5021-hw4.pdf - Complex Analysis Fall 2017 Problem Set 4 Due...

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Complex Analysis, Fall 2017 Problem Set 4 Due: October 3 in class 1. Find the linear fractional transformation which sends the circle | z | = 2 to the circle | z + 1 | = 1, the point -2 to 0 and the point 0 to i . 2. Compute R γ x dz where γ is the line segment from 0 to 1 + i . 3. Find an open set over which 1 + z + 1 - z is holomorphic. 4. Express arctan in terms of log. What is an open set over which arctan is holomor- phic? Justify your answer. 5. (a) Show that if f is a linear fractional transformation which sends the real axis to the imaginary axis and 1 to , then f sends every circle which passes through 1 and has a real number as it center to a line parallel to the real axis.
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