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sfinal

# sfinal - MA 412 Complex Analysis Final Exam Summer II...

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MA 412 Complex Analysis Final Exam Summer II Session, August 9, 2001. 1. Find all the values of ( 8 i ) 1 / 3 . Sketch the solutions. 1 pt. Answer: We start by writing 8 i in polar form and then we’ll compute the cubic root: ( 8 i ) 1 / 3 = (8 e iπ/ 2 ) 1 / 3 = 8 1 / 3 exp ( i ( π 6 + 2 πk 3 ) ) , Hence z 0 = 2 exp( iπ/ 6), z 1 = 2 exp( iπ/ 2) = 2 i , z 2 = 2 exp( i 7 π/ 6). 2. Suppose v is harmonic conjugated to u , and u is harmonic conjugated to v . Show that u and v must be constant functions. 1 pt. Answer By definition, v is harmonic conjugated to u if Δ u = Δ v = 0 and u x = v y , u y = v x hold. On the other hand, as u is harmonic conjugated to v , we also have v x = u y and v y = u x . Then, u y = v x = v x implies v x = 0 and u y = 0 and u x = v y = v y implies v y = 0 and u x = 0 . Hence, u ( x, y ) and v ( x, y ) are constant functions. 3. Show that f ( z ) = | z | 2 is differentiable at the point z 0 = 0 but not at any other point. 1 pt. 1

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z 2 z 0 z 1 2 = 2i Figure 1: Roots of ( 8 i ) 1 / 3 . Answer: We could show that f is not differentiable at any z 0 C 0 by computing different limits for different trajectories. But we’ll use the necessary conditions of differentiability ( i.e. the Cauchy-Riemann equations) to do so. First, we write f ( z ) = u ( x, y )+ iv ( x, y ), so u ( x, y ) = x 2 + y 2 and v ( x, y ) = 0.
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sfinal - MA 412 Complex Analysis Final Exam Summer II...

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