A point is said to be a regular singular point or a Fuchsian singular point of this equation if at least one of the functions has a pole at and, for every value of between and , either is regular at or has a pole of order not greater than .
If is a Fuchsian singular point, then the differential equation may be rewritten as a system of first order equations
in which the coefficient functions are analytic at . This fact helps explain the restiction on the orders of the poles of the ’s.
If an equation has a Fuchsian singularity, then the solution can be expressed as a Frobenius series in a neighborhood of this point.
A singular point of a differential equation which is not a regular singular point is known as an irregular singular point.
The Bessel equation
has a Fuchsian singularity at since the coefficient of has a pole of order and the coefficient of has a pole of order .
On the other hand, the Hamburger equation
has an irregular singularity at since the coefficient of has a pole of order .
|Date of creation||2013-03-22 14:47:26|
|Last modified on||2013-03-22 14:47:26|
|Last modified by||rspuzio (6075)|
|Synonym||Fuchsian singular point|
|Synonym||regular singular point|
|Defines||irregular singular point|