic continuation along a curve
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
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
)
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 intersectionIntersection (set theory)In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
) 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 (that is, 
), one need not have 
equal 
in a neighborhood of 
For example, if one starts at a point 
with 
and the complex logarithmComplex logarithm
In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the natural logarithm ln x is the inverse of the real exponential function ex. Thus, a logarithm of z is a complex number w such that ew = z. The notation for such a w is log z...
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
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 
External links
- Monodromy theorem at MathWorldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...
- Monodromy theorem at PlanetMathPlanetMathPlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is hosted by the Digital...
- Monodromy theorem at the Encyclopaedia of MathematicsEncyclopaedia of MathematicsThe Encyclopaedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM....