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Topic: Sequence of entire functions (Read 3971 times) 

Maria
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Sequence of entire functions
« on: May 15^{th}, 2006, 5:04am » 
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Hi, I need to construct a sequence (h_j) of entire functions with the property that h_j > 1 uniformally on compact subsets of the right half plane, but h_j doesn't converge at any point of the open left half plane. By Runge's theorem I know that such sequence exists, but how do I construct the sequence? I have never constructed sequences of any kind before so I don't know how to approach this problem. Any ideas?


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Icarus
wu::riddles Moderator Uberpuzzler
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Re: Sequence of entire functions
« Reply #1 on: May 15^{th}, 2006, 3:10pm » 
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I have not been familiar with this particular theorem, but the versions I find referenced on the web do not guarantee that the sequence will not converge anywhere on the left halfplane, merely that the rational functions within the sequence will have all their poles there, contained within any set you like. However, Runge's theorem is not really necessary here, and I would have almost guaranteed the existance of such sequences even without it. I suggest you consider the behavior of e^{az} for various real values of a. It's all you need.


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"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? "  Anonymous



Michael Dagg
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Re: Sequence of entire functions
« Reply #2 on: May 17^{th}, 2006, 5:33pm » 
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In addition to Icarus' remarks, notice that the graph of the modulus of the exponential function z > e^(x+iy) \equiv e^x is like a skateboard ramp, of a sort. Let h_j(z) = 1 + e^(j z). If in your question you meant that you want h_j > 1 uniformly on compact subsets of the OPEN righthalf plane, then that should do it. If you like, each h_j can be replaced by a polynomial coming from a Taylor polynomial of the exponential function. If you meant CLOSED righthalf plane, then solutions can be devised from the exponential function or Taylor polynomials which I'll leave to you (using Icarus' remarks and you have it).

« Last Edit: May 17^{th}, 2006, 5:34pm by Michael Dagg » 
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Regards, Michael Dagg



