Re: Elementary representation of complex delta function

From: Robert Israel (israel_at_math.ubc.ca)
Date: 06/21/04 

Date: 21 Jun 2004 23:35:40 GMT

In article <c5e803c2.0406210938.511b4cae@posting.google.com>,
John Creighton <JohnCreighton_@hotmail.com> wrote:
>danl@wolfram.com (Daniel Lichtblau) wrote in message
>news:<efb635a.0406210629.4eea5089@posting.google.com>...
 
>> Actually the mathematics is going against you on this one.
 
>> You cannot represent a putative "complex delta" in any way as a limit
>> of peaked Gaussian functions because they are entire, and all
>> nonconstant entire functions have poles (polynomials) or essential
>> singularities (as do the Gaussians) at infinity. This renders them
>> useless for your purpose.

>I think I am missing something here. Give me an example of a
>polynomial with a pole.

A pole at infinity, he said. Any nonconstant polynomial has one.

> Clearly the reciprocal of a polynomial does.

Not at infinity.

>Similarly isn't the value of a Gaussian function zero at infinity.

No it isn't. exp(-z^2) -> 0 as z -> infinity along the reals, but not
in the complex plane (try z = it where t is real). As Daniel said,
being an entire function and not a polynomial, exp(-z^2) has an essential
singularity at infinity.

I might add that, by the Maximum Modulus Theorem, no nonconstant entire
function has a "peak" at all, local or global.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2

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