SINGULAR POINT
KIAM HEONG KWA
In this and the next sections, we indicate the construction of solutions
to the secondorder homogeneous linear equation
(1)
P
(
x
)
y
00
+
Q
(
x
)
y
0
+
R
(
x
)
y
= 0
in the vicinity of a regular singular point
x
0
. By the transformation
t
=
x

x
0
, we assume that
x
0
= 0. Recall that
x
0
= 0 being a regular
singular point implies that
xp
(
x
) =
xQ
(
x
)
P
(
x
)
and
x
2
q
(
x
) =
x
2
R
(
x
)
P
(
x
)
are
analytic at
x
0
= 0 in the sense that they are representable as convergent
power series
(2)
xp
(
x
) =
∞
X
n
=0
p
n
x
n
and
x
2
q
(
x
) =
∞
X
n
=0
q
n
x
n
on the interval (

ρ,ρ
) for some
ρ >
0, where
(3)
p
0
= lim
x
→
0
xp
(
x
) and
q
0
= lim
x
→
0
x
2
q
(
x
)
.
To make the quantities
xp
(
x
) and
x
2
q
(
x
) appear in (1), we divide (1)
by
P
(
x
) and multiply the resulting equation by
x
2
. The ﬁnal equation
is
(4)
x
2
y
00
+
x
[
xp
(
x
)]
y
0
+ [
x
2
q
(
x
)]
y
= 0
.
It is known that
there is at least one formal power series of the form
y
(
x
) =
x
r
∞
X
n
=0
a
n
x
n
=
∞
X
n
=0
a
n
x
r
+
n
,
where
a
0
6
= 0
, that formally satisﬁes
(4) [1]. As we will see, the values
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 Winter '11
 Kwa
 Differential Equations, Equations

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