HOME

TheInfoList



OR:

In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the monodromy theorem is an important result about
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
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. 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 In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
) \gamma:
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
to \Complex. Let f be an analytic function defined on an
open disk In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius, r, an open disk is usua ...
U centered at \gamma(0). An ''analytic continuation'' of the pair (f, U) along \gamma is a collection of pairs (f_t, U_t) for 0\le t\le 1 such that * f_0=f and U_0=U. * For each t\in
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
U_t is an open disk centered at \gamma(t) and f_t:U_t\to\Complex is an analytic function. * For each t\in
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math> there exists \varepsilon >0 such that for all t'\in
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math> with , t-t', <\varepsilon one has that \gamma(t')\in U_t (which implies that U_t and U_ have a non-empty
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
) and the functions f_t and f_ coincide on the intersection U_t\cap U_.


Properties of analytic continuation along a curve

Analytic continuation along a curve is essentially unique, in the sense that given two analytic continuations (f_t, U_t) and (g_t, V_t) (0\le t\le 1) of (f, U) along \gamma, the functions f_1 and g_1 coincide on U_1\cap V_1. Informally, this says that any two analytic continuations of (f, U) along \gamma will end up with the same values in a neighborhood of \gamma(1). If the curve \gamma is closed (that is, \gamma(0)=\gamma(1)), one need not have f_0 equal f_1 in a neighborhood of \gamma(0). For example, if one starts at a point (a, 0) with a>0 and the
complex logarithm In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
defined in a neighborhood of this point, and one lets \gamma be the circle of radius a centered at the origin (traveled counterclockwise from (a, 0)), then by doing an analytic continuation along this curve one will end up with a value of the logarithm at (a, 0) which is 2\pi i plus the original value (see the second illustration on the right).


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 (a, 0) and the circle centered at the origin and radius a. Then, it is possible to travel from (a, 0) to (-a, 0) 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 (-a, 0) obtained by analytic continuation along these two arcs will differ by 2\pi i. 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 U be an open disk in the complex plane centered at a point P and f:U\to \Complex be a complex-analytic function. Let Q be another point in the complex plane. If there exists a family of curves \gamma_s:
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
to \Complex with s\in
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math> such that \gamma_s(0)=P and \gamma_s(1)=Q for all s\in
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
the function (s, t)\in
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
times
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
to \gamma_s(t)\in \mathbb C is continuous, and for each s\in
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math> it is possible to do an analytic continuation of f along \gamma_s, then the analytic continuations of f along \gamma_0 and \gamma_1 will yield the same values at Q. 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 U be an open disk in the complex plane centered at a point P and f:U\to\Complex be a complex-analytic function. If W is an open
simply-connected set In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
containing U, and it is possible to perform an analytic continuation of f on any curve contained in W which starts at P, then f admits a ''direct analytic continuation'' to W, meaning that there exists a complex-analytic function g:W\to\Complex whose restriction to U is f.


See also

*
Analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
*
Monodromy In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...


References

* * *


External links


Monodromy theorem
at
MathWorld ''MathWorld'' 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 Dig ...
* {{PlanetMath, urlname=MonodromyTheorem, title=Monodromy theorem
Monodromy theorem
at the
Encyclopaedia of Mathematics The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduat ...
Theorems in complex analysis