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.