On Tuesday 26 May 2009, Luke wrote: > Ok, so there is some sort of bug. > > Here is what Matlab (maple) 2008a gives: > >> int((8*pi^2*f^2*w^2+w^4)/(16*pi^4*f^4+w^4), f, 0, inf) > > ans = > > PIECEWISE([NaN, And(0 < (w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^ > (1/2),(w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2) < 0,(w^4)^ > (1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2) < 0,0 < (w^4)^(1/4)*2^ > (1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2))],[NaN, And(0 < (w^4)^(1/4)*2^ > (1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2),(w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn > (w^2)*w^2)^(1/2) < 0,0 < (w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^ > (1/2))],[NaN, And((w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2) < > 0,(w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2) < 0,0 < (w^4)^ > (1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2))],[NaN, And((w^4)^(1/4)*2^ > (1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2) < 0,0 < (w^4)^(1/4)*2^(1/2)*pi-pi* > (-2*csgn(w^2)*w^2)^(1/2))],[NaN, And(0 < (w^4)^(1/4)*2^(1/2)*pi+pi* > (-2*csgn(w^2)*w^2)^(1/2),(w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^ > (1/2) < 0,(w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2) < 0)], > [NaN, And(0 < (w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2),(w^4)^ > (1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2) < 0)],[NaN, And((w^4)^ > (1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2) < 0,(w^4)^(1/4)*2^(1/2)*pi > +pi*(-2*csgn(w^2)*w^2)^(1/2) < 0)],[NaN, (w^4)^(1/4)*2^(1/2)*pi+pi* > (-2*csgn(w^2)*w^2)^(1/2) < 0],[NaN, And(0 < (w^4)^(1/4)*2^(1/2)*pi+pi* > (-2*csgn(w^2)*w^2)^(1/2),(w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^ > (1/2) < 0,0 < (w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2))], > [NaN, And(0 < (w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2),0 < > (w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2))],[NaN, And((w^4)^ > (1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2) < 0,0 < (w^4)^(1/4)*2^ > (1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2))],[NaN, 0 < (w^4)^(1/4)*2^(1/2) > *pi-pi*(-2*csgn(w^2)*w^2)^(1/2)],[NaN, And(0 < (w^4)^(1/4)*2^(1/2)*pi > +pi*(-2*csgn(w^2)*w^2)^(1/2),(w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2) > *w^2)^(1/2) < 0)],[NaN, 0 < (w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2) > *w^2)^(1/2)],[NaN, (w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2) > < 0],[1/8*2^(1/2)*w^2*(2+csgn(w^2))/(w^4)^(1/4), otherwise]) > > > And here is what Mathematica's web integral gives (with 'x' in place > of 'f'): > Integrate[(8*Pi^2*x^2*w^2 + w^4)/(16*Pi^4*x^4 + w^4), x] == > (w*(-6*ArcTan[1 - (2*Sqrt[2]*Pi*x)/w] + 6*ArcTan[1 + (2*Sqrt[2]*Pi*x)/ > w] + Log[-w^2 + 2*Sqrt[2]*Pi*w*x - 4*Pi^2*x^2] - Log[w^2 + 2*Sqrt[2] > *Pi*w*x + 4*Pi^2*x^2]))/ (8*Sqrt[2]*Pi) > > They don't let you do definite integrals there. And the computation > timed out on Wolfram Alpha. > > Anybody actually *know* what this integral should be? > Yes, it's the noise bandwidth of a 2nd order phase-locked-loop with damping sqrt(2)/2, and the maxima answer matches that in Gardner's book.
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