math185f08-hw4 - MATH 185 COMPLEX ANALYSIS FALL 2008/09...

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MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 4 For z C , recall that the argument of z , denoted arg( z ), is any θ R such that z = | z | e . Note that arg( z ) is only defined modulo 2 π . 1. This problem shows that the conditions in Theorem 2.8 in the lectures cannot be omitted. (a) Let f : C C be defined by f ( z ) = ( exp( - z - 4 ) z 6 = 0 , 0 z = 0 . Show that f satisfies the Cauchy-Riemann equation at all z C but f is not analytic on C . Why doesn’t this contradict the second part of Theorem 2.8 ? (b) Let g : C C be defined by g ( z ) = ( | z | - 4 z 5 z 6 = 0 , 0 z = 0 . Show that g satisfies the Cauchy-Riemann equation at z = 0 but g is not complex differen- tiable at z = 0. Why doesn’t this contradict the second part of Theorem 2.8 ? 2. Let Ω be a region and f : Ω C be analytic. (a) Let γ : [ a, b ] Ω be a smooth curve. If f 0 is non-zero on the image of γ , show that σ : [ a, b ] C defined by σ ( t ) = f ( γ ( t )) is also a smooth curve. (b) Let γ 1 : [ a, b ] Ω and γ 2 : [ a, b ] Ω be two smooth curves such that γ 1 ( t 1 ) = z = γ 2 ( t 2 ) for some t 1 , t 2 [ a, b ] and
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