241Final-S11

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

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Unformatted text preview: 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) 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 ( iπ ) 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 )). 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 ....
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2 I If γ is the ellipse x 2 y 2 4 = 1 traced...

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