Mathematics 2 The eleventh and Twelfth Lectures The seventh and eighth weeks 30 - 24/ 6/ 1438 هـ أ / سمر السلمي

Outline for today Office Hours fifth homework due Chapter three : Special functions Beta functions: Beta functions in terms of the gamma Functions Error function Chapter Four :Differential Equations of The Special Functions Legendre's Equation Legendre Polynomials

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 second periodic exam in 11-12-13 / 8 / 1438 h every in her group

The Fifth Homework I put the fifth homework in my website in the university at Friday homework Due Wednesday 15 / 7/ 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 11, pg. 457 Beta functions: Section 6, pg 462 -463 Beta functions in terms of the gamma Functions Section 7, pg 463 -465 Error functions: Section 9, pg 467 -468 Chapter Three: Ch 12, pg. 483 Legendre Equation & Polynomials Section 2, pg 485 - pg 488

Beta Function 1- p > 0 , q > 0 2- prove 3- prove 4- prove

Beta Function Some Relation prove

Beta Function in terms of the gamma Function prove Express the following integrals as Beta functions in terms of gamma functions ? then use gamma function's table in next slide (worksheet)

Error Function You can see it in probability theory in statistical mechanics The error function is the area under part of this curve Definition:

Error Function The other related integrals which often referred to error function a) The Normal or Gaussian distribution function b) The complement error function

Error Function We can use to write erf(x) in terms of Gaussian distribution and complementary error function Some facts about error function prove ( 1 , 2 , 3 ) 1- erf (x) is odd erf (-x) = - erf (x) 2- for large x x > 3 erf ( ∞ ) = 1 3- for small x I x I << 1

Find the value of error function? the Error Function Find the value of error function? erf (0.7) erf (0.003)

the Legender Functions l constant The general solution of the Legendre's differential equation ?

Next class review Legender Functions