mt1-sample-ans - Math 185 Sample Answers to First Sample...

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Math 185. Sample Answers to First Sample Midterm 1. a . A function is analytic on an open set if it has a derivative at every point of that set. b . A set is closed if it contains all of its boundary points. c . An -neighborhood of a point z 0 C is the set { z C : | z - z 0 | < } . d . A branch point is a point that is common to all branch cuts of a function. e . z c = exp( c log z ) . It is a set of complex values. 2. See pp. 71–72. 3. We want to solve z 6 = (1 - i ) / 2 = e - πi/ 4 . The solution is given in polar coordinates by { (1 , θ ) : θ = - π/ 24 + 2 πk/ 6 : k = 0 , 1 , 2 , 3 , 4 , 5 } , or by { e : θ = - π/ 24 + 2 πk/ 6 : k = 0 , 1 , 2 , 3 , 4 , 5 } . 4. a . The function z 7→ (1 + i ) z + 1 is a composition of the map w = (1 + i ) z (rotating by π/ 4 and scaling by a factor of 2 ) and the translation w 7→ w + 1 . Hence it maps rectangles to rectangles. The image of S is therefore the rectangle with vertices 1 , 2 + i , 2 - 2 π + i (2 π + 1) , and 1 - 2 π + 2 πi , which are the images of the vertices 0 , 1 , 1 + 2 πi , and 2 πi , respectively.
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