Complex Analysis: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=134
en-us2017 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSat, 25 Feb 2017 14:08:47 GMTSat, 25 Feb 2017 14:08:47 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: Norm of a Matrix
http://www.maplesoft.com/applications/view.aspx?SID=1430&ref=Feed
The greatest benefits from bringing Maple into the classroom are realized when the static pedagogy of a printed textbook is enlivened by the interplay of symbolic, graphic, and numeric calculations made possible by technology. Getting Maple to compute the correct answer is just the first step. Using Maple to bring insights not easily realized with by-hand calculations should be the goal of everyone who sets a hand to improving the learning experiences of students. In this article we will show how Maple can be used to gain insight on what the norm of a matrix means.<img src="/view.aspx?si=1430/thumb.jpg" alt="Classroom Tips and Techniques: Norm of a Matrix" align="left"/>The greatest benefits from bringing Maple into the classroom are realized when the static pedagogy of a printed textbook is enlivened by the interplay of symbolic, graphic, and numeric calculations made possible by technology. Getting Maple to compute the correct answer is just the first step. Using Maple to bring insights not easily realized with by-hand calculations should be the goal of everyone who sets a hand to improving the learning experiences of students. In this article we will show how Maple can be used to gain insight on what the norm of a matrix means.1430Mon, 13 Feb 2017 05:00:00 ZDr. Robert LopezDr. Robert LopezFractal Leaf Generator
http://www.maplesoft.com/applications/view.aspx?SID=154086&ref=Feed
This application generates Barnsley Fern fractals, using the number of iterations specified by the user.<img src="/view.aspx?si=154086/fractalleafThumb.jpg" alt="Fractal Leaf Generator" align="left"/>This application generates Barnsley Fern fractals, using the number of iterations specified by the user.154086Wed, 20 Apr 2016 04:00:00 ZMaplesoftMaplesoftExponential map fractal viewer
http://www.maplesoft.com/applications/view.aspx?SID=153953&ref=Feed
Static fractal viewer for the Julia sets of the exponential map lambda*exp(z), with variable lambda. Allows adjustable viewing window, zoom-in and display of periodic attractors up to period 5. Dynamic version can generate animated views, like the Knaster explosion of period 3 shown on the figure.<img src="/view.aspx?si=153953/3djulia.png" alt="Exponential map fractal viewer" align="left"/>Static fractal viewer for the Julia sets of the exponential map lambda*exp(z), with variable lambda. Allows adjustable viewing window, zoom-in and display of periodic attractors up to period 5. Dynamic version can generate animated views, like the Knaster explosion of period 3 shown on the figure.153953Tue, 19 Jan 2016 05:00:00 ZRobert Israel, Carl LoveRobert Israel, Carl LoveEscapeTime Fractals
http://www.maplesoft.com/applications/view.aspx?SID=153882&ref=Feed
<P>
The <A HREF="/support/help/Maple/view.aspx?path=Fractals/EscapeTime">Fractals</A> package in Maple makes it easier to create and explore popular fractals, including the Mandelbrot, Julia, Newton, and other time-iterative fractals. The Fractals package can quickly apply various escape time iterative maps over rectangular regions in the complex plane, the results of which consist of images that approximate well-known fractal sets. In the following application, you can explore escape time fractals by manipulating parameters pertaining to the generation of Mandelbrot, Julia, Newton and Burning Ship fractals.</P>
<P>
<B>Also:</B> You can <A HREF="http://maplecloud.maplesoft.com/application.jsp?appId=5690839489576960">interact with this application</A> in the MapleCloud!</P><img src="/view.aspx?si=153882/escapetimefractal.png" alt="EscapeTime Fractals" align="left"/><P>
The <A HREF="/support/help/Maple/view.aspx?path=Fractals/EscapeTime">Fractals</A> package in Maple makes it easier to create and explore popular fractals, including the Mandelbrot, Julia, Newton, and other time-iterative fractals. The Fractals package can quickly apply various escape time iterative maps over rectangular regions in the complex plane, the results of which consist of images that approximate well-known fractal sets. In the following application, you can explore escape time fractals by manipulating parameters pertaining to the generation of Mandelbrot, Julia, Newton and Burning Ship fractals.</P>
<P>
<B>Also:</B> You can <A HREF="http://maplecloud.maplesoft.com/application.jsp?appId=5690839489576960">interact with this application</A> in the MapleCloud!</P>153882Fri, 25 Sep 2015 04:00:00 ZMaplesoftMaplesoftClassroom 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
http://www.maplesoft.com/applications/view.aspx?SID=153552&ref=Feed
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
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 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
http://www.maplesoft.com/applications/view.aspx?SID=126618&ref=Feed
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
http://www.maplesoft.com/applications/view.aspx?SID=102692&ref=Feed
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
http://www.maplesoft.com/applications/view.aspx?SID=99069&ref=Feed
<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
http://www.maplesoft.com/applications/view.aspx?SID=35420&ref=Feed
<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
http://www.maplesoft.com/applications/view.aspx?SID=35196&ref=Feed
<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 Carter
Dr. Michael Angel Carter
Classroom Tips and Techniques: Series Expansions
http://www.maplesoft.com/applications/view.aspx?SID=34685&ref=Feed
<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 Lopez