Necessary and sufficient conditions are be given for the existence of analytic solutions of the nonhomogeneous n-th order differential equation at a singular point. Let be a linear differential operator with coefficients analytic at zero. If denotes the operator conjugate to , then we will show that the dimension of the kernel of is equal to the dimension of the kernel of . Certain representation theorems from functional analysis will be used to describe the space of linear functionals that contain the kernel of . These results will be used to derive a form of the Fredholm Alternative that will establish a link between the solvability of at a singular point and the kernel of . The relationship between the roots of the indicial equation associated with and the kernel of will allow us to show that the kernel of is spanned by a set of polynomials.
Submitted July 28, 2001. Published February 4, 2002.
Math Subject Classifications: 30A99, 34A30, 34M35, 46E15.
Key Words: linear differential equation, regular singular point, analytic solution.
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| Brian Haile |
Department of Mathematics and Statistics
Northwest Missouri state university
800 University Drive
Maryville, MO 64468 USA
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