Examples XII
1. an
Show
that on -TSes, if f(æ) be even, b, = 0; and that if f(æ) be odd,
=
0.
2. \y Obtain Fourier series corresponding to f(æ) =z on (-T,+r] and show that
=2
and
sin t
+
2
hence deduce that 1
sin 4c
sin 3
sin 2
+
4
3
}
+-÷+=
) Show that Fourier series corresponding to z? on -7 < a< T is
2
3
COS n I
+4
n2
and hence deduce that
1
1
1 + 22 + 32
1
1 +
Híi) Prove that f(z) =t t z²
1
32
3
+2
1
12
( ) If f(æ) =
0,
+
1
12
1
+
52
COS
12
COs 2x
sin z
sin 2
22
COs 3x
+
22
1
1
+
1
42
-T < T < I has the Fourier series
On
72
Deduce that
1
1
22 + 32
32
sin 3
2
1
1
32 +
7r2
for
for
then show that Fourier series
corresponding
cos(2n- 1)r
2
4
n=
(2n - 1)2
+
to f(¢) on -T <I < T is
(-1)" sin nr
CHAP 12: FoURIER SERIES
289
() State Dirichlet's conditions for convergence of a Fourier series. Prove that if
the periodic function (with period 27),
f(æ) =
=
then f(æ) =
Deduce that
-1
for
+1
for
for
--T < T <0
0
<
C= 0
< T,
( Bin z 4 sin 8z 4 sin 6z 4.. ).
=1- + - + . .
(C.H. 1996)
What is the value of the series for z =tT and z = 0?
(1) Prove that the even function f(r) = Ja| on -T <Z <I has a cosine series in
Fourier's form as
COs 3T
4
+
COs 5
52
32
Apply Dirichlet's conditions of convergence to show that the series converges
to |z| throughout - <z<T.
Also show that 1 + + +
{vi) Let f(r)
t
=T.
[C.H. 1994]
0<
VÊVIVI
-f(-æ), -TT< T <0.
Verify that f satisfies Dirichlet's condition on [-7, T]. Obtain the Fourier
series for f in -7, T].
[C.H. 1988)
[Ans.
sin T sin nz]
3. Obtain the Fourier series corresponding to the following functions on (-T, n]:
(i)
f(æ)
0,
when
-T < I <0
7T, when
(Ans.
+ 2(sinæ +
sin 3z +
)]
when-T< T<0
(iü) f(z)
H, when
[Ans. sin z+
(ii) f(z) =
0
(Ans.
fo
sin 5z + ]
- 7 <z <K0
at
0<T < T
+(sin z+
T,
=
0<t<T
sin 3 +
i for
(iV) f(x)
sin 5r +
sin 5x +.))
[C.H. 1985]
0 <z < T
2a, for
0,
sin 3z+
for
for
= sin T, for
(Ans.
- 2 C1
cos 2nz+ sin z]
() f(z) = 0, when -7<I <0
=
[Ans.
49
1, when
+
0 < < 1.
02
sin(2n +1)z]
[C.H. 1993]
INTEGRAL CALCULUS
multiples of t, the function defined by
. Expand in the series of sines md cosinCs of
S(r) =
- T when
< T.
0<
when
T
=
\What is the value of the series for t = tT and a = 0?
Ans. f ( ) = - + { o + cos 4}+4{ sinz +
5.
M Represent
[C.H. 1987, '99)
+}}
integer)
f(r) where f(r) = cos ».r on -1 <KI<T (p not being an
in Fourier series. Deduce that
1
sin pr
(C.H. 1983, '91]
n +p
n tp
n=0
[Ans. cos pr =p
COs
Cos 2.T -
COs T +
3.c
+))
Hence deduce
(iü) Obtain the Fourier series expansion of f(æ) =zsin zon [-7, 7].
that
1
1
1.3
Ans. Tsin a=1
3-5
cos T 2>n=2
+
[C.H. 1998]
5.7
n2-1
(iii) Show that Fourier series of t cos T on -7,+] is
T COS I =
- ,sin z +257
(-1)"nsin n.z
22=2
6.
(i) Slhow that e
n2 - 1
on -7 <r< TTepresents
1
e9
2a
(a cos nr
+
n sin nr) }
12=1
(ii) Slhow that on -7 <I < I,
2 sinh T
COS 1T- n sin
1
-e
n
1 + n2
n=
AI f(r) = {r-lz|}2 on (-7, T], prove that Fourier series of f is given by
f(r) =
n=l
8.
n=1
n2
COS nT.
O1
1
Hence deduce that
3
4
+
n2
6
and>
n=l
n4
(C.H. 1997]
90
Find a Fourier series representing f(æ) on -7<T < T when
f(z) = 0,
0
< < T.
= TI,
and deduce that
1 +
(Ans. f(r) = ö+
1
32
+
1
52
+
1
T3t
n (cos n7
=
1) cos nT - )
[C.H. 1984, '86]
cos nT Sin n1
CHAP 12: FoURIER SERIES
291
(H) follows
Find the: Forier series of the periodic function f with period 2r defined as
f(r) = 0, for
-7<2<0
= I, for
0 <z < 7.
What is the sum of the series at z= -5r? Hence deduce tlhat
1+
[Ans. f(r) =
1
72
1
32
+
52 +
(C.H. 1995)
8
- 2 (9+ co + co + )
+ (sinI - sin?r + sin3z - ) : ]
Hii) Find the Fourier series of f(r) with period 27, where
f(a) = 0, -7 <: < a; f(r) = 1, a< r< b; f(r) = 0, b <I< T.
Find the sum of the series for æ = 47 + uand deduce that
sin n(b-a)
T b+
a
2
9. Expand f(z) in Fourier cosine series on 0<sT, where
(1) f(*)
= 1, 0<< 7.
Deduce that.
(Ans. z =
1
12
-
+
1
1
32
52
+
8
(9+ co
+ co
+..))
52
(ii) f(a) = sina, 0 < T < T.
cos 6
cOs 4T
[Ans. f (æ) = ? {1- 2(cos2 + co
15
(iii)
f() = r ,
0<
= }n(7 - ),
(Ans. f(æ) = 6
&
cos 2x +
0<z <
(iv) f(a) =
+)}}
<
7<
2(
35
cos Gr + ..)]
7
0,
and f(4n) = ,
[Ans. f(æ) =
(v) f(æ) =
0,
f(n) = (cos a
cos 5z +
cos 7a -
cos 1lz + . ..))
0< < T
=;T
= }7,T<t<n
(Ans. f(a) = T- (cos a
(vi) f(r) = 1,
0< < 7
0,
cos 3T +
cos 5z - .)]
7<rs
=-1,
[Ans. f(«) ~
l
(sin
+ sin 2yt) cos nr]
[C.H. 2000]
INTEGRAL CALCULUS
292
10. Show that if cbe aconstant then, on 0<z<T,
1
sin z+
C=
1
sin3z +
3
sin 5z +
11. Show that on 0 <z < T,
sin 3z
Sin 2z +
sin z
12. Show that if f(æ) =z for 0 < elr and f(z) = 1 -1 for
COs 2z
f(a) - - (
cOs 6
COs 10z
32
52
T<T< T, then
i+.).
13. Show that
COS TL
log (2 sin )=
if 0 <KI< T.
n=1
14. Expand f(z) in Fourier sine series on 0<<n where
(i) f(æ) =1, 0 <z< m
[Ans. z = 2(sin z
sin 2x +
sin 3 -
sin 4z + ) )
(ii) f(z) = cos T, 0<I<T
Ans. cos T n B( sin
2sin
4..)1
3 2z +
15 4z + 3sinGz
35
(iii) f(æ) =
T,
= m(7 ),
[Ans. f (z) = sin z
(iv)
f(r) =
0<
< 1
sin 3t +
sin 5z
]
0,
= T,
7<T <T
f(m) =0
and f (G7) =T,
[Ans. f(z) = sinz sin 2r + sin 3r +
0<z < 7
f(r) =
sin 5z
sin 6z +..
T<I<7
0,
7,
<T <
T
and f(0)= f(r) = 0, f(}) =
, f(3n) = -
[Ans. f (r) = sin 2z +
sin 8z + ]
sin 4z +
15. Show that
(i) on T
< T < T,
1
1
7-2
1
sin t-sin 2a + 3 sin 3z
(ii) on 0 < < T,
cos 3
1
+
32
cos 5T
52
-.)
CHAP 12: FoURIER SERIES
293
(iii) on 0<T< T,
1
1
I = sin 2.r + , Sin 4T +
2"
16.
(i) Show that for 0 <
1
3
sin 6z +
K+,
sin 2nT
k
lu2 +
n=1
Verify also that the equation does not hold for z =0and z = 7 and explain
why it does not hold.
(ii) Prove that Fourier cosine series for f(z) =T z on 0 <u<I is given by
cos 5T
COs 3
7T-I=
2
+
12 +
32
Show also that the sum of the series at
17.
1
1
12
32
+
52
+.
=1 is 0. Hence deduce that
52 +
(i) Show that the series
2
(
1
sin t
1
sin 2z +
sin 3z +
3
2
represents (7 -c) on the interval 0 <z< 271.
Find the Fourier expansion for f (æ) which is periodic with period 27 and
(C.H. 1989]
which on 0 <t< 27 is given by f() = ².
[Ans. f(æ) =
? +4 (so+
+ oz +..
-47 (sin + in2 + sin8z + )
Find the sum of the series at t = 47 and hence show that
1
12
+
22
1
+
g2
+
6
(i) Find the Fourier series for f() on 0 <t< 2T, where
f(æ) = (z n), 0< z< 7
Hence deduce the value of
.
18.
f(r) =T , T <z< 27.
and
[C.H. 20o1)
(i) Show that Fourier series corresponding to f(¢) on -2 < I<2 where
f(æ) = 0,
-2 < T <0
= 1,
0 <T <2
is given on (-2, 2) by
f(z) ~5+
(sin
+
1
3
sin
(ii) Show that the series
~|k V
represents l - r when 0
< z<l.
sin
3TT
2
1
+sin 5TT
5
2
)