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Subject: Advance Engineering Mathematics (2130002)
Chapter: 02 Fourier Series & Fourier Integral
Department Mechanical Engineering
Name of Subject Teacher
Mr. Dhananjay Chauhan

Team MembersTeam Members
Name Enrollment Number
? Vinay PatelVinay Patel 170990119014
? Dhananjay PatelDhananjay Patel 170990119015
? Dhyey ShuklaDhyey Shukla 170990119016
? Safiuddin SiddiqueSafiuddin Siddique 170990119017
? Aman SinghAman Singh 170990119018

Synthesis
T
nt
b
T
nt
a
a
tf
n
n
n
n
��
+
��
+= �ơ�
��
=
��
=
2
sin
2
cos
2
)(
11
0
DC Part Even Part Odd Part
T is a period of all the above signals
)sin()cos(
2
)( 0
1
0
1
0
tnbtna
a
tf
n
n
n
n ��+��+= �ơ�
��
=
��
=
Let ��0=2��/T.

Decomposition
dttf
T
a
Tt
t��
+
=
0
0
)(
2
0
?,2,1cos)(
2
0
0
0
=��= ��
+
ntdtntf
T
a
Tt
t
n
?,2,1sin)(
2
0
0
0
=��= ��
+
ntdtntf
T
b
Tt
t
n
)sin()cos(
2
)( 0
1
0
1
0
tnbtna
a
tf
n
n
n
n ��+��+= �ơ�
��
=
��
=

Waveform
Symmetry
?Even Functions
?Odd Functions
)()( tftf ?=
)()( tftf ??=

Decomposition
?Any function f(t) can be expressed as
the sum of an even function fe(t) and
an odd function fo(t).
)()()( tftftf oe +=
)]()([)( 2
1
tftftfe ?+=
)]()([)( 2
1
tftftfo ??=
Even Part
Odd Part

Example
?
?
?
<
>
=
?
00
0
)(
t
te
tf
t
Even Part
Odd Part
?
?
?
<
>
=
?
0
0
)(
2
1
2
1
te
te
tf t
t
e
?
?
?
<?
>
=
?
0
0
)(
2
1
2
1
te
te
tf t
t
o

If we are given a function f(x) on an interval [0, L]
and we want to represent f by a Fourier Series we
have two choices - a Cosine Series or a Sine
Series.

Problem-1
EXPAND F(X) = X, 0 < X < 2 IN A HALF-
RANGE (A) SINE SERIES, (B) COSINE
SERIES.

Fourier Cosine & Sine
Integrals

��
��
?
��
��?
==
==
1
1
)sin()(
1
)(
)cos()(
1
)(
dvwvvfwB
dvwvvfwA
��
��
Where,

Thank You
QUESTIONS AND SUGGESTION ARE ACCEPTED

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