mid2 - MATH 6321 Second Midterm April 1, 2009 You can use...

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Unformatted text preview: MATH 6321 Second Midterm April 1, 2009 You can use your book and notes. No laptop or wireless devices allowed. Write clearly and try to make your arguments as linear and simple as possible. The complete solution of one exercise will be considered more that two half solutions. All numbers appearing in the test are complex numbers and all functions are from C to C . Exercises 1-3 are on residue computaions, 4-6 are on Laurent expansions and 7-9 are general. Choose and solve one exercise in each group. Name: Question: 1 2 3 4 5 6 7 8 9 Total Points: 10 12 10 10 10 10 8 8 12 90 Score: MATH 6321 Second Midterm April 1, 2009 1. (10 points) Use complex analysis to evaluate I n = 1 2 Z - sin( n ) sin( ) d (1) for every positive n . Solution: If z = e i we have that: sin( n ) = z n- z- n 2 i (2) so that we can write: I n = 1 2 i Z z n- z- n z ( z- z- 1 ) dz = 1 2 i Z z 2 n- 1 z n ( z 2- 1) dz (3) where = { e i | 2 } . Observe that z 2 n- 1 z 2- 1 = n- 1 X k =0 z 2 k (4) Only the term with 2 k = n- 1 in the previous sum gives a non zero contribution when inserted in the integral. Thus we have: I n = ( n even 1 n odd (5) Page 1 of 9 MATH 6321 Second Midterm April 1, 2009 2. (12 points) Compute Z (log( x )) 2 2 + x 2 dx. (6) ( Hint: Write the integral as an integral on a path containing the complex axis.) Solution: we choose for log( z ) the standard branch. Let be the curve: R r R r We have Z (log( z )) 2 2- z 2 dz = 2 i 2 2 (log 2) 2 (7) since there is only one pole in the curve at z = 2. On the other hand we have Z (log(...
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mid2 - MATH 6321 Second Midterm April 1, 2009 You can use...

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