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[MUG] Winding Numbers and The Fundamental Theorem of Algebra

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[MUG] Winding Numbers and The Fundamental Theorem of Algebra
Author: Les Wright    Posted: Tue, 27 Nov 2001 17:47:13 -0500
>> From: Les Wright "lcwright" Hi there, group-- Anyone out there have or know about any Maple worksheet that graphically demonstrates the concept of winding number in the complex plane? Thanks in advance, Les

[MUG] Re: Winding Numbers and The Fundamental Theorem of Algebra
Author: Maple User Group    Posted: Thu, 29 Nov 2001 09:52:11 -0500
>> From: Maple User Group "maple_gr" | >> From: Les Wright "lcwright" | Anyone out there have or know about any Maple worksheet that graphically | demonstrates the concept of winding number in the complex plane? -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: "Thomas Richard" "t.richard" To: "maple-list" Subject: AW: Winding Numbers and The Fundamental Theorem of Algebra Date: Thu, 29 Nov 2001 10:43:32 +0100 The "Advanced Engineering Mathematics" Powertool seems to contain one; see http://www.mapleapps.com/powertools/engineeringmath/engineeringmath.shtml "views samples" link; see section 36.5: The Nyquist Stability Criterion. I found this by searching for "winding" in the Maple Application Center (http://www.mapleapps.com). -- Mit freundlichen Gruessen / best regards Thomas Richard Tel.: +49-241-40008-52, Fax: -13 Maple Support "mailto:maple.support" Scientific Computers GmbH <http://www.scientific.de> -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: Jason Schattman "jschattman" To: "''" Subject: Winding Numbers and The Fundamental Theorem of Algebra Date: Thu, 29 Nov 2001 09:14:33 -0500 Hi Les, If you go to the Maple Application Center at www.mapleapps.com and enter the search keyword "winding" in the search box, you will see the worksheet "Nyquist Stability Criterion", which demonstrates the concept of winding number quite thoroughly. For a complete set of Maple lessons on Complex Analysis in general, see http://www.mapleapps.com/powertools/complex/complex.shtml. Best regards, Jason Schattman Manager, Applications Marketing Waterloo Maple Inc. http://www.mapleapps.com

[MUG] Re: Winding Numbers and The Fundamental Theorem of Algebra
Author: Les Wright    Posted: Mon, 3 Dec 2001 17:33:30 -0500
>> 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

[MUG] Re: Winding Numbers and The Fundamental Theorem of Algebra
Author: Maple User Group    Posted: Mon, 10 Dec 2001 13:58:55 -0500
>> From: Maple User Group "maple_gr" | >> From: Les Wright "leslie_wright" | 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); | > value(q1); | 0 |... | 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. -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Tue, 4 Dec 2001 12:07:59 -0800 (PST) From: Robert Israel "israel" To: "maple-list" Subject: Winding Numbers and The Fundamental Theorem of Algebra I assume Z is exp(I*t). In both Maple 6 and Maple 7, the correct result 1 is returned. Definite integrals involving multivalued functions, such as the antiderivative ln(exp(I*t)-1/2-1/2*I) in this case, are notoriously difficult for Maple to get right, especially in earlier releases. The problem of deciding whether the path of integration crosses a branch cut of the antiderivative is in general very difficult. There have been significant improvements in this area, but caution is still advisable. Robert Israel "israel" Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Tue, 04 Dec 2001 16:14:54 -0500 From: "Robert J. Lopez" "r.lopez" To: "maple-list" Subject: Winding Numbers and The Fundamental Theorem of Algebra Les, You are running a Maple 7 worksheet in Maple V Release 4. The worksheet was originally written with the older version of Maple, and indeed, many such integrals were incorrectly evaluated by Maple. Often, but not always, applying evalc to the integrand would then provide the correct result. In Maple 7, the calculation is indeed correctly executed, and the Maple 7 version of the worksheet reflects this improvement. However, there are related calculations in "nearby" worksheets that yield similar wrong results, even in Maple 7. Robert -- Robert J. Lopez Department of Mathematics Rose-Hulman Institute of Technology Terre Haute, IN 47803 812 877-8396 (office) 812 877-8883 (Department fax) "r.lopez" -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: "Thomas Richard" "t.richard" To: "maple-list" Subject: Winding Numbers and The Fundamental Theorem of Algebra Date: Wed, 5 Dec 2001 09:41:45 +0100 I assume you wrote Z:=exp(I*t) before. Yes, it was a bug, and it has been fixed in Maple 6. -- Mit freundlichen Gruessen / best regards Thomas Richard Tel.: +49-241-40008-52, Fax: -13 Maple Support "mailto:maple.support" Scientific Computers GmbH <http://www.scientific.de> -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Wed, 05 Dec 2001 11:00:04 +0100 From: Preben Alsholm "ifakpa" To: "maple-list" Subject: Winding Numbers and The Fundamental Theorem of Algebra Releases 6 and 7 get it right, 5.1 does not. You should never trust the value suggested by the exact integrator until you have checked it numerically, as in evalf(q1); -- Preben Alsholm Institut for matematik (Department of Mathematics) DTU (Technical University of Denmark) -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Fri, 7 Dec 2001 09:56:10 -0500 (EST) From: Carl Devore "devore" To: "maple-list" Subject: Winding Numbers and The Fundamental Theorem of Algebra This problem seems to be fixed in Maple 6. An alternative to using evalc is to split the integral at Pi. I suspect that this integrates easier in many cases. Numerical integration also works correctly, if approximations are acceptable.

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