241_fall_2007_final_exam

You may assume the constant a 2 of the wave equation

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You may assume the constant a 2 of the wave equation is equal to 1. Your final answer may contain an integral. 3

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3. Find any two independent solutions u ( x, y ) to the following PDE: 2 u x y = u Neither of your solutions can be the zero function. 4. Find a and b real numbers such that 10 - 5 i 6 + 2 i = a + ib. 5

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5. Let z 1 = 2 cos( π / 8) + 2 i sin( π / 8) z 2 = 4 cos(3 π / 8) + 4 i sin(3 π / 8) Find a and b real numbers such that z 1 z 2 = a + ib. 6
6. Show the complex function f ( z ) = z is not analytic at z = 0. 7

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7. Find all points z in C satisfying the equation sin z = 2 . Write the solutions in the form a + ib for a and b real numbers. 8
8. Compute the contour integral C z z 2 - π 2 dz where C is the circle | z | = 3. 9

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9. Determine the pole(s) of 5 - 6 /z 2 . Find the order(s) of the pole(s). Compute the residue(s) at the pole(s). 10. Determine the pole(s) of 1 1 - e z . Find the order(s) of the pole(s). Compute the residue(s) at the pole(s). 10
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11. Compute the integral π 0 1 5 + 4 cos θ d θ 12
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12. Let C be the curve in the complex plane parametrized by C ( t ) = cos( t )+ i sin( t ), for 0 t π . (Note the π !) Compute the value of the contour integral C dz z 2 14
13. Consider the function f ( x ) = 0 for 0 x 1 1 for 1 < x 2 defined on the interval [0 , 2]. Let n =1 B n sin n π x 2 be a sine series for f ( x ). Using the same values for B n , for all x in the real line define a function g ( x ) = n =1 B n sin n π x 2 . Find g ( - 5 / 2) and g ( - 5). 15

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14. Solve the Laplace equation u xx + u yy = 0 for a function u ( x, y ) with 0 x 2, 0 y 1 and boundary conditions: u (0 , y ) = 0 , u x (2 , y ) = 0 , u ( x, 0) = 0 , u ( x, 1) = 3 sin π x 4 - 2 sin 5 π x 4 . 16
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