1. Mathematics 2 the Fifth and Sixth Lectures
Fourth week
29/ 5/ 1438هـ
3/ 6/ 1438هـ
أ / سمر السلمي

2. Outline for today Office Hours third homework due Chapter One
Fourier Series
Even and Odd Functions
An Applications to Sound

3. 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

4. The Third Homework
I put the third homework in my website in the university at Friday
homework Due Wednesday 9 / 6 / 1438 هـ in my mailbox in Faculty of Physics Department
I will not accept any homework after that , but if you could not come to university you should sent it to me by in the same day than put the paper next day in my mailbox

5. Chapter One: Ch 7, pg. 297
Fourier Series
Even and Odd Functions
Section 9, pg
An Applications to Sound
Section 10, pg 328 – 331

6. Expand the periodic function f(x) in a sine-cosine Fourier Series or in a complex exponentials ?

7. Examples : any function with even power
Even and Odd Functions
Even function f(x)
f(-x) = f(x)
Examples : any function with even power
From the figure, we notice that it is symmetrically about the y-axis

8. Examples : any function with odd power
Even and Odd Functions
odd functions f(x) :
f(-x) = - f(x)
Examples : any function with odd power
From the figure, we notice that it is symmetrically about the y-axis and x-axis in the same time (symmetry about the point of origin). In fact, it is rotation 180 degree

9. Products the (even and odd) function g(x) :
Even and Odd Functions
some functions are not odd nor even as ex ,but it is the sum of odd and even f(x).
ex = cosh x + sinh x
eix = cos x + i sin x
Products the (even and odd) function g(x) :
even * odd = odd g(-x) = f(-x) * f(-x) = f(x) * - f(x) = - g(x)
g(-x) = f(-x) * f(-x) = f(x) * f(x) = g(x) even * even = even
g(-x) = f(-x) * f(-x) =- f(x) * - f(x) = g(x) odd * odd = even
In the case of difference, it be odd f(x). in the case of similarity, it be even f(x).

10. Even and Odd Functions

11. Even and Odd Functions
Expand the periodic function f(x) in a sine-cosine Fourier Series
after deciding whether the function is even or odd function ?

12. An Applications to Sound
We have said that when a sound wave passes through the air and we hear it, the air pressure where we are varies with time.
Examples: The graph below sketched represent one period of the excess pressure p(t) in a sound wave. find
a) the important harmonics? b) their relative intensities ?

13. Next class review
Fourier Transforms
The Dirac Delta Function