MT1_Test_fall2018_185.pdf - Math 185 \u2014 Test Midterm 1 Question 1(3 3 4 points 1 Sketch the set of all complex numbers z 6= \u22121 satisfying z i z 1 < 1

MT1_Test_fall2018_185.pdf - Math 185 — Test Midterm 1...

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Math 185 — Test Midterm 1 September 12, 2018 Question 1 (3+3+4 points) . 1. Sketch the set of all complex numbers z 6 = - 1 satisfying z + i z + 1 < 1 . 2. Given z C , find the real and imaginary parts of exp( e z ). 3. Compute the real and imaginary parts of (1 + i ) 10 . Question 2 (5 points each) . 1. Prove or disprove: The principal argument Arg: C \ { 0 } → ( - π, π ] is continuous. 2. Prove that lim z 0 z e ¯ z/z = 0. Question 3 (5 points each) . 1. Let f ( z ) = x 3 y + iy 3 x , z = x + iy . At which points z C is f differentiable? Prove your statements. 2. Suppose f is analytic in a domain U C and f ( z ) R for all z U . Prove that f is constant throughout U . Question 4 (1+1+5+3 points) . Let U C be a domain and suppose f and g
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Unformatted text preview: are analytic functions on U with f ( z ) 6 = 0 for all z ∈ U . Let’s define the multi-valued function ( f ( z ) ) g ( z ) := exp( g ( z ) log f ( z )) . 1. Show at an example (i.e. your choice of f and g ) that in general this function is multi-valued. 2. Show at another example (i.e. again your choice of f and g ) that this function may be single-valued. 3. Now suppose z ∈ C with Arg f ( z ) 6 = π . Construct a neighborhood of z and a branch of ( f ( z ) ) g ( z ) in this neighborhood. 4. Compute the derivative of this branch. 1...
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