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sfin_95-100a

# sfin_95-100a - Final exam 1(20 Points Obtain all possible...

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O. Bruno ACM 95a/100a Final Exam December 7, 2009 { Due on Friday December 11 at 5pm . { The honor code is in e±ect. { Exam to be taken in one continuous session of at most four hours. { CLOSED BOOK EXAM. The only materials that may be used for reference during the exam are the problem sets and solution sets posted in the course website, as well as your own graded homework sets and any materials written or typed by your own hand. { Use of computers or calculators is not permitted. { Discussion of the exam problems is not permitted until 5pm on December 11. { The exam consists of four problems. { Use standard blue books. Please, write clearly your name and grading section number on the front of each blue book you use. { Please deposit your completed exam in the Firestone 303 door-slot by 5pm on December 11. Don’t forget to write your name and grading section number on the front of each blue book you use.

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ACM 95a/100a, 2009
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Unformatted text preview: Final exam 1. (20 Points.) Obtain all possible Laurent expansions centered at the origin for the function in part (a). Find all Laurent expansions centered at z = 1 for the function in part (b). Specify the regions of convergence of each one of these series. (a) (10 pts.) z 3 cosh ± 1 z ² ; (b) (10 pts.) z ( z ± 1)( z ± 3) : 2. (30 Points.) Evaluate the integrals (a) (15 pts.) Z ± d± 5 + 2 cos( ± ) ; (b) (15 pts.) Z 1 x 1 = 2 x 2 + 1 dx: 3. (30 Points.) (a) (15 pts.) Show that, for n = 0 ; 1 ; 2 ;::: we have Z 1 ±1 1 (1 + x 2 ) n +1 dx = ² (2 n )! 2 2 n ( n !) 2 : (b) (15 pts.) Show that Z 1 sin 2 x x 2 dx = ² 2 : [Hint: Note that 2 sin 2 x = Re(1 ± e 2 ix ).] 4. (20 Points.) Find the electrostatic potential ³ in the semi-disk f x 2 + y 2 < 1 ;y > g with boundary values ³ = 0 for f y = 0 ; ± 1 < x < 1 g , and ³ = 1 for f x 2 + y 2 = 1 ;y > g . 2...
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sfin_95-100a - Final exam 1(20 Points Obtain all possible...

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