1
ECE 6341
Spring 2009
Homework 6
1. Use integration by parts to asymptotically evaluate the following integral:
() ( )
1
0
cos
x
Ie xdx
−
Ω= Ω
∫.
2. Use the stationary-phase method to asymptotically evaluate the Bessel function
()
n
Jx for
large x, starting with
() ()
0
1cos sin
n
J
xnxd
π
θ
θθ
π
=−
∫.
3. An infinite uniform line source carries I Amps along the z axis, at a radian frequency
ω
. By
using the Fourier transform method, the exact vector potential
(
)
,
z
Axy for y > 0 is found to
be
()
01
,4
yx
jk y jk x
zx
y
I
Axy e e dk
jk
μ
π
∞−−
−∞
=∫,
where
()
1/2
22
0yx
kkk=− . Apply the stationary-phase method to find
()
z
A
ρ
in the far field
(1k
ρ
>> ). (Hint: First convert to polar coordinates for the observation point. There is no
need to do a change of variables for kx). Check your result by starting with the known exact
expression
()
(2)
00
4
z
I
AHk
j
μ
ρ
=
and then approximating this expression in the far field.
4. The following integral
()
,
I
r
θ
is a typical “Sommerfeld” type integral that often appears in
the analysis of dipoles in layered-media:
