2
PROBLEM 1
(a) Derive the first two of the (general) orthogonality equations
pT/2
m27r
n2rr
cos() cos(x) 4
-T/2
-
pT/2
m2ir
n2rr
cos(—--x) sin(-T-x)
J—T/2
T/2
1
sin(
-T/2
m27r
n2r
x) sin(—
T
T
)$
ifmn
ifm=n
T/2
.
lo
1T/2
ifm
ifm
=
n
n
You may use the following identities:
cos a cos b
[cos(a
—
b) + cos(a + b)],
sinacosb
[sin(a
—
b) + sin(a + b)].
(b) Assuming f(x) is a piecewise-smooth and continuous T-periodic function, we have the
Fourier series (in the real form)
f(x)
=ao+
Use (a) to show that for n
2
/
n2u
n2r
cos(——x) + b s1n(Tx)
1
T/2
1
n2ir
f(x)cos(x)c b
=
f/2
c>
2
T/2
2
TIT/
n2rr
f(x) siii(--)4
ir
(1v’
—N
d
-r
5,h
(v)2
k
(
—
(I11+h)27r
_75v1cp
(112
Jc.I)
L
-
LLL (a)]
-r
5,
*
(-)c)
x
C
J
,
I
1’
5
-Th
)ç1
—
>c
jc
H1
C
C-’
—
-
3
t
(I
Li
0
(.
s_f,
\j
-
4-
_L
t\
I
I
1
+
N
—
lH
111
4
PROBLEM 3
(a) Find the complex form of the Fourier series of the given 2n-periodic function f(x) =
if —‘it < x < it. (Hint: you might not need any calculation in part (a).)
(b) Find the complex form of the Fourier series of the given 2ir-periodic function e° if
—n <x <‘it. You may use the fact the (complex) Fourier coefficients are
X
2
C_
T/2
1
Cj
(c) Use (b) to find the complex form-&f—t1ie--Fo.u:ier series of the given 2’ir-periodic function
f(x)=cosh2xif—n<x<n.
()
.4
1
f.
(12
(r 2
c
2:
(1
—°
c’
2-7
r.____
L
—
J
ci
‘
/7
77c
-7Tci
)
227
-f
(Cl
iy
1
F
(—I)’
()
n
e
)
ii
—
h