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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 secondorder
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.
 Winter '14
 Equations

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