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These are questions asked by tsunamiBTP

Rayleigh's identity is listed below:

                 )             2      /      2        1      1  \
                /        |f(k)|  = int||f(t)| , t = - - T .. - T|
               -----                  \               2      2  /
            k = -infinity                                        


sum(abs(f(k))^2, k = -infinity .. infinity) = int(abs(f(t))^2, t = -(1/2)*T .. (1/2)*T);

This identity is an extension from Parseval's theorem for the case where the function of interest is periodic.  The link below provides a worksheet that confirms for a finite series that Rayleigh's identity is valid to within so many significant figures as the frequency parameter, k, increases for CASE 1.  However, for CASE 2 concurrence between the integral and the finite series is not that great.  I suspect I have an error somewhere that is causing the discrepancy.  I thought it might be useful if I get other sets of eyes on this to help isolate the discrepancy.  How I came up with Ck for CASE 2 I can create another worksheet with that derivation if requested.


Appreciate any useful feedback


MAPLE will evaluate the 1st integral below, but not the 2nd.  Is it possible to get MAPLE to do the evaluation?  I know it can be done numerically, but what about analytically?

int(sin(Pi*x*n/T), x);


int(sin(Pi*x*n/T), x)

int(sin(Pi*x*n/T)*cos(Pi*x*n/T)/(sin(Pi*x/T)*cos(Pi*x/T)), x);

int(sin(Pi*x*n/T)*cos(Pi*x*n/T)/(sin(Pi*x/T)*cos(Pi*x/T)), x)

This should be a simple problem, but I am having trouble resolving a contradiction in integral evaluation that is included in the results below.  Instead of butting my head further perhaps someone can point out my probable error.

You will see in both cases I have a triangle signal with a slope of (-1).  However in the 2nd case I have linked the period T to the slope parameter alpha.  If I assign alpha=1 then the results of the 2 integrals should = 1/4, but the 2nd one does not.

Appreciate any feedback.


plot(piecewise(-1 < x and x < 0, 0, 0 < x and x < 1, 1-x), x = -2 .. 2, linestyle = solid, thickness = 5, scaling = constrained, title = "Asymmetric Triangle Wave", titlefont = [COURIER, 18])


n := 0; 1; T := 2; 1; a[0] := (int(0*exp(-I*(2*Pi*n*t/T)), t = -(1/2)*T .. 0)+int((1-t)*exp(-I*(2*Pi*n*t/T)), t = 0 .. (1/2)*T))/T




alpha := 1; T := 2/alpha; plot(piecewise(x < 0, 0, 0 < x and x < (1/2)*T, -alpha*x+1), x = -T .. T, linestyle = solid, thickness = 5, scaling = constrained, title = "Asymmetric Triangle Wave", titlefont = [COURIER, 18])


  n=0 term:n := 0; 1; alpha := 1; 1; T := 2; 1; a[0] := (int(0*exp(-I*(2*Pi*n*t/T)), t = -(1/2)*T .. 0)+int((1-alpha*T)*exp(-I*(2*Pi*n*t/T)), t = 0 .. (1/2)*T))/T





Download integral_contradiction.mw

I am seeking the limit of a series representation of a waveform to within so many significant figures.  Unfortunately, as the series contains more & more terms the demands on computation memory become a problem.  Perhaps there is a method within MAPLE I am not familiar with that can possibly circumvent this problem?

Below is the link to my worksheet.  The presumed theoretical limit is supposed to be 0.0894945 (this is based on what I have read)  (perhaps it may be wrong).  So far I have 3 of the sig figs agree, but would like to take it further.  Is this possible with different MAPLE commands than what I am currently employing?  Currently, MAPLE crashes if I venture beyond 100000 terms in the series.


I was recently cleaning up a worksheet to make things more succinct.  In that process I modified how I expressed the series coefficients, Ck.  What seemed to be an innocous change apparently upset MAPLE to the point it cannot process my results.  In the abbreviated worksheet link below I process the results in the previous manner, Ck1, as opposed to my current modification, Ck2.

In the previous manner the results are generated under 3 minutes.  After modifying the expression for Ck, MAPLE cannot seem to process the results at all.  As far as I can tell the Ck1 & Ck2 concur.  So I am perplexed.  Can anyone see what is wrong?  The only thing I can think of is that sin(2*pi*k/T*x) in the denominator might cause the problem.  However, the sin term is cancelled out by the same sin term in S4.


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