ps4sol - ACM 95a/100a Problem Set 4 Solutions Prepared by:...

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ACM 95a/100a Problem Set 4 Solutions Prepared by: Zhiyi Li October 26, 2007 Total: 92 points Include grading section: 2 points Problem 1 (10 points) Consider the function f ( z ) = z 1 / 2 = re iθ/ 2 , - π/ 2 < θ 3 π/ 2 and the contour C corresponding to the positively oriented boundary of the half disk 0 r 1, 0 θ π in the upper half plane. Show by separate parametric evaluation of the semi-circle and the two radii constituting the boundary that Z C f ( z ) dz = 0 . Does the Cauchy-Goursat theorem apply here? Solution 1 Let C C 1 + C 2 + C 3 , where C 1 ' z = re : r = 0 to r = 1 , θ = 0 (1) C 2 ' z = re : r = 1 , θ = 0 to θ = π (2) C 3 ' z = re : r = 1 to r = 0 , θ = π (3) For C 2 , dz = ire = ie . For C 1 and C 3 , dz = e , ( θ = 0 or π ). So, Z C f ( z ) dz = Z C 1 f ( z ) dz + Z C 2 f ( z ) dz + Z C 3 f ( z ) dz = Z 1 0 re i 0 2 e i 0 dr + Z π 0 1 e i θ 2 ie + Z 0 1 re i π 2 e dr = 2 3 r 3 2 1 r =0 + 2 3 e i 3 θ 2 π θ =0 + 2 3 r 3 2 e i 3 π 2 0 r =1 = ± 2 3 - 0 + ± - i 2 3 - 2 3 + ± 0 - ± - i 2 3 ¶¶ = 0 (4) Though this integral is zero, Cauchy-Goursat doesn’t apply . The integrand has a singularity (a branch point) on C and therefore isn’t analytic on C . Problem 2 (10 points) Evaluate the following contour integral using anti-derivatives and justify your approach. Z C z i dz = 1 + e - π 2 (1 - i ) 1
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with z i = e i Log z , - π < Arg z π. C joins z 1 = - 1 and z 2 = 1, lying above the real axis except at the end points. (Hint: redefine z i so that it remains unchanged above the real axis and is defined continuously on the real axis.) Solution 2
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ps4sol - ACM 95a/100a Problem Set 4 Solutions Prepared by:...

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