Complex Analysis: New Applications
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en-us2015 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemTue, 03 Mar 2015 11:12:48 GMTTue, 03 Mar 2015 11:12:48 GMTNew applications in the Complex Analysis categoryhttp://www.mapleprimes.com/images/mapleapps.gifComplex Analysis: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=134
Classroom Tips and Techniques: Real and Complex Derivatives of Some Elementary Functions
http://www.maplesoft.com/applications/view.aspx?SID=153726&ref=Feed
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
http://www.maplesoft.com/applications/view.aspx?SID=153697&ref=Feed
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: The Explore Command in Maple 18
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The Explore functionality, which provides an interactive experience with parameter-dependent plots and expressions, has been significantly enhanced in Maple 18. In this Tips and Techniques article, I will focus on some key usage points of using the Explore command with plots, including explorations based on simple Maple plots as well as user-defined plotting procedures.<img src="/view.aspx?si=153552/thumb.jpg" alt="Classroom Tips and Techniques: The Explore Command in Maple 18" align="left"/>The Explore functionality, which provides an interactive experience with parameter-dependent plots and expressions, has been significantly enhanced in Maple 18. In this Tips and Techniques article, I will focus on some key usage points of using the Explore command with plots, including explorations based on simple Maple plots as well as user-defined plotting procedures.153552Wed, 16 Apr 2014 04:00:00 ZDave LinderDave LinderClassroom Tips and Techniques: Mathematical Thoughts on the Root Locus
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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 LopezPolynomial System Solving in Maple 16
http://www.maplesoft.com/applications/view.aspx?SID=132208&ref=Feed
Computing and manipulating the real solutions of a polynomial system is a requirement for many application areas, such as biological modeling, robotics, program verification, and control design, to name just a few. For example, an important problem in computational biology is to study the stability of the equilibria (or steady states) of biological systems. This question can often be reduced to solving a parametric system of polynomial equations and inequalities. In this application, these techniques are used to perform stability analysis of a parametric dynamical system and verify mathematical identities through branch cut analysis.<img src="/view.aspx?si=132208/thumb.jpg" alt="Polynomial System Solving in Maple 16" align="left"/>Computing and manipulating the real solutions of a polynomial system is a requirement for many application areas, such as biological modeling, robotics, program verification, and control design, to name just a few. For example, an important problem in computational biology is to study the stability of the equilibria (or steady states) of biological systems. This question can often be reduced to solving a parametric system of polynomial equations and inequalities. In this application, these techniques are used to perform stability analysis of a parametric dynamical system and verify mathematical identities through branch cut analysis.132208Tue, 27 Mar 2012 04:00:00 ZMaplesoftMaplesoftCoriolis Effect
http://www.maplesoft.com/applications/view.aspx?SID=1437&ref=Feed
The Coriolis effect is a force that modifies the trajectory of falling object on Earth. It is due to the rotation of the referential and, thereby, it is not a real force. The mathematical expression of this effect is obtained from the crossproduct of Earth's angular velocity (omega) with the object's linear velocity (v). The exact equation is F = 2m(v x omega). This worksheet demonstrates the action of the Coriolis effect on a projectile launched from our planet. It includes a graphic of the projectile's path as well as a procedure that determines how far the projectile will travel.<img src="/view.aspx?si=1437/coriolis_sm.jpg" alt="Coriolis Effect" align="left"/>The Coriolis effect is a force that modifies the trajectory of falling object on Earth. It is due to the rotation of the referential and, thereby, it is not a real force. The mathematical expression of this effect is obtained from the crossproduct of Earth's angular velocity (omega) with the object's linear velocity (v). The exact equation is F = 2m(v x omega). This worksheet demonstrates the action of the Coriolis effect on a projectile launched from our planet. It includes a graphic of the projectile's path as well as a procedure that determines how far the projectile will travel.1437Fri, 09 Dec 2011 05:00:00 ZPascal Thériault LauzierPascal Thériault LauzierThe Origin of Complex Numbers
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The origin of complex numbers starts with the contributions of Scipione del Ferro, Nicolo Tartaglia, Girolamo Cardano, and Rafael Bombelli. This Maple worksheed details the methods and formulas they used. It explores these formulas using Maple and shows how they can be extended. Numerous examples, exercises and illustrations make this a useful teaching module for an introduction of complex numbers.<img src="/applications/images/app_image_blank_lg.jpg" alt="The Origin of Complex Numbers" align="left"/>The origin of complex numbers starts with the contributions of Scipione del Ferro, Nicolo Tartaglia, Girolamo Cardano, and Rafael Bombelli. This Maple worksheed details the methods and formulas they used. It explores these formulas using Maple and shows how they can be extended. Numerous examples, exercises and illustrations make this a useful teaching module for an introduction of complex numbers.126618Fri, 14 Oct 2011 04:00:00 ZDr. John MathewsDr. John MathewsClassroom Tips and Techniques: Yet More Gems from the Little Red Book of Maple Magic
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Five more bits of accumulated "Maple magic" are shared: the limit of Picard iterates, combining radicals, factoring, yet another trig identity, and sorting strategies.<img src="/view.aspx?si=102692/thumb.jpg" alt="Classroom Tips and Techniques: Yet More Gems from the Little Red Book of Maple Magic" align="left"/>Five more bits of accumulated "Maple magic" are shared: the limit of Picard iterates, combining radicals, factoring, yet another trig identity, and sorting strategies.102692Mon, 21 Mar 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Maple Meets Marden's Theorem
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<p>Dan Kalman ascribes to Marden, and describes it as the most amazing theorem: If three noncollinear points in the complex plane forming the vertices of a triangle are interpolated by a (monic) cubic polynomial <em>p(z)</em>, the zeros of <em>p'(z)</em> are the foci of a unique ellipse (the Steiner in-ellipse) that is tangent to the sides of the triangle at the midpoints of the sides. The charm of this easy-to-state theorem that joins algebra, geometry, and complex analysis is explored with the tools of Maple.<br /></p><img src="/view.aspx?si=99069/thumb.jpg" alt="Classroom Tips and Techniques: Maple Meets Marden's Theorem" align="left"/><p>Dan Kalman ascribes to Marden, and describes it as the most amazing theorem: If three noncollinear points in the complex plane forming the vertices of a triangle are interpolated by a (monic) cubic polynomial <em>p(z)</em>, the zeros of <em>p'(z)</em> are the foci of a unique ellipse (the Steiner in-ellipse) that is tangent to the sides of the triangle at the midpoints of the sides. The charm of this easy-to-state theorem that joins algebra, geometry, and complex analysis is explored with the tools of Maple.<br /></p>99069Tue, 16 Nov 2010 05:00:00 ZRobert LopezRobert LopezHarmonic Analysis
http://www.maplesoft.com/applications/view.aspx?SID=96900&ref=Feed
<p>Harmonic Analysis combines innovative numerical tools for signal processing with rich analytical tools for studying problems of physics and the mathematics of complex variables. Harmonic Analysis provides a Maple package with worked examples in signal filtering, finance, and conformal mapping. The rich documentation includes an in-depth explanation of the theory of harmonic analysis. <br /> <br /> The algorithms in Harmonic Analysis are the culmination of new research by the author in the field of harmonic analysis. They have a US Patent pending (#10/856,453), and a paper on the methods behind the technology is published in the September 2004 issue of IEEE Explorer.</p><img src="/view.aspx?si=96900/harmoniclogo.gif" alt="Harmonic Analysis" align="left"/><p>Harmonic Analysis combines innovative numerical tools for signal processing with rich analytical tools for studying problems of physics and the mathematics of complex variables. Harmonic Analysis provides a Maple package with worked examples in signal filtering, finance, and conformal mapping. The rich documentation includes an in-depth explanation of the theory of harmonic analysis. <br /> <br /> The algorithms in Harmonic Analysis are the culmination of new research by the author in the field of harmonic analysis. They have a US Patent pending (#10/856,453), and a paper on the methods behind the technology is published in the September 2004 issue of IEEE Explorer.</p>96900Wed, 15 Sep 2010 04:00:00 ZVladimir ClueVladimir ClueThe CayleyDickson Algebra from 4D to 256D
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<p>There are higher dimensional numbers besides complex numbers. There are also hypercomplex numbers, such as, quaternions (4 D), octonions (8 D), sedenions (16 D), pathions (32 D), chingons (64 D), routons (128 D), voudons (256 D), and so on, without end. These names were coined by Robert P.C. de Marrais and Tony Smith. It is an alternate naming system providing relief from the difficult Latin names, such as:<br /> trigintaduonions (32 D), sexagintaquatronions (64 D), centumduodetrigintanions (128 D), and ducentiquinquagintasexions (256 D).</p><img src="/applications/images/app_image_blank_lg.jpg" alt="The CayleyDickson Algebra from 4D to 256D" align="left"/><p>There are higher dimensional numbers besides complex numbers. There are also hypercomplex numbers, such as, quaternions (4 D), octonions (8 D), sedenions (16 D), pathions (32 D), chingons (64 D), routons (128 D), voudons (256 D), and so on, without end. These names were coined by Robert P.C. de Marrais and Tony Smith. It is an alternate naming system providing relief from the difficult Latin names, such as:<br /> trigintaduonions (32 D), sexagintaquatronions (64 D), centumduodetrigintanions (128 D), and ducentiquinquagintasexions (256 D).</p>35420Fri, 23 Apr 2010 04:00:00 ZMichael CarterMichael CarterQuaternions, Octonions and Sedenions
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<p>This Hypercomplex package provides the algebra of the quaternion, octonion and sedenion hypercomplex numbers.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Quaternions, Octonions and Sedenions" align="left"/><p>This Hypercomplex package provides the algebra of the quaternion, octonion and sedenion hypercomplex numbers.</p>35196Fri, 16 Apr 2010 04:00:00 ZDr. Michael Angel CarterDr. Michael Angel CarterClassroom Tips and Techniques: Series Expansions
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<p>Maple has the ability to provide various series expansions and their truncations, as well as complete formal series for a variety of elementary and special functions. In this month's article, we examine the relevant commands and interface devices that access these functionalities.</p><img src="/view.aspx?si=34685/thumb.jpg" alt="Classroom Tips and Techniques: Series Expansions" align="left"/><p>Maple has the ability to provide various series expansions and their truncations, as well as complete formal series for a variety of elementary and special functions. In this month's article, we examine the relevant commands and interface devices that access these functionalities.</p>34685Thu, 05 Nov 2009 05:00:00 ZDr. Robert LopezDr. Robert LopezInteractive Multi-Threaded Mandelbrot Set Explorer
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<p>This application lets you interatively explore the Mandelbrot set. If this application detects that you're using Maple 13 or higher on a multi-core computer, it multi-threads calculations across as many cores as it finds.</p><img src="/view.aspx?si=32605/thumb.png" alt="Interactive Multi-Threaded Mandelbrot Set Explorer" align="left"/><p>This application lets you interatively explore the Mandelbrot set. If this application detects that you're using Maple 13 or higher on a multi-core computer, it multi-threads calculations across as many cores as it finds.</p>32605Tue, 28 Apr 2009 04:00:00 ZMaplesoftMaplesoftFractal Fun!
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<p>A simple search in your favorite search engine will attest to the sudden popularity of fractal art. That said, many people are often shocked to learn that these visually stunning images are created by iterating a simple complex formula to create a fractal object. A fractal object is any geometric object that posses the property of self-similarity. Self-similarity is a term attributed to Benoît Mandelbrot, to describe any object that appears roughly the same at any level of magnification. Fractal objects are readily prevalent in nature and can be easily seen by examining the intricate shape of sea shells, snowflakes and lightning bolts. <br />
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This application illustrates how Maple can be used to generate the two most famous fractal objects: the Mandelbrot Set and the Julia Set. </p><img src="/view.aspx?si=32594/frctl1.jpg" alt="Fractal Fun!" align="left"/><p>A simple search in your favorite search engine will attest to the sudden popularity of fractal art. That said, many people are often shocked to learn that these visually stunning images are created by iterating a simple complex formula to create a fractal object. A fractal object is any geometric object that posses the property of self-similarity. Self-similarity is a term attributed to Benoît Mandelbrot, to describe any object that appears roughly the same at any level of magnification. Fractal objects are readily prevalent in nature and can be easily seen by examining the intricate shape of sea shells, snowflakes and lightning bolts. <br />
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This application illustrates how Maple can be used to generate the two most famous fractal objects: the Mandelbrot Set and the Julia Set. </p>32594Mon, 27 Apr 2009 04:00:00 ZMaplesoftMaplesoftDC Motor Control Design
http://www.maplesoft.com/applications/view.aspx?SID=1458&ref=Feed
Given a model of a DC motor as a set of differential equations, we want to obtain both the transfer function and the state space model of the system. Then, we want to use the state space model to design a LQR controller and study the effect the R parameter in the LQR control design has on the controlled performance of the system.<img src="/view.aspx?si=1458/thumb.gif" alt="DC Motor Control Design" align="left"/>Given a model of a DC motor as a set of differential equations, we want to obtain both the transfer function and the state space model of the system. Then, we want to use the state space model to design a LQR controller and study the effect the R parameter in the LQR control design has on the controlled performance of the system.1458Tue, 06 May 2008 00:00:00 ZMaplesoftMaplesoftDigital Filter Design
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A FIR filter is derived from the impulse response of the desired filter and then sampled to convert it to a discrete time filter. The infinitely long impulse response must be truncated to be implemented. If the impulse response is nonzero for negative time (the filter is anti-causal) the response must also be shifted to the right until all of the impulse response coefficients are located in the positive time region.<img src="/view.aspx?si=1459/thumb.jpg" alt="Digital Filter Design" align="left"/>A FIR filter is derived from the impulse response of the desired filter and then sampled to convert it to a discrete time filter. The infinitely long impulse response must be truncated to be implemented. If the impulse response is nonzero for negative time (the filter is anti-causal) the response must also be shifted to the right until all of the impulse response coefficients are located in the positive time region.1459Tue, 06 May 2008 00:00:00 ZMaplesoftMaplesoftFrequency Domain System Identification
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System identification deals with the problem of identifying a model describing some physical system by measuring the response of the system. This example illustrates a problem where the structure of the model is known, and parameters of the model are identified. This is done by first designing the input signal, which is applied to the system. The measured output is converted to the frequency domain and the parameters are estimated in this domain.<img src="/view.aspx?si=1460/thumb.gif" alt="Frequency Domain System Identification" align="left"/>System identification deals with the problem of identifying a model describing some physical system by measuring the response of the system. This example illustrates a problem where the structure of the model is known, and parameters of the model are identified. This is done by first designing the input signal, which is applied to the system. The measured output is converted to the frequency domain and the parameters are estimated in this domain.1460Tue, 06 May 2008 00:00:00 ZMaplesoftMaplesoft