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ComplexAnalysis2008Aug

# ComplexAnalysis2008Aug - Qualifying Exam Complex Analysis...

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Unformatted text preview: Qualifying Exam, Complex Analysis, August 2008 i. Let f be an entire function, a E (C and 7“ > la]. Show that 1 (1/2) 2m dz = f(0). werz— Z—i 3+1” 2. Find the image of the ﬁrst quadrant {as > 0, y > 0} under the Mobius map in = 3. Find all the continuous functions 11 : (C —> R which have the property that for every rectangle R C C with sides parallel to the coordinate axes /vd3:=—areaR, /vdy=0, .1 6'}? where 8B is traversed counterclockwise. (Hint: Consider the function f(2) = a: + iv(:r, y), where z = a: + 1y.) 4. Suppose that f(z) =l+clz+0222+... is a holornorphic function on the closed unit disc X such that lf(z)| S M for |zf = 1. ll 20 E A is a zero of f show that 1 > . Vd— 4+1 ...
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