5021-hw1.pdf

# 5021-hw1.pdf - Complex Analysis Fall 2017 Problem Set 1 Due...

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Complex Analysis, Fall 2017 Problem Set 1 Due: September 12 in class 1. Find arg( i - 2 - 2 i ). 2. Find all the points at which f ( z ) = ¯ z 2 + z is complex differentiable. 3. Find the following roots. (i) All the fourth roots of - 1 - 3 i . Express the roots in rectangular coordinates and exhibit them as vertices of a certain square. (ii) The square roots of 8 i . 4. If z 1 , z 2 C , then show that z 1 ¯ z 2 = - 1 if and only if the pre-image of z 1 and z 2 under the stereographic projection correspond to diametrically opposite points on the Riemann sphere. 5. Let U 1 and U 2 be open subset of C and f : U 1 U 2 and g : U 2 C functions which are real differentiable. Prove the following chain rule: ( f g ) ∂z = ∂f ∂z ∂g ∂z + ∂f ¯ z ¯ g ∂z . (If f = u + iv , then ¯ f = u - iv ). Similarly ( f g ) ¯ z = ∂f ∂z ∂g ¯ z + ∂f ¯ z ¯ g ¯ z . (You don’t need to prove the second identity.) 6. Let U C be a connected open subset and

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