review - P Vojta Math 185 Review Problems Fall 2007 This...

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P. Vojta Math 185: Review Problems Fall 2007 This sheet is adapted from a similar review sheet by M. Christ, which in turn contains some problems from Notes on Complex Function Theory by Donald Sarason. 1. Suppose that n a n z n and n b n z n have radii of convergence R 1 ,R 2 respectively. Suppose that there exists a finite constant M such that | a n | ≤ M | b n | for all but finitely many n . Prove that R 1 R 2 . 2. Evaluate n =1 n 2 z n . 3. Evaluate (a) R [ - i,i ] | z | dz and (b) R C + | z | dz where C + is the right half of the unit circle, oriented counterclockwise. You should get two different answers. Explain why this doesn’t contradict the theorem on page 135. 4. Let a > 0 . Calculate R 0 t 2 t 4 + a 4 dt and R 0 t 2 cos t t 4 + a 4 dt . 5. Show that Z 0 e - t 2 cos( t 2 ) dt = π 4 q 1 + 2 . 6. Let C be a circle bounding an open disk D , and suppose that f is analytic on D . Suppose that z 0 / D . Evaluate R C f ( z ) z - z 0 dz . 7. Show that if f is analytic on the disk of radius R centered at z 0 then for any 0 < r < R , f ( z 0 ) = ( πr 2 ) - 1 RR | z - z 0 | <r f ( z ) dxdy where z = x + iy . 8. Evaluate
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This note was uploaded on 09/01/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.

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review - P Vojta Math 185 Review Problems Fall 2007 This...

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