4150-bonus problems

4150-bonus problems - Math 4150/6150 Bonus Problems —...

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Unformatted text preview: Math 4150/6150: Bonus Problems — Spring 2011 Instructor: Dr. Shuzhou Wang Each problem is worth an extra 1% of the course. Note : If you turn in solutions of these problems for credit, you must work independently and must not discuss them with anyone else except me. Your work on the bonus problems will not count if your homework average is below 70%. Part I: Due 3/4/2011 1. Let z k ( k = 1 , ··· ,n ) be complex numbers lying on the same side of a straight line passing through the origin . Show that z 1 + z 2 + ··· + z n 6 = 0 , 1 /z 1 + 1 /z 2 + ··· + 1 /z n 6 = 0 . 2. Use n-th roots of unity (i.e. solutions of z n- 1 = 0) to show that 2 n- 1 sin π n sin 2 π n ··· sin ( n- 1) π n = n 3. Let f ( z ) = z + 1 /z . Describe the images of both the circle | z | = r of radius r ( r 6 = 0) and the ray arg z = θ under f in terms of well known curves. 4. Show that f ( z ) = z 2 ¯ z is not analytic anywhere....
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