241Final-S11

# 1 0 e 2 compute u 1 1 a exp 1 4 b exp 3 4 c exp 5 4 d

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(1 , 0) = e 2 . Compute u (1 , 1): a) exp ( 1 4 ) b) exp ( 3 4 ) c) exp ( 5 4 ) d) exp ( 7 4 ) e) exp ( 9 4 ) VIII) A certain function u ( x, y ) is defined on the square whose vertices are (0,0), (1,0), (0,1) and (1,1). The function u satisfies the two PDE’s: u xx = 0 and u yy = 0 and has values at the corners of the square given by u (0 , 0) = 0 , u (1 , 0) = 3 , u (0 , 1) = - 2 , u (1 , 1) = 1. For this function u , the value u ( 1 2 , 1 2 ) is: a) 1 b) -2 c) 0 d) -3/2 e) 1/2

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5 IX) We expand the function f ( z ) = e z ( z - 1) 2 in a Laurent Series valid in the annulus 1 < | ( z - 1) | < 2. This series has the form m =1 b m 1 ( z - 1) m + k = o a k ( z - 1) k . Then the value of the product b 2 b 1 a o a 1 a 2 is: a) e 5 b) e 5 / 288 c) e 5 / 120 d) e 5 / 84 e) e 5 / 316. X) For the function f ( z ) = 2 z - 1 z ( z - 5) 2 , find its residue at z = 5. a) 0 b) i/2 c) 2/5 d) π /4 e) 1/25 THE NEXT 5 PROBLEMS ARE LONG ANSWER–WORK IS REQUIRED TO BE SHOWN.
6 XI) Compute the integral R -∞ sin 2 ( x ) 1+ x 2 dx .

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7 XII) If u ( x, y ) is a harmonic function (that is, it satisfies Laplace’s DE: Δ u = 0) defined on the whole plane, then it is known that there is another harmonic function, call it v ( x, y ), so that the complex function f ( z ) = f ( x, y ) = u ( x, y ) + i v ( x, y ) is an entire function (i.e., is holomorphic (= analytic) on the entire complex plane). Write g ( z ) for the function g ( z ) = e - f ( z ) . a) If u ( x, y ) is always 0, show, with an explicit upper bound, that | g ( z ) | is bounded. b) Again assume u ( x, y ) 0 for all x, y . Use a) to find all such (every- where non-negative harmonic) functions—be careful, clear and explicit in your reasoning.
8 XIII) Consider the wave equation u tt = 4 u xx on the interval [0 , π ] with boundary values u (0 , t ) = u ( π, t ) = 0. If the initial conditions are u ( x, 0) = 1 3 sin (3 x ) + 1 4 sin (4 x ) u t ( x, 0) = 1 5 sin (5 x ) + 1 6 sin (6 x ), find (explicitly) the solution, u ( x, t ), of the problem.

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