1
Physics 384 2016 Assignment 8
Due in class Friday November 25
Problem I
Consider the inhomogeneous ODE
d
2
y
(
x
)
dx
2

k
2
y
(
x
) =
f
(
x
)
where the boundary conditions are that
y
(
±∞
) =
finite. Use the Fourier transform method
to show that the Green’s function
G
(
x
;
x
0
)
for this problem, which satisfies
d
2
G
(
x
;
x
0
)
dx
2

k
2
G
(
x
;
x
0
) =
δ
(
x

x
0
)
can be expressed as
G
(
x
;
x
0
) =

1
2
π
∞
ˆ
∞
e
ik
0
(
x

x
0
)
k
0
2
+
k
2
dk
0
Evaluate the integral and show that
G
(
x
;
x
0
) =

1
2
k
e

k

x

x
0

∀
x
Problem II
Use Fourier transform methods to show that the Green’s function for Poisson’s equation
~
∇
2
G
(
~
r
;
~
r
0
) = 4
πδ
3
(
~
r

~
r
0
)
is given by
G
(
~
r
;
~
r
0
) =
1

~
r

~
r
0

Note: This requires a bit more care than some of the examples we’ve already treated. In
particular, the Fourier transform of
G
does not actually exist, without modification, since
the existence of a transform requires that the function be absolutely integrable. In this case,
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