Name: Dhyey Shukla

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

2. 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

3. Fourier Series
FOURIER SERIES

4. Synthesis
T
nt
b
T
nt
a
a
tf
n
n
n
n
π
+
π
+= ∑∑
∞
=
∞
=
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 ω+ω+= ∑∑
∞
=
∞
=
Let ω0=2π/T.

5. Decomposition
dttf
T
a
Tt
t∫
+
=
0
0
)(
2
0
,2,1cos)(
2
0
0
0
=ω= ∫
+
ntdtntf
T
a
Tt
t
n
,2,1sin)(
2
0
0
0
=ω= ∫
+
ntdtntf
T
b
Tt
t
n
)sin()cos(
2
)( 0
1
0
1
0
tnbtna
a
tf
n
n
n
n ω+ω+= ∑∑
∞
=
∞
=

6. Waveform
Symmetry
Even Functions
Odd Functions
)()( tftf −=
)()( tftf −−=

7. 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

8. 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

15. Half Range
Fourier Series

16. 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.

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

18. (a) Sine Series: (L=2)

19. (b) Cousine series

20. Fourier Cosine & Sine
Integrals

22. ∫
∫
−
∞
∞−
==
==
1
1
)sin()(
1
)(
)cos()(
1
)(
dvwvvfwB
dvwvvfwA
π
π
Where,

23. Thank You
QUESTIONS AND SUGGESTION ARE ACCEPTED