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Title: Sequence of entire functions Post by Maria on May 15th, 2006, 5:04am 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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Title: Re: Sequence of entire functions Post by Icarus on May 15th, 2006, 3:10pm 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 half-plane, 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 eaz for various real values of a. It's all you need. |
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Title: Re: Sequence of entire functions Post by Michael_Dagg on May 17th, 2006, 5:33pm 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 right-half 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 right-half 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). |
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