This document provides information on Fourier series and Fourier coefficients over various intervals. It defines the Fourier series representation of a periodic function f(x) over a general interval (–l, l) and the interval (–π, π). It also gives the formulas for the Fourier coefficients in the cases when f(x) is even or odd. The document concludes with a table of common Fourier series representations and a list of frequently used formulas for computing Fourier coefficients.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
In this slide fourier series of Engineering Mathematics has been described. one Example is also added for you. Hope this will help you understand fourier series.
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
In this slide fourier series of Engineering Mathematics has been described. one Example is also added for you. Hope this will help you understand fourier series.
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
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Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
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2. MAT2002 - ADDE
Module 1
Page 1
Fourier series over a general interval: Let f (x) be a periodic function, with period 2l
satisfying the following Dirichlet conditions on each interval (–l, l):
(a) )
(x
f has only a finitely many finite discontinuities;
(b) )
(x
f has only at most finite number of maxima or minima.
Then )
(x
f can be represented by the Fourier series
0
1
cos sin ,
2
n n
n
a n x n x
f x a b
l l
where
0
1
,
l
l
a f x dx
l
1
cos
n
l
l
n x
a f x dx
l l
for 1, 2, 3,
n
and
1
sin
l
n
l
n x
b f x dx
l l
for 1, 2, 3,
n .
are called the Fourier coefficients of f.
At a point of finite discontinuity: If f is discontinuous at some interior point c of
(–l, l), then the Fourier series converges to the average
( 0) ( 0)
2
f c f c
of left
hand and the right hand limits of f at c
At the end points: The Fourier series converges to the average
( 0) ( 0)
2
f l f l
of left hand limit of f at the right end point l and the right hand limit of f at the left end
point
The nth harmonic of f is given by
cos sin
n n n
n x n x
H a b
l l
for 1, 2, 3,
n .
Case (a): f (x) is an even function: The Fourier coefficients of f are
0
0
2
,
l
a f x dx
l
0
2
cos
n
l n x
a f x dx
l l
, 0
n
b for 1, 2, 3,
n
Case (b): f (x) is an odd function: The Fourier coefficients of f are
0 0,
a 0,
n
a
0
2
sin
l
n
n x
b f x dx
l l
for 1, 2, 3,
n .
3. MAT2002 - ADDE
Module 1
Page 2
Fourier series over ( , )
: Let f (x) be a periodic function, with period 2 on ( , )
.
The Fourier series of f (x) is
0
1
cos sin
2
n n
n
a
f x a nx b nx
where
0
1
,
a f x dx
1
cos
n
a f x nx dx
for 1, 2, 3,
n
and
1
sin
n
b f x nx dx
for 1, 2, 3,
n
The nth harmonic of f is given by
cos sin
n n n
H a nx b nx
for 1, 2, 3,
n .
Case (a): f (x) is an even function: The Fourier coefficients of f are
0
0
2
,
a f x dx
0
2
cos
n
a f x nx dx
, 0
n
b for 1, 2, 3,
n .
Case (b): f (x) is an odd function: The Fourier coefficients of f are
0 0,
a 0,
n
a
0
2
sin
n
b f x nx dx
for 1, 2, 3,
n .
4. MAT2002 - ADDE
Module 1
Page 1
Table of Fourier Coefficients
Interval Fourier Series of ( )
f x Fourier coefficients
(0, 2l),
l > 0
0
th harmonic
1
cos sin
2
n x n x
l l
n
n n
n
a
a b
π π
=
∞
+ +
∑
( )
0
2
0
1
,
l
a f x dx
l
= ∫
( )
2
0
1
cos ,
n x
n l
l
a f x dx
l
π
= ∫
( )
2
0
1
sin ,
n x
l
l
n
b f x dx
l
π
= ∫
for 1, 2, 3,
n =
(0, 2π) ( )
0
th harmonic
1
cos sin
2
n
n n
n
a
a nx b nx
=
∞
+ +
∑
( )
0
2π
0
1
,
π
a f x dx
= ∫
( )
2π
0
1
cos ,
π
n
a f x nx dx
= ∫
( )
2π
0
1
sin ,
π
n
b f x nxdx
= ∫
for 1, 2, 3,
n =
(–l, l),
l > 0
0
th harmonic
1
cos sin
2
n x n x
l l
n
n n
n
a
a b
π π
=
∞
+ +
∑
( )
0
1
,
l
l
a f x dx
l −
= ∫
( )
1
cos ,
n x
n l
l
l
a f x dx
l
π
−
= ∫
( )
1
sin ,
n x
l
l
n
l
b f x dx
l
π
−
= ∫
for 1, 2, 3,
n =
Even f
(–l, l),
l > 0
( )
0
th harmonic
1
cos /
2
n
n
n
a
a n x l
=
∞
+ π
∑
( )
0
0
2
,
l
a f x dx
l
= ∫
( ) ( )
0
2
cos / ,
n
l
a f x n x l dx
l
= π
∫
0,
n
b = for 1, 2, 3,
n =
Odd f
(–l, l),
l > 0
( )
th harmonic
1
sin /
n
n
n
a n x l
=
∞
π
∑
0 0,
a = 0,
n
a =
( ) ( )
0
2
sin / ,
n
l
b f x n x l dx
l
= π
∫
for 1, 2, 3,
n =
(–π, π) ( )
0
th harmonic
1
cos sin
2
n
n n
n
a
a nx b nx
=
∞
+ +
∑
( )
0
π
π
1
,
π
a f x dx
−
= ∫
( )
π
π
1
cos ,
π
n
a f x nx dx
−
= ∫
( )
π
π
1
sin ,
π
n
b f x nxdx
−
= ∫
for 1, 2, 3,
n =
Even f
(–π, π)
0
th harmonic
1
cos
2
n
n
n
a
a nx
=
∞
+ ∑
( )
0
π
0
2
,
π
a f x dx
= ∫
( ) ( )
π
0
2
cos ,
π
n
a f x nx dx
= ∫
0,
n
b = for 1, 2, 3,
n =
Odd f
(–π, π) th harmonic
1
sin
n
n
n
a nx
=
∞
∑
0 0,
a = 0,
n
a =
( )
0
2
sin ,
n
b f x nx dx
π
=
π ∫ 1, 2, 3,
n =
5. MAT2002 - ADDE
Module 1
Page 2
Formulas frequently used in computing the Fourier coefficients:
1. Leibnitz rule of integration:
a. Version 1 d d
U V UV V U
= −
∫ ∫
:
b. Version 2 1 2 3 4
d ' '' ''' ,
UV x UV U V U V U V
= − + − + ⋅⋅⋅
∫
: where ', '', ''',
U U U ⋅⋅⋅
are the successive derivatives of U, and 1 2 3 4
, , , ,
V V V V ⋅⋅⋅ are the successive
integrals of V
2.
( )
0
2 , if is even
( )
0, if is even
a
f x dx f
f x
f
=
∫
3. cos
sin px
p
px dx = −
∫ , sin
cos px
p
px dx =
∫
4. [ ]
2 2
sin sin cos
Ax
Ax e
A B
e Bx dx A Bx B Bx
+
= −
∫ ,
5. [ ]
2 2
cos cos sin
Ax
Ax e
A B
e Bx dx A Bx B Bx
+
= +
∫
6. Property of Absolute value function:
x x
= − if 0
x < , x x
= if 0
x >
7. sin0 0,cos0 1 cos2
= = = π
8. sin 0,cos ( 1)n
n n
π = π = − for all n
9. (2 1) 1
2
sin ( 1) , 1,2,3,...
k k
k
− π −
=
− =
for all n
10. ( )
2
1, if 1(mod4)
sin
1, if 3(mod4)
n
n
n
π
≡
=
− ≡
11. ( )
2
cos 0
nπ
= for all odd values of n
12. 1
2
sin cos [sin( ) sin( )]
A B A B A B
⋅ = + + −
13. 1
2
cos sin [sin( ) sin( )]
A B A B A B
⋅ = + − −
14. 1
2
cos cos [cos( ) cos( )]
A B A B A B
⋅ = + + −
15. 1
2
sin sin [cos( ) cos( )]
A B A B A B
⋅ = − − −
16. 1
2
sin cos sin 2
A A A
⋅ =
17. 2 2 2 2
cos2 2cos 1 cos sin 1 2sin
A A A A A
= − = − = −