MATH 185: COMPLEX ANALYSISFALL 2008/09PROBLEM SET 9Forf: Ω→Candn∈N, recall thatg=fnis the function defined byg(z) = [f(z)]2for allz∈Ω.A function on Ω⊆Cis said to bemeromorphicif it has only removable singularities or poles in Ω(i.e. no essential singularities or non-isolated singularities).1.Let Ω⊆Cbe a region and letf: Ω→C. Supposeg=f2andh=f3are both analytic on Ω.Show thatfis analytic on Ω.2.Is there a polynomialp(z) such thatp(z)e1/zis an entire function?3.Letf:C→Cbe a nonconstant entire function. Prove thatf(C) is dense inC.4.Show that each of the following series define a meromorphic function onCand determine theset of poles and their orders.f(z) =∞Xn=0(-1)nn!(z+n),g(z) =∞Xn=0
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NZ, 1 g, Entire function, nonconstant entire function