# The point of these denitions is that in the case

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Unformatted text preview: he form z = e−x /2 Pn (x), where Pn (x) is a polynomial. 1.3 Singular points Our ultimate goal is to give a mathematical description of the vibrations of a circular drum. For this, we will need to solve Bessel’s equation, a second-order homogeneous linear diﬀerential equation with a “singular point” at 0. A point x0 is called an ordinary point for the diﬀerential equation d2 y dy + P (x) + Q(x)y = 0 2 dx dx (1.13) if the coeﬃcients P (x) or Q(x) are both real analytic at x = x0 , or equivalently, both P (x) or Q(x) have power series expansions about x = x0 with positive radius of convergence. In the opposite case, we say that x0 is a singular point ; thus x0 is a singular point if at least one of the coeﬃcients P (x) or Q(x) fails to be real analytic at x = x0 . A singular point is said to be regular if (x − x0 )P (x) and 15 (x − x0 )2 Q(x) are real analytic. For example, x0 = 1 is a singular point for Legendre’s equation d2 y p(p + 1) 2x dy + − y = 0, dx2 1 − x2 dx 1 − x2 because 1 − x...
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## This document was uploaded on 01/12/2014.

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