I made this presentation for my own college assignment and i had referred contents from websites and other presentations and made it presentable and reasonable hope you will like it!!!
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
I made this presentation for my own college assignment and i had referred contents from websites and other presentations and made it presentable and reasonable hope you will like it!!!
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
Mathematical description of Legendre Functions.
Presentation at Undergraduate in Science (math, physics, engineering) level.
Please send any comments or suggestions to improve to solo.hermelin@gmail.com.
More presentations can be found on my website at http://www.solohermelin.com.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
Mathematical description of Legendre Functions.
Presentation at Undergraduate in Science (math, physics, engineering) level.
Please send any comments or suggestions to improve to solo.hermelin@gmail.com.
More presentations can be found on my website at http://www.solohermelin.com.
Successive Differentiation is the process of differentiating a given function successively times and the results of such differentiation are called successive derivatives. The higher order differential coefficients are of utmost importance in scientific and engineering applications.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
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.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
4. 4
Overview
• What is Fourier series?
• What is periodic function?
• What are integration rules?
• How to solve Fourier series with given periodic function.
Through examples.
• Applications of Fourier series.
5. “
5
Periodic Functions
A function f: R →R is said to be periodic if there exists a positive
number T such that f(x+T) = f(x) for all x ϵ R. T is called the period
of f(x).
Examples:
i. Trigonometric functions sin x, cos x, cosec x are periodic
functions with primitive period 2π.
ii. Sin 2x, cos 2x are periodic functions with primitive period π.
iii. Tan x has primitive period π.
6. Fourier Series
Then, f(x)=
6
Let f(x) be a periodic function with period 2π. Then we can obtain the
Fourier series in any interval of length 2 π.
0
1 1
cos sin
2
n n
n n
a
a nx b nx
,
0
1
( )
a f x dx
1
( )cos
n
a f x nxdx
1
( )sin
n
b f x xdx
The formulae for the coefficients a0, an, bn are known as Euler’s formulae.
7. 7
Integration Properties
' '' '''
1 2 3 4 ......
uvdx uv uv u v u v
(suffixes(v) represent successive integrals and dashes(u’) represent
successive derivatives)
2 2
sin [ sin cos ]
ax
ax e
e bxdx a bx b bx
a b
2 2
cos [ cos sin ]
ax
ax e
e bxdx a bx b bx
a b
8. 8
Example# 01
Obtain a Fourier series to represent f(x)=x2 in (0, 2𝛑) and hence deduce that
2
2 2 2 2
1 1 1 1
........
1 2 3 4 12
2
2 0
1 1
2
2
0
0
2
3
0
2
2
2
2
0
0
2
2
Let ( )
cos sin
2
be the Fourier series of ( ) in (0,2 ).
1
then
1
3
8
3
4 1
= cos
2 3
1 sin cos
-2x
n
n n
n
f x x
a
x a nx bn nx
f x
a x dx
x
a
a x nxdx
nx nx
x
n n
2
3
0
cos
+2
(using generalised rule of integeration by parts)
nx
n
9. 9
2
3 3
1 4 2 2
4
n n n
n
2 2 2
2
2
0
2
2
2 3
0
sin 2 sin0 0
1 4 4
= =
cos2 n=(-1) 1
1
sin
1 cos sin cos
2 2
n
n
n
n n
b x nxdx
nx nx nx
x x
n n n
10. 10
0
2
2
2
1 1
2
2 2 2
Substituting the values of a , a , b in (1), we get,
4 4 4
cos sin
3
4 cos cos2 cos3 sin sin 2 sin3
= +4 ..... -4 ......
3 1 2 3 1 2 3
n n
n n
x nx nx
n n
x x x x x x
2
......(2)
is the Fourier expansion of f(x) = x in (0, 2 )
Take x = in (2), so that,
2
2
2 2 2 2
2
2 2 2 2
2
2 2 2 2
4 1 1 1 1
= +4 .....
3 1 2 3 4
1 1 1 1
= -4 .....
3 1 2 3 4
1 1 1 1
.....
12 1 2 3 4
14. Exercise # 9b
14
•Obtain the Fourier series expansions for the following function in the
Intervals given against them.
( ) ( ) in (0, 2 )
x
a f x e
0
1 1
Let cos sin ...(i)
2
x
n n
n n
a
e a nx b nx
2
2
0 0
0
1 1
[ ]
x
x
a e
e dx
2
2
0
1
[ 1]
1
e
e
a
15. 15
2
0
2
2
0
2 0
2
2
2
2
2
1
cos
1
cos sin here a=1 and b=n
1
1
(cos2 sin2 ) (cos0 sin0 )
( 1)
sin2 sin0 0
1
[ (1 0) 1(1 0)]
cos2 ( 1) 1
( 1)
1
[
( 1)
x
n
x
n
e nxdx
a
e
nx n nx
n
e n n n e n n n
n
n
e
n
n
n
2
1]
e
2
2
1
( 1)
n
e
a
n
16. 16
2
0
2
2
0
2 0
2
2
2
2
2
2
2
2
2
1
sin
1
sin cos here a=1 and b=n
1
1
[ sin(2 ) cos(2 ) sin0 cos0 ]
( 1)
1
[ 0 (1) 1 0 (1) ]
( 1)
1
[ 1 ]
( 1)
1
[ ]
( 1)
( 1
x
n
x
n
e nxdx
b
e
nx n nx
n
e n n n e n
n
e n n
n
e n n
n
n ne
n
n ne
b
n
)
17. 17
0
a , a , b
n n
Putting values of in equation (i)
2 2 2
2 2
1 1
2 2 2
2 2
1 1
2
2 2
1 1
1 1
cos sin
2 ( 1) ( 1)
1 1 ( 1)
= cos sin
2 ( 1) ( 1)
1 1 cos sin
2 ( 1) ( 1)
x
n n
n n
n n
e e n ne
e nx nx
n n
e e n e
nx nx
n n
e nx nx
n
n n
28. Applications
28
• A Fourier Series has many applications in mathematical analysis
as it is defined as the sum of multiple sines and cosines.
• Thus, it can be easily differentiated and integrated, which
usually analyses the functions such as saw waves which are
periodic signals in experimentation.
• It also provides an analytical approach to solve the
discontinuity problem. In calculus, this helps in solving complex
differential equations.