Mathematics 2 the First and Second Lectures Third week 26 - 22/ 5/ 1438هـ أ / سمر السلمي

Outline for today Office Hours first homework due Chapter One Fourier Series Solving examples of Fourier Coefficients Complex form of Fourier Series Solving examples of Complex form of Fourier Series Other Intervals Even and Odd Functions

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 Second Homework I put the second homework in my website in the university at Friday homework Due Wednesday 2 / 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 Fourier Coefficients Section 5, pg 307 – 312 Complex form of Fourier Series Section 7, pg 315 – 317 Other Intervals Section 8, pg 317 - 321

Complex form of Fourier series we can write the general formula for Fourier Series with the exponential form or complex form By replace exponential function with sine –cosine function Thus To find the value of coefficient c0 all the integrals on the right – hand side are zero except the first term (zero term), if we use the equation below :

Complex form of Fourier series Continue finding the value of coefficient c0 all the integrals on the right – hand side are zero except the first term (zero term), then take the average value for all terms on the same interval (-π, π) : Thus thus

Complex form of Fourier series To find the value of coefficient cn If we use the equation below : all terms in right – hand will be zero except the nth term because k = (n-n) =0

Complex form of Fourier series Continue finding the value of coefficient cn 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 einx , we will know the other values The same for zero term

Complex form of Fourier series Solving Fourier Series’ problems First, we find the value complex coefficients from 2 eq. Second, we put them in The general complex formula for Fourier Series

Fourier Coefficients Expand the periodic function f(x) in a complex exponentials from of Fourier Series 1)

Other intervals we can replaced the interval (-π, π) to other intervals (0, 2π) , (-2π, 0) , (π, 3π) , (2π, 4π) , (-π, -3π) . (a , 2π + a) ......etc as long as the Length is 2π

Other intervals We will change the from with different angle and interval from π to l we can replaced the interval (-l, l) to other intervals (0, 2l) , (-2l, 0) , (l, 3l) , (2l, 4l) , (-l, -3l) . (a , 2l + a) ......etc as long as the Length is 2l

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

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

Next class review Even and Odd Functions An Applications to Sound