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Maria
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 Sequence of entire functions   « on: May 15th, 2006, 5:04am » Quote Modify

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
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 Re: Sequence of entire functions   « Reply #1 on: May 15th, 2006, 3:10pm » Quote Modify

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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Michael Dagg
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 Re: Sequence of entire functions   « Reply #2 on: May 17th, 2006, 5:33pm » Quote Modify

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).
 « Last Edit: May 17th, 2006, 5:34pm by Michael Dagg » IP Logged

Regards,
Michael Dagg
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