Unformatted text preview: this would be a contradiction. Thus ± 1 and ± 2 are not continuous deformations of each other in D: (b) No. Consider the example g ( z ) = 1 z 2 : In D , g ( z ) = f ( z ) with f ( z ) = & 1 z : Thus R j z j =1 g ( z ) dz = 0 : Note that lim z ! g ( z ) = 1 , so g cannot be analytic at z : 3. Let f : D ! C : (a) f is bounded on D if there exists a constant M such that j f ( z ) j ± M for all z 2 D: (b) f is an entire function if it analytic at every point of C : (c) No. An example of a bounded nonanalytic function is f ( x + iy ) = 1 1+ x 2 : (d) The &aw is in the assertion lim z !1 e & z 2 = 0 : For example, lim y !1 e & ( yi ) 2 = lim y !1 e y 2 = 1 : 1...
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 Spring '11
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 Math, Topology, Trigraph, Deformation Invariance theorem

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