Math_301_Section_201_April_2005

Math_301_Section_201_April_2005 - z and w planes. It might...

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The University of British Columbia Math 301 (201) Final Examination - April 2005 Closed book exam. No notes or calculators allowed. Answer all 5 questions. Time: 2.5 hours. 1. [15] The flow field given by a source located at z = 1 is modified by the introduction of an infinite barrier at x = 0. For what values of y is the speed on the barrier k times the speed at the same location without the barrier? What is the possible range of k ? Explain. 2. [30] (a) Evaluate: I = Z 0 xdx 8 + x 3 . Hint: you should consider using a contour that includes the ray θ = 2 π 3 . (b) By integrating around the finite branch cut [ - 1 , 1] (and using symmetry), evaluate J = Z 1 0 x 4 dx (1 - x 2 ) 1 2 (1 + x 2 ) . (c) Show by considering the two cases x > 0 and x < 0 that p.v. Z -∞ e iωx ω 2 - 1 = - π sin | x | . 3. [20] (a) Find a conformal mapping w = f ( z ) that takes the region {| z - 1 | < 2 } ∩ {| z + 1 | < 2 } into a portion of the right half plane. Draw rough sketches of the regions in both the
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Unformatted text preview: z and w planes. It might be useful to check the image of z = 0. (b) Find ( x, y ) that satises 2 = 0 in {| z-1 | &lt; 2 } {| z + 1 | &lt; 2 } with : = 1 on | z + 1 | = 2 , and = 2 on | z-1 | = 2 . 4. [20] Let f ( x ) and g ( x ) be two absolutely integrable functions. Solve the boundary-value problem using Fourier transform, assuming | u ( x, y ) | decays rapidly as ( x, y ) . u xx + u yy = f ( x ) e-y ,- &lt; x &lt; , &lt; y, u ( x, 0) = g ( x ) ,- &lt; x &lt; . 5. [15] Solve the following ODE using Laplace transform and Bromwich formula: y 000 + y = 1 , ( t &gt; 0); y (0) = y (0) = 0 , y 00 (0) = 1 . Do not replace exponential functions by trigonometric functions in your solution....
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