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
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.
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
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.
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
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.
Laurent's Series & Types of SingularitiesAakash Singh
Detailed explanation of Laurent's series and various types of singularities like Essential Singularity, Removable Singularity, Poles, Isolated Singularity, etc.
Periodic Function, Dirichlet's Condition, Fourier series, Even & Odd functions, Euler's Formula for Fourier Coefficients, Change of Interval, Fourier series in the intervals (0,2l), (-l,l) , (-pi, pi), (0, 2pi), Half Range Cosine & Sine series Root mean square, Complex Form of Fourier series, Parseval's Identity
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
This presentation contributes towards understanding the periodic function of a Laplace Transform. A sum has been included to relate the method for this topic and a video also so that the learning can be easy.
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
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
Laurent's Series & Types of SingularitiesAakash Singh
Detailed explanation of Laurent's series and various types of singularities like Essential Singularity, Removable Singularity, Poles, Isolated Singularity, etc.
Periodic Function, Dirichlet's Condition, Fourier series, Even & Odd functions, Euler's Formula for Fourier Coefficients, Change of Interval, Fourier series in the intervals (0,2l), (-l,l) , (-pi, pi), (0, 2pi), Half Range Cosine & Sine series Root mean square, Complex Form of Fourier series, Parseval's Identity
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
This presentation contributes towards understanding the periodic function of a Laplace Transform. A sum has been included to relate the method for this topic and a video also so that the learning can be easy.
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
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
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
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Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
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Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
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• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
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Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
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(C) 2024 Robbie E. Sayers
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
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A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
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In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
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3. FOURIER SERIES
Definition of a Fourier
series
A Fourier series may be defined as an expansion of a function in a series
of sine's and cosine’s such as
(1)
0
1
( ) ( cos sin ).
2
n n
n
a
f x a nx b nx
∞
=
= + +∑
The coefficients are related to the periodic function f(x)
by definite integrals in equation 1.
Henceforth we assume f satisfies the following conditions:
(1) f(x) is a periodic function;
(2) f(x) has only a finite number of finite discontinuities;
(3) f(x) has only a finite number of extreme values, maxima and minima in the
interval [0,2π].
Fourier series are named in honour of Joseph Fourier (1768-1830), who made important
contributions to the study of trigonometric series, in connection with the solution of the
heat equation
4. ( ) ( )f t f t T= +
where T is a constant and is called the period of the function.
A function f(x) which satisfies the relation
f(x) = f(x + T) for all real x and some fixed T is called Periodic function. The smallest
positive number T, for which this relation holds, is called the period of f(x).
Any function that satisfies
Periodic Function
5. Euler’s Formula
The Fourier series for the function f(x) in the interval
Is given by -
These values of a0,an&bn are known as Euler’s Formula.
παα 2+<< x
{ }
∫
∫
∫
∑
+
+
+
=
∞
=
=
=
++=
πα
α
πα
α
πα
α
π
π
π
2
2
2
0
1
0
sin)(
1
cos)(
1
)(
1
sincos
2
)(
nxdxxfb
nxdxxfa
dxxfa
nxbnxa
a
xf
n
n
n
nn
6. Problems
Q1.Obtain the Fourier series for f(x)=e-x
in the interval 0<x<2
.
Sol. We know that,
π
{ }∑
∞
=
++=
1
0
sincos
2
)(
n
nn nxbnxa
a
xf
{ }
π
π
π
π
π
x
x
x
n
nn
x
e
ea
ea
nxbnxa
a
e
2
2
00
2
0
0
1
0
1
1
1
sincos
2
−
−
−
∞
=
−
−
=
−=
=
++=
∫
∑
7. ( )
( ) ( ) π
π
ππ
π
π
π
π
π
ππ
π
π
π
2
02
2
0
2
2
2
1
2
2
2
02
2
0
cossin
1
1
sin
1
,....
5
11
&
2
11
1
1
.
1
sincos
)1(
1
cos
1
nxnnxe
n
b
nxdxeb
e
a
e
a
n
e
a
nxnnxe
n
a
nxdxea
x
n
x
n
n
x
n
x
n
−−
+
=
=
−
=
−
=∴
+
−
=
+−
+
=
=
−
−
−−
−
−
−
∫
∫
21. Half Range Series
The Fourier series which contains terms sine or cosine only is
known as half range Fourier sine series or half range Fourier
cosine series.
The function will be defined in range of 0 to but in order to
obtain half range Fourier cosine series or half range Fourier
sine series we extend the range of the function f(x) or
in general (-l,l). So, that the function is either converted in form
of even function of even or odd function.
Case-1 Half range Fourier cosine series:
For the half range Fourier cosine series of the function f(x) in
the range (0,l), we extend the function f(x) over the range (-l,l).
So that the function become even function.
π
),( ππ−
22. ∑
∞
=
+=
1
0 cos
2
)(
n l
xnana
xf
π ∫=
l
dxxf
l
a
0
0 )(
2
∫
=
l
n dx
l
xn
xf
l
a
0
cos)(
2 π
Where,
Case-2 Half range Fourier sine series:
For half range fourier sine series of function f(x),in the
range(0,l), we extend the function f(x) over the range (-l,l); so,
that the function becomes odd function.
∫
∑
=
=
∞
=
l
n
n
n
dx
l
xn
xf
l
b
l
xn
bxf
0
1
sin)(
2
sin)(
π
π
23. Problems
Q1. Find the Fourier cosine series for the function
f(x)=x2
in the range .
Sol.
The given function f(x)=x2
is a even function.
So, we apply case 1
i.e.
π≤≤ x0
∑
∞
=
+=
1
0 cos
2
)(
n l
xnana
xf
π
∫
=
l
n dx
l
xn
xf
l
a
0
cos)(
2 π
[ ]
3
2
3
2
2
1
2
0
0
3
0
0
2
0
2
0
π
π
π
π
π
π
π
π
=
=
=
=
∫
∫−
a
xa
dxxa
dxxa
29. Even and Odd Functions:-Even and Odd Functions:-
Even function:-
1. The function is said to be even function if
2. The function f(x) is said to be Odd function if
30. • The graph of odd function is symmetric about origin.
• The graph of even function is symmetric about Y-axis.
• First we have to check whether the domain of the function is symmetric
about the y-axis.
How To Determine Whether The Function IsHow To Determine Whether The Function Is
Even Or OddEven Or Odd
f(x)=sin x
-3π -5π/2 -2π -3π/2 -π -π/2 π/2 π 3π/2 2π 5π/2 3π
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
y
2
5
2:sin
π
π <<− xx
f(x)=sin x
-3π -5π/2 -2π -3π/2 -π -π/2 π/2 π 3π/2 2π 5π/2 3π
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
y
41. Half Range Series
The Fourier series which contains terms of sine or cosine only
is known as half range Fourier sine series or half range Fourier
cosine series.
The function will be defined in range of 0 to but in order to
obtain half range Fourier cosine series or half range Fourier
sine series we extend the range of the function f(x) or
in general (-l,l). So, that the function is either converted in form
of even function of even or odd function.
Case-1 Half range Fourier cosine series:
For the half range Fourier cosine series of the function f(x) in
the range (0,l), we extend the function f(x) over the range (-l,l).
So that the function become even function.
π
),( ππ−
42. ∑
∞
=
+=
1
0 cos
2
)(
n l
xnana
xf
π ∫=
l
dxxf
l
a
0
0 )(
2
∫
=
l
n dx
l
xn
xf
l
a
0
cos)(
2 π
Where,
Case-2 Half range Fourier sine series:
For half range fourier sine series of function f(x),in the
range(0,l), we extend the function f(x) over the range (-l,l); so,
that the function becomes odd function.
∫
∑
=
=
∞
=
l
n
n
n
dx
l
xn
xf
l
b
l
xn
bxf
0
1
sin)(
2
sin)(
π
π
43. Problems
Q1. Find the Fourier cosine series for the function
f(x)=x2
in the range .
Sol.
The given function f(x)=x2
is a even function.
So, we apply case 1
i.e.
π≤≤ x0
∑
∞
=
+=
1
0 cos
2
)(
n l
xnana
xf
π
3
22
0
0
3
3
2
0
0
22
0
21
0
π
a
πx
π
a
π dxx
π
a
π
π dxx
π
a
=
=
∫=
∫−=
49. Harmonic Analysis
The process of finding the Fourier series corresponding to the function when the
function by numerical values is known as harmonic analysis. The Fourier series for
the function f(x) in the interval is given by-
If the given function is not the explicit function of the independent variable x and
the function is defined by the numerical values then the formula of a0,an and bn are
given by the following relations.
∫
∫
∫
∑
+
+
+
∞
=
=
=
=
+
+=
c
n
c
n
c
n
nn
dx
c
xn
xf
c
b
dx
c
xn
xf
c
a
dxxf
c
a
c
xn
b
c
xn
a
a
xf
2
2
2
0
1
0
sin)(
1
cos)(
1
)(
1
sincos
2
)(
α
α
α
α
α
α
π
π
ππ
Where,
cx 2+<< αα
50. a0 = 2x[Mean value of f(x) in the interval ( )]
an = 2x[Mean value of in the interval( )]
bn = 2x[Mean value of in the interval ( )]
In formula 1 the first term of expansion is known as first or
fundamental harmonic.
The second term is known as second harmonic and the term
is known as third harmonic and so on…..
c2, +αα
⋅
c
xn
xf
π
cos)( c2, +αα
⋅
c
xn
xf
π
sin)( c2, +αα
c
xb
c
xa n ππ sincos1
+
c
xb
c
xa ππ 2sin2cos 22
+
c
xb
c
xa ππ 3sin3cos 33
+
51. Problems
Q1. In a machine the displacement y of a given point is given for a certain angle as
follows.
Find the coefficient of in the Fourier series representing the above variations.
Sol.
The Fourier series for the function y in the given interval 0-360o
or ( ) is given by
θ
0 30 60 90 120 150 180 210 240 270 300 350
7.9 8 7.2 5.6 3.6 1.7 0.5 0.2 0.9 2.5 4.7 6.8
°θ
y
( ).sincos
2 1
0
∑
∞
=
++=
n
nn xnbxna
a
y ππ
π2,0
θ2sin
