380_hw_8[1] - & 1 cannot be deformed in D to& 2(b...

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Math 380 Assignment 8 due 4/7/2010 1. Let G be a subset of the complex plane. (a) Following the language used in the text, what properties must G satisfy in order to qualify as a domain ? (b) Assuming G is a domain, must it be simply connected? If not, give an example of a domain G which is not simply connected. (c) Is every simply connected open subset of the plane a domain? If not, give an example. 2. For all parts of this problem, let D = C nf 0 g : (a) Let 1 traverse the unit circle of the complex plane once in the counter-clockwise direction. Let 2 traverse the circle twice in the counter-clockwise direction. Use the Deformation Invariance Theorem (Thm. 8) with the function f ( z ) = 1 =z to prove that
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Unformatted text preview: & 1 cannot be deformed in D to & 2 : (b) Suppose g is analytic on D and R j z j =1 g ( z ) dz = 0 : Must g be analytic at z = 0? If not, give an example. 3. Let f be a complex valued function. (a) What does it mean for f to be a bounded function? (b) What does it mean for f to be an entire function? (c) Does Liouville&s Theorem say that every bounded function is entire? If not, give an example. (d) Find the ±aw in the following "proof" that e & z 2 is constant. Proof. Let g ( z ) = e & z 2 : Then g is an entire function. Also, lim z !1 e & z 2 = 0 ; so g is bounded. By Liouville&s theorem, g must be constant. 1...
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This note was uploaded on 09/16/2011 for the course MATH 380 taught by Professor Staff during the Spring '11 term at S.F. State.

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