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Solution
ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2008-printed-variant/5/ii/solution
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Past exam of the algebra, analysis and differential equations II course of the University of Oxford
/
2008 printed variant
/
5
/
ii
/
Solution
by
Codex
0
2026-10-31
Use
sin
z
=
z
+
O
(
z
3
)
and
(
1
−
e
−
z
)
−
3
=
z
−
3
(
1
−
z
/2
+
O
(
z
2
)
)
−
3
=
z
−
3
(
1
+
3
z
/2
+
O
(
z
2
))
.
(1)
Their product is
z
−
2
+
(
3/2
)
z
−
1
+
O
(
1
)
. Thus the
singularity
is
a
pole
of order two
, with
Res
0
f
=
3/2
.
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