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Unformatted text preview: x2 ) d2 y
dy
+ p(p + 1)y = 0,
− 2x
dx2
dx (1.11) where p is a parameter. This equation is very useful for treating spherically
symmetric potentials in the theories of Newtonian gravitation and in electricity
and magnetism.
To apply our theorem, we need to divide by 1 − x2 to obtain
d2 y
p(p + 1)
2x dy
+
−
y = 0.
2
2 dx
dx
1−x
1 − x2
Thus we have
P (x) = − 2x
,
1 − x2 Q(x) =
12 p(p + 1)
.
1 − x2 Now from the preceding section, we know that the power series
1 + u + u2 + u3 + · · · converges to 1
1−u for u < 1. If we substitute u = x2 , we can conclude that
1
= 1 + x2 + x4 + x6 + · · · ,
1 − x2
the power series converging when x < 1. It follows quickly that
P (x) = − 2x
= −2x − 2x3 − 2x5 − · · ·
1 − x2 and p(p + 1)
= p(p + 1) + p(p + 1)x2 + p(p + 1)x4 + · · · .
1 − x2
Both of these functions have power series expansions about x0 = 0 which converge for x < 1. Hence our theorem implies that any solution to Legendre’s
equation will be expressible as a power series about x0 = 0 which converges for...
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 Winter '14
 Equations

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