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Unformatted text preview: /z ) around 0, and give its domain of convergence. (3) (This problem counts double.) Calculate using residue calculus the follow-ing integrals (a) H γ ( e z-1)(1 + z ) /z ( z-2 i ) 2 ( z + 3) cos( πz/ 2) dz for the contour γ as depicted below (on the back side); (b) R ∞ 1 / (1 + x 3 ) dx ; (c) R ∞-∞ x/ sinh( x ) dx (recall sinh( x ) = ( e x-e-x ) / 2); (d) R π-π 1 / ( a + b cos( θ )) dθ for a > | b | and a,b ∈ R (Hint: Unit circle); Finally, calculate the sum ∑ n ∈ Z 1 / ( u + n ) 2 ( u 6∈ Z ) by integrating π cot( πz ) / ( u + z ) 2 over the circle ∂D (0 ,N + 1 / 2) for integer N . 1 2 HOMEWORK 4 FOR MA108C: COMPLEX ANALYSIS DUE DATE: 4PM, APRIL 28, 2009 Figure 1. The contour used in problem 3a...
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This document was uploaded on 12/08/2010.
- Spring '09