Mathematics 2 the Seventh and Eighth Lectures Fifth week 10 - 6 / 6/ 1438هـ أ / سمر السلمي

Outline for today Office Hours fourth homework due Chapter One Fourier Transforms Chapter Two Dirac Delta Function

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 Fourth Homework I put the fourth homework in my website in the university at Friday homework Due Wednesday 16 / 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 Three: Ch 15, pg. 647 Integral Transforms Fourier Transforms Section 4, pg 647 – 654 (notice: in the 3rd ed it will be ch 7 Section 12) The Dirac Delta Function Section 7, pg 669 (notice: in the 3rd ed it will be ch 8 Section 11) Also, (will take Dirac Delta Function from another book (Introduction to Electrodynamics by David Griffiths & Reed College Ch 1 Section 5.2,) (this part will be as a PDF file in my website in Lectures https://drive.uqu.edu.sa/_/smsolamy/files/Mathematics2%201437%20-%201438%20-%20S2/The%20Dirac%20Delta%20Function.pdf)

Definition of fourier transforms The f(x) and g(α) a pair of fourier transforms g(α) is called the fourier transforms of f(x) f(x) is called the inverse fourier transforms of g(α) Common simply to ca either a fourier transforms of the other

fourier sine transforms fourier cosine transforms Fourier Transforms fourier sine transforms The fs(x) and gs(α) a pair of fourier sine transforms Representing odd function fourier cosine transforms The fc(x) and gc(α) a pair of fourier cosine transforms Representing even function

Fourier Transforms Let f(x) is a nonperiodic function , Find the exponential Fourier transform of f(x) and write f(x)as a Fourier integral ? or the Fourier cosine transform of f(x) and write it as a Fourier integral ?

Fourier Transforms Let f(x) is a nonperiodic function , Find the exponential Fourier transform of f(x) and write f(x)as a Fourier integral ? or the Fourier sine transform of f(x) and write it as a Fourier integral ?

Dirac Delta Function can be pictured as an infinitely high infinitesimally narrow “spike” with area 1 is generalized f(x) or distribution

Dirac Delta Function Example: Evaluate the following integral

Dirac Delta Function Proof of the At x=0 so

Dirac Delta Function Proof of the At x=a so

Dirac Delta Function Properties of delta function 1) if delta function of 2) k ≠ 0 3) 4) a ≠ b

Dirac Delta Function Proof of the y=kx dy = k dx x = y / k limit integer - to for k is positive limit integer to - for k is negative So it is true that

Dirac Delta Function Proof of the u=x du = dx = 0 - So it is true that

Dirac Delta Function Fourier transform of a delta function Inverse transforms

Dirac Delta Function Example: Evaluate the following integral (Worksheet )

Next class review Gamma Function Beta Function