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[MUG] Re: Winding Numbers and The Fundamental Theorem of Algebra
| [MUG] Re: Winding Numbers and The Fundamental Theorem of Algebra |
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Author: Les Wright
Posted: Mon, 3 Dec 2001 17:33:30 -0500
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>> From: Les Wright "leslie_wright"
Thank you for directing me to the Nyquist worksheet. Most of the
treatment is at my novice level, and I find it helpful.
But I have found an error, or is it a bug?
The following should give 1, since it is the winding number of the unit circle
z=e^it about the point (1 + i)/2, which is clearly inside the circle. However:
> q1 := (1/2/Pi/I)*Int(diff(Z,t)/(Z-(1+I)/2),t=0..2*Pi);
2 Pi
/
| I exp(I t)
I | ---------------------- dt
| exp(I t) - 1/2 - 1/2 I
/
0
q1 := - 1/2 -----------------------------------
Pi
> value(q1);
0
Yet, when I subject the integrand first to evalc(), we get the right answer:
> q1 := (1/2/Pi/I)*Int(evalc(diff(Z,t)/(Z-(1+I)/2)),t=0..2*Pi);
2 Pi
/
| sin(t) (cos(t) - 1/2) cos(t) (sin(t) - 1/2)
q1 := - 1/2 I | - --------------------- + ---------------------
| %1 %1
/
0
/cos(t) (cos(t) - 1/2) sin(t) (sin(t) - 1/2)\
+ I |--------------------- + ---------------------| dt/Pi
\ %1 %1 /
2 2
%1 := (cos(t) - 1/2) + (sin(t) - 1/2)
> value(q1);
1
I am using MVR4, that oldie but goodie, and the worksheet was clearly
written more recently than that, so maybe the newer versions get it
right?
I do find this disconcerting! If I didn't know the right answer in
advance, I would have trusted Maple's original answer.
Thanks,
Les
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