•
Integrating power series
If
f
(
x
) =
∑
n
a
n
(
x

x
0
)
n
has radius
R >
0 then
f
can be integrated on any subinterval
of (
x
0

R,
x
0
+
R
) by integrating under the summation sign.
•
Taylor series
If
f
(
x
) =
∑
n
a
n
(
x

x
0
)
n
has radius
R >
0 then
a
n
=
f
(
n
)
(
x
0
)
n
!
.
•
x
0
is an “ordinary point” for
P
(
x
)
y
±±
+
Q
(
x
)
y
±
+
R
(
x
)
y
= 0 if
Q
(
x
)
P
(
x
)
and
R
(
x
)
P
(
x
)
have Taylor
series
expansions
about
x
=
x
0
with
positive
radii of convergence.
Otherwise
x
0
is a
“singular point”.
•
Power series solutions about ordinary points
If
x
0
is an ordinary point of
P
(
x
)
y
±±
+
Q
(
x
)
y
±
+
R
(
x
)
y
= 0 then every solution
y
(
x
)
has a Taylor series expansion about
x
=
x
0
with positive radius of convergence.
•
x
0
is a “regular singular point” for
P
(
x
)
y
±±
+
Q
(
x
)
y
±
+
R
(
x
)
y
= 0 if it is a singular point
and
(
x

x
0
)
Q
(
x
)
P
(
x
)
and
(
x

x
0
)
2
R
(
x
)
P
(
x
)
have Taylor series expansions about
x
=
x
0
with positive
radii of convergence.
•
Indicial equation
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 Spring '08
 Fonken
 Power Series, Taylor Series, Regular singular point

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