monodromy group

Consider an ordinary linear differential equation


in which the coefficients are polynomials. If cn is not constant, then it is possible that the solutions of this equation will have branch pointsMathworldPlanetmath at the zeros of cn. (To see if this actually happens, we need to examine the indicial equationMathworldPlanetmath.)

By the persistence of differential equations, the analytic continuation of a solution of this equation will be another solution. Pick a neighborhoodMathworldPlanetmath which does not contain any zeros of cn. Since the differential equation is of order n, there will be n independent solutions y=f1(x),,y=fn(x). (For example, one may exhibit these solutions as power seriesMathworldPlanetmath about some point in the neighborhood.)

Upon analytic continuation back to the original neighborhood via a chain of neghborhoods, suppose that the solution y=fi(x) is taken to a solution y=gi(x). Because the solutions are linearly independentMathworldPlanetmath, there will exist a matrix {mij}i,j=1n such that


Now consider the totality of all such matrices corresponding to all possible ways of making analytic continuations along chains which begin and end wit the original neighborhood. They form a group known as the monodromy group of the differential equation. The reason this set is a group is some basic facts about analytic continuation. First, there is the trivial analytic continuation which simply takes a function element to itself. This will correspond to the identity matrixMathworldPlanetmath. Second, we can reverse a process of analytic continuation. This will correspond to the inverse matrix. Third, we can follow continuation along one chain of neighborhoods by continuation along another chain. This will correspond to multiplying the matrices corresponding to the two chains.

Title monodromy group
Canonical name MonodromyGroup
Date of creation 2013-03-22 16:21:51
Last modified on 2013-03-22 16:21:51
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 11
Author rspuzio (6075)
Entry type Definition
Classification msc 30A99
Related topic Monodromy