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

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

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 (https://uqu.edu.sa/smsolamy) , so please check it my email is smsolamy@uqu.edu.sa for any question. every Wednesday a homework will be submit at my mailbox (or email 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 20- 21 -22 / 6 / 1438 h every in her group The second periodic exam in 11-12-13 / 8 / 1438 h every in her group

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 email in the same day than put the paper next day in my mailbox

Chapter One: Ch 7, pg. 297 Fourier Series Even and Odd Functions Section 9, pg 321 - 328 An Applications to Sound Section 10, pg 328 – 331

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

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

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

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).

Even and Odd Functions

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 ?

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 ?

Next class review Fourier Transforms The Dirac Delta Function