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BÄ°LKENT COMPLEX CALCULUS3pdf

BÄ°LKENT COMPLEX CALCULUS3pdf - Date...

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Date: May 18, 2007, Friday NAME: ..................................................... Time: 9:00-11:00 ¨ Ozg¨uler & Sert¨oz STUDENT NO: ..................................................... Math 206 Complex Calculus – Final Exam – Solutions 1 2 3 4 TOTAL 25 25 25 25 100 Please do not write anything inside the above boxes! PLEASE READ: Check that there are 4 questions on your exam booklet. Write your name on the top of every page. Show your work in reasonable detail. A correct answer without proper reasoning may not get any credit. Q-1) Evaluate the integral Z R cot z z 4 dz , where R is the positively oriented boundary of the rectangle whose corners are at the points 2 + 4 i , - 2 + 4 i , - 2 - 4 i and 2 - 4 i . Solution: There is only one pole at z = 0 in this region. The value of the integral is then equal to 2 πi times the residue of cot z z 4 at z = 0. We first find this residue: cot z z 4 = cos z (sin z )( z 4 ) = cos z ( z - z 3 6 + z 5 120 - · · · )( z 4 ) = cos z (1 - z 2 6 + z 4 120 - · · · )( z 5 ) = (1 - z 2 2 + z 4 24 - · · · )(1 + z 2 6 + 7 z 4 360 + · · · ) z 5 = · · · - z 4 45 + · · · 1 z 5 from where we see that the residue is - 1 45 .
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