1. Mathematics 2 the First and Second Lectures
Second week
/ 5/ 1438 هـ
أ / سمر السلمي

2. Outline for today Office Hours Syllabus References Chapter One
Grades
Office Hours
Syllabus
References
Chapter One
Fourier Series
Simple harmonic motion and wave motion; periodic functions
Applications of Fourier Series
Average Value of a Function
Fourier Coefficients
Solving examples of Fourier Coefficients

3. Grades 6 % Attendance and participation % 20
% 34
% 40 %
Attendance and participation
Worksheets, Homework & Online quizzes
Periodic Exam ( 2 exams )
Final Exam:

4. Office Hours Time of Periodic Exams
Sunday, Tuesday and Thursday from 11 to 12 p.m.
you can put any paper or homework in my mailbox in Faculty of Physics Department
I will put any announcement or apology in my website ( , so please check it
my is for any question.
every Wednesday a homework will be submit at my mailbox (or if you did not came to university )
every week a worksheet will be submit in class
Time of Periodic Exams
The first periodic exam in / 6 / 1438 h every in her group
The second periodic exam in / 8 / 1438 h every in her group

5. Syllabus Chapter One, Fourier Series & Transforms
Chapter Two: Dirac Delta Function
Chapter Three , Special Function:
Gamma Function , Beta Function & the Error Function
Chapter Four, Series Solutions of Differential Equations:
Legender , Bessel , Hermite & laguerre Functions

6. References Mathematical Methods in the Physical Sciences by Mary Boas
(main book) we will study from it ch 7 , ch15 section )4, 7 ( ch11 , ch 12) , There is a electronic copy in my website in the university
(extra books)
Mathematical Methods for Physicists by Arfken and Weber
Mathematical Physics, World Student Series by Butkov
Advanced Engineering Mathematics by Zill & Cullen
Mathematical Handbook of Formulas and Tables by Schaum’s

7. Lectures will be from
From the book Mathematical Methods in the Physical Sciences By Mary Boas: I will put electronic copy in my website in the university.
we will study from it (ch 7 , ch15 section )4, 7 ( ch11 , ch 12) , Respectively.
From the class: I will write examples & solving problems of homework in board.

8. From the electronic copy of the book Mathematical Methods in the Physical Sciences By Mary Boas
Chapter One : Ch 7, pg. 297
Fourier Series
Simple harmonic motion and wave motion; periodic functions
Section 2, pg
Applications of Fourier Series
Section 3, pg
Average Value of a Function
Section 4, pg
Fourier Coefficients
Section 5, pg 307 – 312

9. Simple harmonic motion and wave motion; periodic functions
Let particle P move at constant speed around a circle of radius A. At the same time, let particle ṕ move up and down along the straight line segment RS in such a way that the y coordinates of P & ṕ are always equal.
Here w is the angular velocity of P in radians per second. θ = wt
Then in the y coordinates y = A sin θ = A sin wt here we can say that particle ṕ move Simple Harmonic Motion R S

10. Z = x + iy = A(cos wt + i sin wt) = A eiwt
Simple harmonic motion and wave motion; periodic functions
If we look at particle P the y coordinates of it y = A sin θ = A sin wt
& the x coordinates of it x = A cos wt
where A is amplitude of the vibration or the function or the wave
& T= 2π /w is the periodic time
If we want look at this particle in complex plane, we write
Z = x + iy = A(cos wt + i sin wt) = A eiwt
also if we want to take about the wave motion, we should remember φ phase difference y = A sin (wt + φ)

11. Simple harmonic motion and wave motion; periodic functions
In the Wave Motion or Wave Properties, we should remember
The frequency w=2π f & the wavelength λ

12. Simple harmonic motion and wave motion; periodic functions
What is the difference here
amplitude A
the angular velocity w
phase difference φ

13. Simple harmonic motion and wave motion; periodic functions
the function f(x) called periodic function with period of T and we write
f ( x ± T) = f(x)
Where T is constant called period of function
Example : sinx
sin (x + 2π) = sin (x)
Where T= 2π
-∏ -∏/2
∏/2 ∏

14. Simple harmonic motion and wave motion; periodic functions
f ( x± T) = f(x) where T period of function
Other examples
cot x , tan x
tan (x + π) = tan (x) ، cot (x + π) = cot (x) where T= π
sin 2π x
sin 2π(x +1) =sin (2πx + 2π) = sin 2πx where T= 1
sin (πx / l)
sin (π/ l)(x +2l) =sin (πx / l + 2π) = sin πx / l where T= 2l
Note:
f ( x± nT) = f(x) where n is integer & T period of function

15. Applications of Fourier Series
Fourier series deals with problems involving vibrations or oscillations or repeating motion.
Examples: a vibrating tuning fork, a pendulum, a weight attached to a spring, water waves, sound waves, etc..
All to these examples are most probably periodic i.e. it repeats itself with time. These motions that repeat themselves regularly with time are also called harmonic motion.
Fourier series is based on the idea that it approximates the function under study to sines and cosines.

16. Applications of Fourier Series
As the tuning fork vibrates it pushes against the air molecules, creating alternately regions of high and low pressure.
If we measure the pressure as the function of x and t from the tuning fork to us, we find the pressure is of the form of
If we measure the pressure where we are as a function of t as the wave passes, we find the pressure is a periodic function of t.
The sound wave is a pure sine wave of a definite frequency. Now suppose that several pure tones are heard simultaneously. In the resultant sound wave, the pressure will not be a single sine function but sum of several sine functions

17. Applications of Fourier Series
There several types or shapes of waves ( Sine , square , Triangle , … elc)

18. Average Value of a Function
For the average value for number example (the average value of homeworks for Mathematical Methods 2 = (HW1 +HW2 + HW3) / 3
In the same way , we can find the average value of a function f(x) on interval from a = x1 to b = xn .
Where is the length of the
interval, When & ,\
the numerator is , then we get

19. Average Value of a Function
The average value of function f(x) on interval from a = x1 to b = x2
Examples: the average value of
1- sin x on interval (-π, π)
(note : the average value of sinx in any periodic interval is zero)
2- sin2nx on interval (-π, π)
(note : the average value of sin2nx & cos2nx in any periodic interval is half)
3- sin mx cos nx on interval (-π, π)
4- x - cos26x on interval ( 0, π/6 ) (Worksheet )

20. Average Value of a Function
We will use the average value for below on interval (-π, π)

21. Fourier Coefficients The general formula for Fourier Series
We want to find the formula for coefficients a0 , an & bn

22. Fourier Coefficients To find the value of coefficient a0
We want all the integrals on the right – hand side are zero except the first term (zero term), so we use the two equations below :
We assume that n=0 و m ≠ 0 & m = 1,2 , 3, …..

23. Continue finding the value of coefficient a0
Fourier Coefficients
Continue finding the value of coefficient a0
all the integrals on the right – hand side are zero except the first term
after we take the average value for all terms on interval (-π, π)

24. To find the value of coefficient an
Fourier Coefficients
To find the value of coefficient an
The same way, we first Multiply all terms with cos nx, then take the average value for all terms on the same interval (-π, π)

25. Fourier Coefficients Continue finding the value of coefficient an
we see that all terms will be zero except the nth term for cos nx if we use the two equations below
the nth term

26. Fourier Coefficients Continue finding the value of coefficient an
all terms in right – hand will be zero except the nth term for cos nx if , then
Thus,
Because we now know the nth term for cos nx , we will know the other values

27. Fourier Coefficients To find the value of coefficient bn
The same way, we first Multiply all terms with sin nx, then take the average value for all terms on the same interval (-π, π)

28. Fourier Coefficients Continue finding the value of coefficient bn
we see that all terms will be zero except the nth term for sin nx if we use the two equations below
the nth term

29. Fourier Coefficients Continue finding the value of coefficient bn
all terms in right – hand will be zero except the nth term for sin nx if , then
Thus,
Because we now know the nth term for sin nx , we will know the other values

30. Solving Fourier Series’ problems
Fourier Coefficients
Solving Fourier Series’ problems
First, we find the value coefficients from 3 eq.
Second, we put them in The general formula for Fourier Series

31. Fourier Coefficients
Expand the periodic function f(x) in a sine-cosine Fourier Series? 1} 2}

32. Next class review
Complex form of Fourier Series
Other Intervals