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[MUG] Fourier Series, Kronecker-Delta ?
| [MUG] Fourier Series, Kronecker-Delta ? |
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Author: Andreas Wolf
Posted: Tue, 19 Nov 2002 12:11:50 +0100
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>> From: Andreas Wolf "Andreas.Wolf"
Hello,
I'm trying to deal with Fourier-Series (instead of Fourier-Integrals
from the inttrans package) there the following problems/questions
arise:
1. Why is there no Kronecker-Delta in Maple? (a symbol delta(n,m)
which is 1 for n=m, 0 otherwise)
Ok, i could just define this. BUT consider the following:
an elementary operation with fourier-series is the following Integral:
int(exp(2*Pi*I*(n-m)*x),x=0..1);
which - as every physicist knows - equates to delta(n,m)
maple cannot (me using maple cannot) compute this.
if i assume(n,integer) and assume(m,integer) maple can do the integral
and gives me ZERO. but it SHOULD be zero for any n,m except n=m
2. if i write any function f(x) as a fourier-series
sum(exp(2*Pi*n*x)*f[n],n=-N..N)
and want to recover f[n] out of that, i have to integrate.
e.g. f[0] = int(f(x),x=0..1);
or f[n] = int(exp(-2*Pi*I*n*x),x=0..1);
BUT HOW can i tell maple to interchange the INT and the SUM.
and how could it ever get back f[n] with the problem decribed in 1.
can anybody help me here?
a.w.
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| [MUG] Re: Fourier Series, Kronecker-Delta ? |
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Author: Maple User Group
Posted: Fri, 22 Nov 2002 16:39:58 -0500
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>> From: Maple User Group "maple_gr"
-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-
Date: Tue, 19 Nov 2002 16:46:28 -0800 (PST)
From: Robert Israel "israel"
To: "maple-list"
Subject: Fourier Series, Kronecker-Delta ?
On Tue, 19 Nov 2002, Andreas Wolf wrote:
|> I'm trying to deal with Fourier-Series (instead of Fourier-Integrals
|> from the inttrans package) there the following problems/questions
|> arise:
|> 1. Why is there no Kronecker-Delta in Maple? (a symbol delta(n,m)
|> which is 1 for n=m, 0 otherwise)
|> Ok, i could just define this. BUT consider the following:
|> an elementary operation with fourier-series is the following Integral:
|> int(exp(2*Pi*I*(n-m)*x),x=0..1);
|> which - as every physicist knows - equates to delta(n,m)
|> maple cannot (me using maple cannot) compute this.
|> if i assume(n,integer) and assume(m,integer) maple can do the integral
|> and gives me ZERO. but it SHOULD be zero for any n,m except n=m
The specialization problem. Basically, a computer algebra system is
working over the field of meromorphic functions of the variables
(including n and m in this case), and the results are true generically
but not necessarily for specific values of those variables. This is
the case even if you assume n and m are integers (which is sometimes
used for simplification). So Maple computes
> int(exp(2*Pi*I*n*x),x=0..1) ;
2
-1/2 I (exp(Pi n I) - 1)
-------------------------
Pi n
which is valid for n <> 0, but not for n=0. If n was assumed to be
an integer, it will then simplify this under that assumption and get 0
(because the numerator simplifies to 0, but the denominator doesn't).
Robert Israel "israel"
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-
From: "Thomas Richard" "t.richard"
To: "maple-list"
Subject: AW: Fourier Series, Kronecker-Delta ?
Date: Wed, 20 Nov 2002 18:18:06 +0100
I think some of these problems can be solved with the worksheet
"Symbolic computation of Fourier series" by Prof. Wilhelm Werner.
Please search for "fourier" in the Maple Application Center,
http://www.mapleapps.com.
--
Mit freundlichen Gruessen / best regards
Thomas Richard Tel.: +49-241-40008-52, Fax: -13
Maple Support "mailto:maple.support"
Scientific Computers GmbH <http://www.scientific.de>
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