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Re: Help with finding zeros
Posted:
Sep 14, 2010 3:35 PM
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On Sep 14, 3:21 am, Robert Israel
<isr...@math.MyUniversitysInitials.ca> wrote:
> > I could use a little help trying to solve the following engineering
> > problem (no, I am not a student looking for homework help!)
>
> > I need to find the values of w which cause the following summation to
> > go to zero
>
> > sum (n = 1 to N) of exp(-i*w*ln(n))
>
> > where i is the usual sqrt(-1)
>
> > For a given N, there are infintely many solutions due to the
> > periodicity of exp(-i ...)
>
> Not so simple, because the periods of these terms are incommensurate.
> Nevertheless I think it's true that there are infinitely many solutions.
> By the way, it's easy to show that there are no solutions for
> Im(w) > ln(N-1)/ln(N/(N-1)) or Im(w) < -ln(N-1)/ln(2).
>
> > How do I find these solutions?
>
> For example, in Maple 14:
>
> f:= add(exp(-I*w*ln(n)),n=1..3);
> RootFinding[Analytic](f, w = -50-I .. 50+1.71*I);
>
> 24.8457184705258 + 0.456504696119652 I,
>
> 37.8051239930230 + 0.785954842746225 I,
>
> 49.1771732033222 - 0.610621746068730 I,
>
> 42.0124169890384 - 0.235739713797234 I,
>
> 31.5814052419879 - 0.779819971679015 I,
>
> 13.9743928714389 - 0.733234829565427 I,
>
> 20.9796413441853 + 0.132624219794781 I,
>
> 3.59817149399476 - 0.454397008195022 I,
>
> 8.23070954841730 + 0.940592183315165 I,
>
> -24.8457184705258 + 0.456504696119677 I,
>
> -37.8051239930230 + 0.785954842746195 I,
>
> -49.1771732033222 - 0.610621746068725 I,
>
> -42.0124169890384 - 0.235739713797184 I,
>
> -31.5814052419879 - 0.779819971678945 I,
>
> -13.9743928714389 - 0.733234829565420 I,
>
> -20.9796413441853 + 0.132624219794766 I,
>
> -3.59817149399476 - 0.454397008195023 I,
>
> -8.23070954841730 + 0.940592183315170 I
> --
> Robert Israel isr...@math.MyUniversitysInitials.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada
Thanks for the help!
I was searching for a discrete fourier type of forward and reverse
transform pair, but with the time variable n replaced by ln(n); but I
don't see an easy way to do this.
Bob
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