Complex Analysis 3.pdf-Theme 8. Taylor a...
Complex_Analysis_3.pdf-Theme 8. Taylor and Laurent Series.
Showing 22-26 out of 26
Complex Analysis 3.pdf-Theme 8. Tay...
Complex_Analysis_3.pdf-Theme 8. Taylor and Laurent Series.
##### Page 22
(5) Now
Z
-∞
f
(
x
)
dx
= lim
R
→∞
Z
L
R
f
(
z
)
dz
= lim
R
→∞
Z
L
R
f
(
z
)
dz
+ lim
R
→∞
Z
C
R
f
(
z
)
dz
= lim
R
→∞
Z
L
R
f
(
z
)
dz
+
Z
C
R
f
(
z
)
dz
(by the Algebra of Limits)
= lim
R
→∞
Z
L
R
+
C
R
f
(
z
)
dz
= lim
R
→∞
Z
γ
R
f
(
z
)
dz
=2
πi
lim
R
→∞
(sum of residues of
f
in upper half plane)
=2
πi
(sum of residues of
f
in upper half plane)
.
22

Example:
Evaluate
I
=
Z
-∞
x
2
x
4
+1
dx
.
23

Example:
Evaluate
J
=
Z
-∞
cos 2
x
x
2
+4
x
+6
dx
.
24

##### Page 25
Theme 12.
Trigonometric Integrals.
The method of residues is also useful in the evaluation of deﬁnite integrals of the type
I
=
Z
2
π
0
F
(sin
θ,
cos
θ
)
d
0
where
F
is a rational function in sin
θ
and cos
θ
.
Idea: consider
θ
as the argument of
z
on the unit circle
|
z
|
= 1.
As
θ
increases from 0
to 2
π
,
z
=
e
traces out the unit circle once anti-clockwise. Note that
dz
=
ie
=
izdθ
and
sin
θ
=
e
-
e
-
2
i
=
z
-
1
/z
2
i
,
cos
θ
=
e
+
e
-
2
=
z
+1
/z
2
So
I
is a contour integral of a rational function around the unit circle.
Example:
Find
Z
2
π
0
5 + 4 cos
θ
.
25

##### Page 26
Example:
Find
Z
2
π
0
cos 2
θ
13 + 5 cos
θ
dθ.
26

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