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Math_301_Section_201_April_2007

Math_301_Section_201_April_2007 - The University of British...

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The University of British Columbia Math 301 (201) Final Examination - April 2007 Closed book exam. No notes or calculators allowed Answer all 5 questions. Time: 2.5 hours 1. [20] The complex potential w ( z ) for a source of strength 2 π located at z = a in a steady inviscid fl ow is w ( z ) = log( z a ) A source of strength 2 π is located at z = 1 + i, and the x axis is a solid barrier. For this fl ow, fi nd: (a) The complex potential of this fl ow w 1 ( z ) in the region Im( z ) > 0 . (b) The velocity components along the x axis. (c) The velocity components at the point x = 0 , y = 2 . A second solid barrier is now introduced along the line x = 0 . (d) What is the complex potential of the fl ow, w 2 ( z ) , in the fi rst quadrant? (e) Find the velocity components at the point x = 0 , y = 2 . 2. [20] Evaluate, carefully explaining all steps, J = Z 0 ( Log x ) 2 1 + x 2 dx Note that R 0 dx 1+ x 2 = π 2 . 3. [20] (a) A function is de fi ned as g ( z ) = z 1 2 (1 z ) 1 2 with a fi nite branch cut for y = 0 , 0 < x < 1 , and g ( 1+ i 2 ) > 0 . Find g ( i ) and g ( 1 i 2 ) . (b) Evaluate, carefully explaining all steps, I = Z 1 0 xdx x 1 2 (1 x ) 1 2 (4 + x ) . 1
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