Complex Analysis: New Applications
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en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSun, 21 Dec 2014 23:15:54 GMTSun, 21 Dec 2014 23:15:54 GMTNew applications in the Complex Analysis categoryhttp://www.mapleprimes.com/images/mapleapps.gifComplex Analysis: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=2881
Classroom Tips and Techniques: Real and Complex Derivatives of Some Elementary Functions
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The elementary functions include the six trigonometric and hyperbolic functions and their inverses. For all but five of these 24 functions, Maple's derivative (correct on the complex plane) agrees with the real-variable form found in the standard calculus text. For these five exceptions, this article explores two issues: (1) Does Maple's derivative, restricted to the real domain, agree with the real-variable form; and (2), to what extent do both forms agree on the complex plane.<img src="/view.aspx?si=153726/thumb.jpg" alt="Classroom Tips and Techniques: Real and Complex Derivatives of Some Elementary Functions" align="left"/>The elementary functions include the six trigonometric and hyperbolic functions and their inverses. For all but five of these 24 functions, Maple's derivative (correct on the complex plane) agrees with the real-variable form found in the standard calculus text. For these five exceptions, this article explores two issues: (1) Does Maple's derivative, restricted to the real domain, agree with the real-variable form; and (2), to what extent do both forms agree on the complex plane.153726Wed, 10 Dec 2014 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots
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Naive simplification of f(z) = sqrt(z - 1) sqrt(z + 1) to F(z) = sqrt(z<sup>2</sup> - 1) results in a pair of functions that agree on only part of the complex plane. The enhanced ability of Maple 18 to find and display branch cuts of composite functions is used in this article to explore the branch cuts and regions of agreement/disagreement of f and F.<img src="/view.aspx?si=153697/thumb.jpg" alt="Classroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots" align="left"/>Naive simplification of f(z) = sqrt(z - 1) sqrt(z + 1) to F(z) = sqrt(z<sup>2</sup> - 1) results in a pair of functions that agree on only part of the complex plane. The enhanced ability of Maple 18 to find and display branch cuts of composite functions is used in this article to explore the branch cuts and regions of agreement/disagreement of f and F.153697Tue, 11 Nov 2014 05:00:00 ZDr. Robert LopezDr. Robert LopezHopalong Attractor
http://www.maplesoft.com/applications/view.aspx?SID=153557&ref=Feed
<p>Hopalong attractors are fractals, introduced by Barry Martin of Aston University in Birmingham, England. This application allows you to explore the Hopalong by varying the parameters, the number of iterations, the iterates' symbol size, and the background color choice. You can also change the starting values of each of the three orbits by dragging the cross symbols appearing in the plot. Full details on how this application was created using the Explore command with a user-defined module are included.</p><img src="/view.aspx?si=153557/95fa944692de1fb724cb7e758e6c56e5.gif" alt="Hopalong Attractor" align="left"/><p>Hopalong attractors are fractals, introduced by Barry Martin of Aston University in Birmingham, England. This application allows you to explore the Hopalong by varying the parameters, the number of iterations, the iterates' symbol size, and the background color choice. You can also change the starting values of each of the three orbits by dragging the cross symbols appearing in the plot. Full details on how this application was created using the Explore command with a user-defined module are included.</p>153557Mon, 28 Apr 2014 04:00:00 ZDave LinderDave LinderClassroom Tips and Techniques: Mathematical Thoughts on the Root Locus
http://www.maplesoft.com/applications/view.aspx?SID=153452&ref=Feed
Under suitable assumptions, the roots of the equation <em>f</em>(<em>z, c</em>) = 0, namely, <em>z</em> = <em>z</em>(<em>c</em>), trace a curve in the complex plane. In engineering feedback-control, such curves are called a <em>root locus</em>. This article examines the parameter-dependence of roots of polynomial and transcendental equations.<img src="/view.aspx?si=153452/thumb.jpg" alt="Classroom Tips and Techniques: Mathematical Thoughts on the Root Locus" align="left"/>Under suitable assumptions, the roots of the equation <em>f</em>(<em>z, c</em>) = 0, namely, <em>z</em> = <em>z</em>(<em>c</em>), trace a curve in the complex plane. In engineering feedback-control, such curves are called a <em>root locus</em>. This article examines the parameter-dependence of roots of polynomial and transcendental equations.153452Tue, 29 Oct 2013 04:00:00 ZDr. Robert LopezDr. Robert Lopez