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ADVANCE ENGINEERING MATHEMATICS
BOOK ERN N KTREYSZ G
Exercise (10.2) only (5 16)
Que. 5 f(x) = x , ?? < x < ??
We know that, the fourier series is
f(x) a
? .a
cos
b
sin
/
where a
? f(x)dx
a
? f(x) cosnx dx
b n=
? f(x)sinnxdx
now, a 0 =
? f(x)dx
=
? xdx
= 0 (? f(x) x is odd fun
n
)
a
? f(x) cosnx dx
? xcosnx dx
=
02x .
/
? 1 .
dx
31
= [
0
* sinn sin( n )+
(cosnx)
1]
=
00
*cosn cos( n )+1
=
0
*cosn cosn +1
a n = 0
b n =
? f(x)sinnxdx
b n =
? xsinnxdx
= 0 2
3
?
dx
1
=0
* cosnx cosn +
(sinnx)
1
=
.
/2 cosnx 0
=
( )
( )
b n = 2.
( )
the fourier series is
f(x) a
?(a
cosnx b
sinnx)
= 0 ? 0 2.
( 1 )
sinnx
= 2 ?
( )
sinnx
f(x) = 20
..1
Que: (6) f(x) = x 0 < x < 2??
Here,
a 0 =
? f(x)dx
=
? xdx
=
.
/
=
.
(4
)
a 0 = ??
a n =
? f(x)cosnxdx
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ADVANCE ENGINEERING MATHEMATICS
BOOK ERN N KTREYSZ G
Exercise (10.2) only (5 16)
Que. 5 f(x) = x , ?? < x < ??
We know that, the fourier series is
f(x) a
? .a
cos
b
sin
/
where a
? f(x)dx
a
? f(x) cosnx dx
b n=
? f(x)sinnxdx
now, a 0 =
? f(x)dx
=
? xdx
= 0 (? f(x) x is odd fun
n
)
a
? f(x) cosnx dx
? xcosnx dx
=
02x .
/
? 1 .
dx
31
= [
0
* sinn sin( n )+
(cosnx)
1]
=
00
*cosn cos( n )+1
=
0
*cosn cosn +1
a n = 0
b n =
? f(x)sinnxdx
b n =
? xsinnxdx
= 0 2
3
?
dx
1
=0
* cosnx cosn +
(sinnx)
1
=
.
/2 cosnx 0
=
( )
( )
b n = 2.
( )
the fourier series is
f(x) a
?(a
cosnx b
sinnx)
= 0 ? 0 2.
( 1 )
sinnx
= 2 ?
( )
sinnx
f(x) = 20
..1
Que: (6) f(x) = x 0 < x < 2??
Here,
a 0 =
? f(x)dx
=
? xdx
=
.
/
=
.
(4
)
a 0 = ??
a n =
? f(x)cosnxdx
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=
? xcos nx dx
=
[2
3
? 1 .
dx
]
=
[0
.
/
]
=
.
(cos2n cos0 )
=
(1 1 ) 0
a n = 0
b n =
? f(x)sinnxdx
=
? xsinnx dx
=
[ .
/
? 1 .
dx
]
=
0
(2 cos2n 0 )
(sinnx)
1
=
0
.2 .1 0 1
b n =
the fourier series is
f(x) a
?(a
cosnx b
sinnx)
= ? .0
sinnx/
= 2 ? .
/
f(x) = ?? 2 0
1
Exercise 7, 8, 9 same as it is
Que 10 x |x| , ?? < x < ??
Sol
n
a 0 =
? f(x)dx
=
?
(x |x|)dx
=
[? xdx ?
|x|dx
]
=
[0 2 ? xdx
]
=
[2 2
3
]
a 0 = ?? /2
a n =
? f(x)cosnxdx
=
?
(x |x|)cosnxdx
=
? xcosnx
?
|x|cosnxdx
= 0
.2 ? xcosnxdx *? xcosnxdx 0
by que 5
=
0.
/
? 1 .
dx
1
=
0
(cosnx)
1
=
.
(cosn cos0 )
,( 1 )
1 -
a n =
,( 1 )
1 -
b n =
? f(x)cosnxdx
=
?
(|x| x) sinnx dx
=
[? xsinnx ?
|x|sinnxdx
]
=
0
( 1 )
2 ? xsinnx dx ? xsinnxdx
31
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ADVANCE ENGINEERING MATHEMATICS
BOOK ERN N KTREYSZ G
Exercise (10.2) only (5 16)
Que. 5 f(x) = x , ?? < x < ??
We know that, the fourier series is
f(x) a
? .a
cos
b
sin
/
where a
? f(x)dx
a
? f(x) cosnx dx
b n=
? f(x)sinnxdx
now, a 0 =
? f(x)dx
=
? xdx
= 0 (? f(x) x is odd fun
n
)
a
? f(x) cosnx dx
? xcosnx dx
=
02x .
/
? 1 .
dx
31
= [
0
* sinn sin( n )+
(cosnx)
1]
=
00
*cosn cos( n )+1
=
0
*cosn cosn +1
a n = 0
b n =
? f(x)sinnxdx
b n =
? xsinnxdx
= 0 2
3
?
dx
1
=0
* cosnx cosn +
(sinnx)
1
=
.
/2 cosnx 0
=
( )
( )
b n = 2.
( )
the fourier series is
f(x) a
?(a
cosnx b
sinnx)
= 0 ? 0 2.
( 1 )
sinnx
= 2 ?
( )
sinnx
f(x) = 20
..1
Que: (6) f(x) = x 0 < x < 2??
Here,
a 0 =
? f(x)dx
=
? xdx
=
.
/
=
.
(4
)
a 0 = ??
a n =
? f(x)cosnxdx
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=
? xcos nx dx
=
[2
3
? 1 .
dx
]
=
[0
.
/
]
=
.
(cos2n cos0 )
=
(1 1 ) 0
a n = 0
b n =
? f(x)sinnxdx
=
? xsinnx dx
=
[ .
/
? 1 .
dx
]
=
0
(2 cos2n 0 )
(sinnx)
1
=
0
.2 .1 0 1
b n =
the fourier series is
f(x) a
?(a
cosnx b
sinnx)
= ? .0
sinnx/
= 2 ? .
/
f(x) = ?? 2 0
1
Exercise 7, 8, 9 same as it is
Que 10 x |x| , ?? < x < ??
Sol
n
a 0 =
? f(x)dx
=
?
(x |x|)dx
=
[? xdx ?
|x|dx
]
=
[0 2 ? xdx
]
=
[2 2
3
]
a 0 = ?? /2
a n =
? f(x)cosnxdx
=
?
(x |x|)cosnxdx
=
? xcosnx
?
|x|cosnxdx
= 0
.2 ? xcosnxdx *? xcosnxdx 0
by que 5
=
0.
/
? 1 .
dx
1
=
0
(cosnx)
1
=
.
(cosn cos0 )
,( 1 )
1 -
a n =
,( 1 )
1 -
b n =
? f(x)cosnxdx
=
?
(|x| x) sinnx dx
=
[? xsinnx ?
|x|sinnxdx
]
=
0
( 1 )
2 ? xsinnx dx ? xsinnxdx
31
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=
[
( 1 )
[ {.
/
?
}]]
Procced
b n = 4
( )
the required fourier series is
f(x) =
? 0
*( 1 )
1 +cosnx 2
( )
sinnx31
Que: (11) f(x) = 2
1 if x 0
1 if 0 x
3
Sol
n
: a 0 =
? f(x)dx
0? f(x)dx ? f(x)dx
1
=
0? dx ? dx
1
=
,* x|
*x|
-
=
, - 0
a n =
? f(x)cosnx dx
=
0? cosnx dx ? cosnxdx
1
=
.
/.0 0 0
a n = 0
b n =
? f(x)sinnxdx
=
? sinnxdx ? sinnx dx
=
[
|
(
|
]
=
0
(1 ( 1 )
( 1 )
1 )1
b n =
,1 ( 1 )
-
the fourier series is
f(x) = a 0 +? (a
cosnx b
sinnx)
= 0 + ?
,1 ( 1 )
-sinnx
=
0
0
0
1
f(x) =
0
1
Que (13): f(x) = {
1 if
x
1 if
x
}
Sol
n
a 0 =
0? f(x)dx ? f(x)dx
1
=
0? dx ? dx
1
= 0 (x)
dx (x)
1
=
02
.
/3 2
31
=
, - 0
a n = 0
a n = 0
? f(x)cosnx dx ? f(x)cosnxdx
1
= 0? cosnx dx ? cosnxdx
1
=[.
/
.
/
]
=
0sin
sin.
/ 2sin
sin
31
=
.
03 sin
sin
1
Page 4
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ADVANCE ENGINEERING MATHEMATICS
BOOK ERN N KTREYSZ G
Exercise (10.2) only (5 16)
Que. 5 f(x) = x , ?? < x < ??
We know that, the fourier series is
f(x) a
? .a
cos
b
sin
/
where a
? f(x)dx
a
? f(x) cosnx dx
b n=
? f(x)sinnxdx
now, a 0 =
? f(x)dx
=
? xdx
= 0 (? f(x) x is odd fun
n
)
a
? f(x) cosnx dx
? xcosnx dx
=
02x .
/
? 1 .
dx
31
= [
0
* sinn sin( n )+
(cosnx)
1]
=
00
*cosn cos( n )+1
=
0
*cosn cosn +1
a n = 0
b n =
? f(x)sinnxdx
b n =
? xsinnxdx
= 0 2
3
?
dx
1
=0
* cosnx cosn +
(sinnx)
1
=
.
/2 cosnx 0
=
( )
( )
b n = 2.
( )
the fourier series is
f(x) a
?(a
cosnx b
sinnx)
= 0 ? 0 2.
( 1 )
sinnx
= 2 ?
( )
sinnx
f(x) = 20
..1
Que: (6) f(x) = x 0 < x < 2??
Here,
a 0 =
? f(x)dx
=
? xdx
=
.
/
=
.
(4
)
a 0 = ??
a n =
? f(x)cosnxdx
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=
? xcos nx dx
=
[2
3
? 1 .
dx
]
=
[0
.
/
]
=
.
(cos2n cos0 )
=
(1 1 ) 0
a n = 0
b n =
? f(x)sinnxdx
=
? xsinnx dx
=
[ .
/
? 1 .
dx
]
=
0
(2 cos2n 0 )
(sinnx)
1
=
0
.2 .1 0 1
b n =
the fourier series is
f(x) a
?(a
cosnx b
sinnx)
= ? .0
sinnx/
= 2 ? .
/
f(x) = ?? 2 0
1
Exercise 7, 8, 9 same as it is
Que 10 x |x| , ?? < x < ??
Sol
n
a 0 =
? f(x)dx
=
?
(x |x|)dx
=
[? xdx ?
|x|dx
]
=
[0 2 ? xdx
]
=
[2 2
3
]
a 0 = ?? /2
a n =
? f(x)cosnxdx
=
?
(x |x|)cosnxdx
=
? xcosnx
?
|x|cosnxdx
= 0
.2 ? xcosnxdx *? xcosnxdx 0
by que 5
=
0.
/
? 1 .
dx
1
=
0
(cosnx)
1
=
.
(cosn cos0 )
,( 1 )
1 -
a n =
,( 1 )
1 -
b n =
? f(x)cosnxdx
=
?
(|x| x) sinnx dx
=
[? xsinnx ?
|x|sinnxdx
]
=
0
( 1 )
2 ? xsinnx dx ? xsinnxdx
31
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=
[
( 1 )
[ {.
/
?
}]]
Procced
b n = 4
( )
the required fourier series is
f(x) =
? 0
*( 1 )
1 +cosnx 2
( )
sinnx31
Que: (11) f(x) = 2
1 if x 0
1 if 0 x
3
Sol
n
: a 0 =
? f(x)dx
0? f(x)dx ? f(x)dx
1
=
0? dx ? dx
1
=
,* x|
*x|
-
=
, - 0
a n =
? f(x)cosnx dx
=
0? cosnx dx ? cosnxdx
1
=
.
/.0 0 0
a n = 0
b n =
? f(x)sinnxdx
=
? sinnxdx ? sinnx dx
=
[
|
(
|
]
=
0
(1 ( 1 )
( 1 )
1 )1
b n =
,1 ( 1 )
-
the fourier series is
f(x) = a 0 +? (a
cosnx b
sinnx)
= 0 + ?
,1 ( 1 )
-sinnx
=
0
0
0
1
f(x) =
0
1
Que (13): f(x) = {
1 if
x
1 if
x
}
Sol
n
a 0 =
0? f(x)dx ? f(x)dx
1
=
0? dx ? dx
1
= 0 (x)
dx (x)
1
=
02
.
/3 2
31
=
, - 0
a n = 0
a n = 0
? f(x)cosnx dx ? f(x)cosnxdx
1
= 0? cosnx dx ? cosnxdx
1
=[.
/
.
/
]
=
0sin
sin.
/ 2sin
sin
31
=
.
03 sin
sin
1
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Similarly, b n = 0
the fourier series is
f(x) = a 0+? a
cosnx b
sinnx
f(x) = ? a
cosnx
=
?
03 sin
sin
1cosnx
=
04cosx 4cos
1
Que (15) f(x) = 2
x
0
if 2 x 2
if 2 x 3 2
a 0 =
0? x dx ? 0 dx
1
=
.
/
= 0
a n =
?
xcosnx dx 0
=
[.x
/
?
dx
]
=
0
2
sin
sin.
/3 0 1
=
[? xsinnx dx 0
]
=
[ .
/
?
dx
]
0
(sinnx)
=
0sin
sin.
/1
=
.
.2 .sin
the fourier series is
f(x) =
? .
sin
/sinnx
ex (10.3)
if a fun
c
f(x) of period p =2?? has a fourier series , then
f (x) = a
? a
cos
b
sin
with the fourier coefficient of f(x) given by the Euler formula
a 0 =
? f(x)dx
a n =
? f(x)cos
dx
b n =
? f(x)sin
dx
f(x) = 2
1 if 1 x 0
1 if 0 x 1
, p =2?? 2 ? ?? =1
Sol
n
a 0 =
? f(x)dx
.
? f(x)dx
=
0? dx ? dx
1
=
, (x)
(x)
-
=
, (1 ) 1 - 0
a n =
? f(x)cos
dx
= ? cosn xdx ? cosn dx
= .
/
.
/
= 0
(sin0 sin( n ))1
,sinn sin0 -
= 0
b n =
? f(x)sin
dx ? f(x)sin
dx
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ADVANCE ENGINEERING MATHEMATICS
BOOK ERN N KTREYSZ G
Exercise (10.2) only (5 16)
Que. 5 f(x) = x , ?? < x < ??
We know that, the fourier series is
f(x) a
? .a
cos
b
sin
/
where a
? f(x)dx
a
? f(x) cosnx dx
b n=
? f(x)sinnxdx
now, a 0 =
? f(x)dx
=
? xdx
= 0 (? f(x) x is odd fun
n
)
a
? f(x) cosnx dx
? xcosnx dx
=
02x .
/
? 1 .
dx
31
= [
0
* sinn sin( n )+
(cosnx)
1]
=
00
*cosn cos( n )+1
=
0
*cosn cosn +1
a n = 0
b n =
? f(x)sinnxdx
b n =
? xsinnxdx
= 0 2
3
?
dx
1
=0
* cosnx cosn +
(sinnx)
1
=
.
/2 cosnx 0
=
( )
( )
b n = 2.
( )
the fourier series is
f(x) a
?(a
cosnx b
sinnx)
= 0 ? 0 2.
( 1 )
sinnx
= 2 ?
( )
sinnx
f(x) = 20
..1
Que: (6) f(x) = x 0 < x < 2??
Here,
a 0 =
? f(x)dx
=
? xdx
=
.
/
=
.
(4
)
a 0 = ??
a n =
? f(x)cosnxdx
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=
? xcos nx dx
=
[2
3
? 1 .
dx
]
=
[0
.
/
]
=
.
(cos2n cos0 )
=
(1 1 ) 0
a n = 0
b n =
? f(x)sinnxdx
=
? xsinnx dx
=
[ .
/
? 1 .
dx
]
=
0
(2 cos2n 0 )
(sinnx)
1
=
0
.2 .1 0 1
b n =
the fourier series is
f(x) a
?(a
cosnx b
sinnx)
= ? .0
sinnx/
= 2 ? .
/
f(x) = ?? 2 0
1
Exercise 7, 8, 9 same as it is
Que 10 x |x| , ?? < x < ??
Sol
n
a 0 =
? f(x)dx
=
?
(x |x|)dx
=
[? xdx ?
|x|dx
]
=
[0 2 ? xdx
]
=
[2 2
3
]
a 0 = ?? /2
a n =
? f(x)cosnxdx
=
?
(x |x|)cosnxdx
=
? xcosnx
?
|x|cosnxdx
= 0
.2 ? xcosnxdx *? xcosnxdx 0
by que 5
=
0.
/
? 1 .
dx
1
=
0
(cosnx)
1
=
.
(cosn cos0 )
,( 1 )
1 -
a n =
,( 1 )
1 -
b n =
? f(x)cosnxdx
=
?
(|x| x) sinnx dx
=
[? xsinnx ?
|x|sinnxdx
]
=
0
( 1 )
2 ? xsinnx dx ? xsinnxdx
31
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=
[
( 1 )
[ {.
/
?
}]]
Procced
b n = 4
( )
the required fourier series is
f(x) =
? 0
*( 1 )
1 +cosnx 2
( )
sinnx31
Que: (11) f(x) = 2
1 if x 0
1 if 0 x
3
Sol
n
: a 0 =
? f(x)dx
0? f(x)dx ? f(x)dx
1
=
0? dx ? dx
1
=
,* x|
*x|
-
=
, - 0
a n =
? f(x)cosnx dx
=
0? cosnx dx ? cosnxdx
1
=
.
/.0 0 0
a n = 0
b n =
? f(x)sinnxdx
=
? sinnxdx ? sinnx dx
=
[
|
(
|
]
=
0
(1 ( 1 )
( 1 )
1 )1
b n =
,1 ( 1 )
-
the fourier series is
f(x) = a 0 +? (a
cosnx b
sinnx)
= 0 + ?
,1 ( 1 )
-sinnx
=
0
0
0
1
f(x) =
0
1
Que (13): f(x) = {
1 if
x
1 if
x
}
Sol
n
a 0 =
0? f(x)dx ? f(x)dx
1
=
0? dx ? dx
1
= 0 (x)
dx (x)
1
=
02
.
/3 2
31
=
, - 0
a n = 0
a n = 0
? f(x)cosnx dx ? f(x)cosnxdx
1
= 0? cosnx dx ? cosnxdx
1
=[.
/
.
/
]
=
0sin
sin.
/ 2sin
sin
31
=
.
03 sin
sin
1
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Similarly, b n = 0
the fourier series is
f(x) = a 0+? a
cosnx b
sinnx
f(x) = ? a
cosnx
=
?
03 sin
sin
1cosnx
=
04cosx 4cos
1
Que (15) f(x) = 2
x
0
if 2 x 2
if 2 x 3 2
a 0 =
0? x dx ? 0 dx
1
=
.
/
= 0
a n =
?
xcosnx dx 0
=
[.x
/
?
dx
]
=
0
2
sin
sin.
/3 0 1
=
[? xsinnx dx 0
]
=
[ .
/
?
dx
]
0
(sinnx)
=
0sin
sin.
/1
=
.
.2 .sin
the fourier series is
f(x) =
? .
sin
/sinnx
ex (10.3)
if a fun
c
f(x) of period p =2?? has a fourier series , then
f (x) = a
? a
cos
b
sin
with the fourier coefficient of f(x) given by the Euler formula
a 0 =
? f(x)dx
a n =
? f(x)cos
dx
b n =
? f(x)sin
dx
f(x) = 2
1 if 1 x 0
1 if 0 x 1
, p =2?? 2 ? ?? =1
Sol
n
a 0 =
? f(x)dx
.
? f(x)dx
=
0? dx ? dx
1
=
, (x)
(x)
-
=
, (1 ) 1 - 0
a n =
? f(x)cos
dx
= ? cosn xdx ? cosn dx
= .
/
.
/
= 0
(sin0 sin( n ))1
,sinn sin0 -
= 0
b n =
? f(x)sin
dx ? f(x)sin
dx
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= ? sinn xdx ? sinn xdx
=.
/
.
/
=
,*cos0 cos( n )+ *cosn cos0 +-
=
2,1 ( 1 )
-
the required fourier series is
f(x) = 0 + ? 0
2
( ( )
)
3
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FAQs on Engineering Mathematics - Fourier Series - Topic-wise Tests & Solved Examples for Mathematics
| 1. What is Fourier series and how is it used in engineering mathematics? |  |
Ans. Fourier series is a mathematical tool that allows us to represent periodic functions as a sum of sine and cosine functions. In engineering mathematics, Fourier series is used to analyze and describe periodic phenomena such as vibrations, electrical signals, and sound waves. It helps in understanding the frequency components present in a given signal and is widely used in fields such as signal processing, telecommunications, and control systems.
| 2. What are the applications of Fourier series in engineering? | |
Ans. Fourier series has numerous applications in engineering. Some of the key applications include:
- Signal Analysis: Fourier series is used to analyze and process signals in fields such as telecommunications, audio processing, and image processing. It helps in understanding the frequency content of a signal and extracting useful information.
- Control Systems: Fourier series is used in control systems to analyze and design filters, amplifiers, and other components. It helps in shaping the frequency response of systems and improving their performance.
- Heat Transfer: Fourier series is used to solve heat transfer problems, such as determining temperature distributions in materials or analyzing heat conduction in different geometries. It helps in understanding how heat is transferred in various systems.
- Vibrations and Acoustics: Fourier series is used to analyze and describe vibrations and sound waves. It helps in understanding the harmonic components of vibrations and designing structures to minimize unwanted vibrations and noise.
- Image Compression: Fourier series is used in image compression algorithms such as JPEG. It helps in representing images using fewer coefficients, reducing the storage space required.
| 3. How are Fourier series coefficients calculated? | |
Ans. Fourier series coefficients can be calculated using integration techniques. For a periodic function f(x) with period T, the Fourier series representation is given by:
f(x) = a0 + Σ(an*cos(nω0*x) + bn*sin(nω0*x))
where ω0 = 2π/T and an, bn are the Fourier coefficients. The coefficients can be calculated using the following formulas:
an = (2/T) * ∫[f(x)*cos(nω0*x)] dx
bn = (2/T) * ∫[f(x)*sin(nω0*x)] dx
Integrate the product of the function and the respective trigonometric function over one period, from -T/2 to T/2, and multiply the result by 2/T to obtain the coefficients.
| 4. What are the limitations of Fourier series? | |
Ans. Fourier series has a few limitations that should be considered:
- The function must be periodic: Fourier series can only be applied to functions that are periodic over a finite interval. If the function is not periodic, its Fourier series representation will not accurately capture its behavior.
- Convergence issues: Some functions may not converge to their Fourier series representation, especially if they have discontinuities or sharp changes. In such cases, the Fourier series may exhibit Gibbs phenomenon, where oscillations occur near the discontinuities.
- Limited frequency resolution: Fourier series provides information about the frequency components present in a function, but it does not provide detailed time information. This means that it may not accurately represent functions with rapidly changing behavior.
- Complex calculations: Calculating Fourier series coefficients can be computationally intensive, especially for complex functions or functions with complicated boundary conditions. It may require advanced mathematical techniques or numerical methods for accurate calculations.
| 5. Can Fourier series be used for non-periodic functions? | |
Ans. Fourier series is specifically designed for periodic functions. For non-periodic functions, a related mathematical tool called the Fourier transform is used. The Fourier transform allows us to analyze non-periodic functions and provides a continuous spectrum of frequency components. It is widely used in fields such as signal processing, image processing, and quantum mechanics. However, it should be noted that the Fourier transform requires functions to have finite energy and certain mathematical properties for accurate analysis.
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