
*a
3g13p~,g&
44a+35
lwl
%
16
a%
3
FI
**m
C%K~I&+I
fij+%
00
io
x
12. The Fourier transform of
f
(x) is defined by
F(f
(x)}
=
f
(x)
e
h.
If
f
("'(x) is the
nth
00
I
derivative,
F{
f
'"'
(x)) is equal to
I
(d)
(io)"
f
(x))
(e) none of the above
13. If
H(x)
is the Heaviside step hction
and
f
=
H(x
+
1)


.
The Fourier transform
I
F(
f
is equal to
(b)
2%
0
sin w
(c)

W
COS
W
(4

W
(e) none of the above
14. Consider the wave equation
c2u,
=
u,,,
u
(0,t)
=
u(L,
t)
=
0,
u(x,
0)
=
f
(x), u,(x,O)
=
g(x) for
I
0
5 x S
L,
0
5
t.
By
the method of separation of variables,
we
have
u(x,
=
X(x)T(t) and
1
115. Following the above problem, u(x,,t) is the superposition ofthe possible
X(x)T(t)
has
the
form
CO
u(x, t)
=
A,X, (x)T,
(t)
.
Then
X,
(x) is equal to
n=l
(2n

1)n
(a) sin
x
L
(2n

1)n
(b)
cos
x
2L
2nn
(c) sin
x
L
16. Which of the following complex hctions is
NOT
differentiable?
nn
(d) sinx
L
I
(e) cosx
(a) sinz