Total 1 2 i if γ is the ellipse x 2 y 2 4 1 traced

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—————————————————————————————— TOTAL: —————————————————————————————— 1
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2 I) If γ is the ellipse x 2 + y 2 4 = 1, traced counterclockwise, evaluate H γ e z z 2 ( z - 2) dz . a) 5 πi 2 b) - 3 πi 2 c) 0 d) 2 πi e) - 4 πi II) If H ( x ) = - 1 for - π x < 0 and H ( x ) = 1 for 0 x < π and if we extend H to be 2 π periodic, then when we expand H ( x ) in a complex Fourier Series k = -∞ c k e ikx , we find the sum c - 2 + c - 1 + c o equals a) 2 πi b) 3 πi c) i π d) 2 i π e) - 1 III) The complex number ( ) i may have many values. One of its val- ues is: a) e - π 2 ( cos ( logπ ) + i sin ( logπ )) b) logπ + i logπ c) e π ( cos ( logπ ) + i sin ( logπ )) d) cos ( logπ ) + i sin ( logπ ) e) e - π ( cos ( e π ) + i sin ( e π )).
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3 IV) For the Sturm-Liouville Problem y 00 + λy = 0 , y (0) = y ( π ) = 0, the eigenvalues are: a) λ = k 2 , k = 1 , 2 , 3 , · · · b) λ = (2 k +1) 2 4 , k = 1 , 2 , 3 , · · · c) λ = k, k = 1 , 2 , 3 , · · · d) λ = k 2 2 , k = 1 , 2 , 3 , · · · e) λ = (2 k + 1) 2 , k = 1 , 2 , 3 , · · · . V) Consider the function f ( x ) = | x | for - π < x π . We extend f to be 2 π periodic and compute its Fourier Series a o 2 + k =1 a k cos ( kx ) + b k sin ( kx ). Then the sum a o + a 1 + b 1 + a 2 + b 2 + a 3 + b 3 is: a) π 2 2 b) 3 π 2 - 10 3 π c) π 2 - 8 π d) 7 π 2 - 22 7 π e) 9 π 2 - 40 9 π . VI) Consider the heat equation u t = u xx with boundary values u (0 , t ) = u (1 , t ) = 0 and with initial condition u ( x, 0) = sin (6 πx ). Compute u ( 1 4 , 1 4 ): a) - exp ( - 5 π 2 ) b) exp ( - 7 π 2 ) c) - exp ( - 9 π 2 ) d) exp ( - 3 π 2 ) e) - exp ( - π 2 )
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4 VII) For the PDE: u x + 3 u y = 0, we are interested only in solutions of the form u ( x, y ) = X ( x ) Y ( y ). For such a solution, suppose we know u (0 , 1) = e and u
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