Find the Laurent series of f ( z ) = e z / ( 1 − z ) for | z | < 1 and | z | > 1. Hints: For | z | < 1 , multiply two power series; you should find f ( z ) = ∑ n = 0 ∞ a n z n with a n = ∑ k = 0 n 1 / k ! . For | z | > 1 , use (4.3) where C is a circle | z | = a with a > 1 Evaluate the integrals by finding residues at 1 and 0. You should find f ( z ) = ∑ n = 0 ∞ a n z n + ∑ n = 1 ∞ b n z − n where all b n = − e and a n = − e + ∑ k = 0 n 1 / k ! .
Find the Laurent series of f ( z ) = e z / ( 1 − z ) for | z | < 1 and | z | > 1. Hints: For | z | < 1 , multiply two power series; you should find f ( z ) = ∑ n = 0 ∞ a n z n with a n = ∑ k = 0 n 1 / k ! . For | z | > 1 , use (4.3) where C is a circle | z | = a with a > 1 Evaluate the integrals by finding residues at 1 and 0. You should find f ( z ) = ∑ n = 0 ∞ a n z n + ∑ n = 1 ∞ b n z − n where all b n = − e and a n = − e + ∑ k = 0 n 1 / k ! .
Find the Laurent series of
f
(
z
)
=
e
z
/
(
1
−
z
)
for
|
z
|
<
1
and
|
z
|
>
1.
Hints: For
|
z
|
<
1
,
multiply two power series; you should find
f
(
z
)
=
∑
n
=
0
∞
a
n
z
n
with
a
n
=
∑
k
=
0
n
1
/
k
!
.
For
|
z
|
>
1
,
use (4.3) where
C
is a circle
|
z
|
=
a
with
a
>
1
Evaluate the integrals by finding residues at 1 and 0. You should find
f
(
z
)
=
∑
n
=
0
∞
a
n
z
n
+
∑
n
=
1
∞
b
n
z
−
n
where all
b
n
=
−
e
and
a
n
=
−
e
+
∑
k
=
0
n
1
/
k
!
.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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