Serge Ballif
MATH 502 Homework 5
February 22, 2008
(1)
Evaluate the integral
Z
∞
∞
x
2
cos
x
4 +
x
4
d
x
by means of contour integration. Be sure to justify carefully all the steps in your argument.
To compute this integral, we will consider the function
h
(
z
) =
z
2
e
iz
4+
z
4
, which is
bounded in the upper half plane and has real part equal to the integrand in our
problem. We will integrate over the curve
γ
1
and
γ
2
as pictured below.
R

R
1 +
i

1 +
i
γ
2
γ
1
Call the entire path Γ. Then by the residue theorem
1
2
πi
Z
Γ
f
(
z
) d
z
= Res(
h,
1 +
i
) + Res(
h,

1 +
i
)
.
Since

e
iz
 ≤
1 in the upper half plane and tends to 0 almost everywhere as
R
→
∞
, we see that
R
γ
2
z
2
e
z
4+
z
4
d
z
tends to 0 as
R
→ ∞
by the dominated convergence
theorem.
Therefore, by the residue theorem
Z
Γ
z
2
e
z
4 +
z
4
d
z
=
Z
γ
1
z
2
e
z
4 +
z
4
d
z
= 2
πi
(Res(
h,
1 +
i
) + Res(
h,

1 +
i
))
.
We are finished when we compute this, because
R
∞
∞
x
2
cos
x
4+
x
4
d
x
= lim
R
→∞
R
γ
1
z
2
e
z
4+
z
4
d
z
To calculate the residues we note that
z
4
+ 4 = (
z

1

i
)(
z

1 +
i
)(
z
+ 1

i
)(
z
+1+
i
), so
h
(
z
) has simple poles at
z
=
±
1
±
i
. We can compute the residues
using the formula for simple poles: Res(
h,
1 +
i
) =
(1+
i
)
2
e
i
(1+
i
)
4(1+
i
)
3
= (1

i
)
e

1+
i
/
8.
Similarly Res(
h,

1 +
i
) = (

1

i
)
e

1

i
/
8. Thus
Z
∞
∞
x
3
cos
x
4 +
x
4
d
x
= Re
2
πi
e

1
(
(
e
i

e

i
) +
i
(

e
i

e

i
)
)
8
!
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 Summer '08
 JOHNRIDENER
 Math, Methods of contour integration, dz, Serge Ballif

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