math185f09-hw5

# math185f09-hw5 - MATH 185: COMPLEX ANALYSIS FALL 2009/10...

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MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 5 1. Let a R and z C . (a) Evaluate the following integrals Z 1 0 e it cos atdt and Z 1 - 1 dt t 2 + i . (b) Show that if Re z > - 1, then the integral R 1 0 t z dt exists and ± ± ± ± Z 1 0 t z dt ± ± ± ± 1 1 + Re z . (c) Show that if a < 1, then ± ± ± ± Z 1 - 1 cos it t a dt ± ± ± ± 2 Z 1 - 1 dt t a and thus the (improper) integral R 1 - 1 t - a cos itdt converges absolutely. 2. (a) For k = 1 , 2 , 3, evaluate the following integrals Z Γ k Re( z ) dz, Z Γ k z 2 dz, Z Γ k dz z along the curves from the point z 0 = 1 to z 1 = i in the counter clockwise direction as described in Figure 1. Figure 1. Left : Γ 1 is along the boundary of the square: { x + iy | 0 x 1 , 0 y 1 } . Center : Γ 2 is along the boundary of the circle: { e it | 0 t π/ 2 } . Right : Γ 3 is along the line segment: { (1 - t ) + it | 0 t 1 } . (b) Let a,b > 0. By considering a path along the ellipse { a cos t + ib sin t | 0 t 2 π } or otherwise, show that Z 2 π 0 dt

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## This note was uploaded on 02/17/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.

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math185f09-hw5 - MATH 185: COMPLEX ANALYSIS FALL 2009/10...

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