Complex Analysis: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=134
en-us2020 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemWed, 05 Aug 2020 13:53:06 GMTWed, 05 Aug 2020 13:53:06 GMTNew applications in the Complex Analysis categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgComplex Analysis: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=134
Fractal Leaf Generator
https://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="https://www.maplesoft.com/view.aspx?si=154086/fractalleafThumb.jpg" alt="Fractal Leaf Generator" style="max-width: 25%;" align="left"/>This application generates Barnsley Fern fractals, using the number of iterations specified by the user.https://www.maplesoft.com/applications/view.aspx?SID=154086&ref=FeedWed, 20 Apr 2016 04:00:00 ZMaplesoftMaplesoftEscapeTime Fractals
https://www.maplesoft.com/applications/view.aspx?SID=153882&ref=Feed
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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>
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<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="https://www.maplesoft.com/view.aspx?si=153882/escapetimefractal.png" alt="EscapeTime Fractals" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=153882&ref=FeedFri, 25 Sep 2015 04:00:00 ZMaplesoftMaplesoftClassroom Tips and Techniques: Real and Complex Derivatives of Some Elementary Functions
https://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="https://www.maplesoft.com/view.aspx?si=153726/thumb.jpg" alt="Classroom Tips and Techniques: Real and Complex Derivatives of Some Elementary Functions" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=153726&ref=FeedWed, 10 Dec 2014 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots
https://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="https://www.maplesoft.com/view.aspx?si=153697/thumb.jpg" alt="Classroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=153697&ref=FeedTue, 11 Nov 2014 05:00:00 ZDr. Robert LopezDr. Robert LopezHopalong Attractor
https://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="https://www.maplesoft.com/view.aspx?si=153557/95fa944692de1fb724cb7e758e6c56e5.gif" alt="Hopalong Attractor" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=153557&ref=FeedMon, 28 Apr 2014 04:00:00 ZDave LinderDave LinderClassroom Tips and Techniques: The Explore Command in Maple 18
https://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="https://www.maplesoft.com/view.aspx?si=153552/thumb.jpg" alt="Classroom Tips and Techniques: The Explore Command in Maple 18" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=153552&ref=FeedWed, 16 Apr 2014 04:00:00 ZDave LinderDave LinderThe Origin of Complex Numbers
https://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="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="The Origin of Complex Numbers" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=126618&ref=FeedFri, 14 Oct 2011 04:00:00 ZDr. John MathewsDr. John MathewsThe CayleyDickson Algebra from 4D to 256D
https://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="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="The CayleyDickson Algebra from 4D to 256D" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=35420&ref=FeedFri, 23 Apr 2010 04:00:00 ZMichael CarterMichael CarterQuaternions, Octonions and Sedenions
https://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="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Quaternions, Octonions and Sedenions" style="max-width: 25%;" align="left"/><p>This Hypercomplex package provides the algebra of the quaternion, octonion and sedenion hypercomplex numbers.</p>https://www.maplesoft.com/applications/view.aspx?SID=35196&ref=FeedFri, 16 Apr 2010 04:00:00 ZDr. Michael Angel Carter
Dr. Michael Angel Carter
Fractal Fun!
https://www.maplesoft.com/applications/view.aspx?SID=32594&ref=Feed
<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>
<P>
<B>Also: </B> Make sure to check out the <A HREF="/applications/view.aspx?SID=153882">EscapeTime Fractals</A> application, which has been updated for Maple 2015.
</P><img src="https://www.maplesoft.com/view.aspx?si=32594/frctl1.jpg" alt="Fractal Fun!" style="max-width: 25%;" 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 />
<br />
This application illustrates how Maple can be used to generate the two most famous fractal objects: the Mandelbrot Set and the Julia Set. </p>
<P>
<B>Also: </B> Make sure to check out the <A HREF="/applications/view.aspx?SID=153882">EscapeTime Fractals</A> application, which has been updated for Maple 2015.
</P>https://www.maplesoft.com/applications/view.aspx?SID=32594&ref=FeedMon, 27 Apr 2009 04:00:00 ZMaplesoftMaplesoftFractal Dimension and Space-Filling Curves (with iterated function systems)
https://www.maplesoft.com/applications/view.aspx?SID=4869&ref=Feed
By using complex numbers to represent points in the plane, and the concept of iterated function system, we efficiently describe fractal sets of any dimension from 0 to 2 and continuous curves that pass through them. Maple's animation feature allows us to make "movies" that show the transition through different dimensions.<img src="https://www.maplesoft.com/view.aspx?si=4869/thumb.png" alt="Fractal Dimension and Space-Filling Curves (with iterated function systems)" style="max-width: 25%;" align="left"/>By using complex numbers to represent points in the plane, and the concept of iterated function system, we efficiently describe fractal sets of any dimension from 0 to 2 and continuous curves that pass through them. Maple's animation feature allows us to make "movies" that show the transition through different dimensions.https://www.maplesoft.com/applications/view.aspx?SID=4869&ref=FeedFri, 16 Feb 2007 00:00:00 ZProf. Mark MeyersonProf. Mark MeyersonComplex Analysis Project
https://www.maplesoft.com/applications/view.aspx?SID=4846&ref=Feed
Complex analysis Maple 12 worksheets to accompany the textbook: COMPLEX ANALYSIS: for Mathematics and Engineering, Fifth Edition, 2006
by John H. Mathews and Russell W. Howell
ISBN: 0-7637-3748-8
Jones and Bartlett Pub. Inc.<img src="https://www.maplesoft.com/view.aspx?si=4846/0763737488.01._AA240_SCLZZZZZZZ_.jpg" alt="Complex Analysis Project" style="max-width: 25%;" align="left"/>Complex analysis Maple 12 worksheets to accompany the textbook: COMPLEX ANALYSIS: for Mathematics and Engineering, Fifth Edition, 2006
by John H. Mathews and Russell W. Howell
ISBN: 0-7637-3748-8
Jones and Bartlett Pub. Inc.https://www.maplesoft.com/applications/view.aspx?SID=4846&ref=FeedMon, 27 Nov 2006 00:00:00 ZDr. John MathewsDr. John MathewsHomographie Complexe - Polynomes
https://www.maplesoft.com/applications/view.aspx?SID=1710&ref=Feed
We study the differences between the direct image and the reciprocal image of a point of a unit by a homographic function. Includes student exercise and worked solutions.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Homographie Complexe - Polynomes" style="max-width: 25%;" align="left"/>We study the differences between the direct image and the reciprocal image of a point of a unit by a homographic function. Includes student exercise and worked solutions.https://www.maplesoft.com/applications/view.aspx?SID=1710&ref=FeedSun, 19 Feb 2006 05:00:00 ZProf. KERNIVINEN SebastienProf. KERNIVINEN SebastienAesthetic Plots in Complex Plane
https://www.maplesoft.com/applications/view.aspx?SID=1453&ref=Feed
Complex functions can create beautiful and wonderful graphs in the complex plane. In this worksheet we some interesting graphs and animations will be presented. As well as a mathematical fun, the results can be of interest in educational field. Similar graphics or animations can help students to visualize practical phenomenons in some engineering fields such as fluid mechanics or electrical engineering.<img src="https://www.maplesoft.com/view.aspx?si=1453/thumb.gif" alt="Aesthetic Plots in Complex Plane" style="max-width: 25%;" align="left"/>Complex functions can create beautiful and wonderful graphs in the complex plane. In this worksheet we some interesting graphs and animations will be presented. As well as a mathematical fun, the results can be of interest in educational field. Similar graphics or animations can help students to visualize practical phenomenons in some engineering fields such as fluid mechanics or electrical engineering.https://www.maplesoft.com/applications/view.aspx?SID=1453&ref=FeedWed, 06 Apr 2005 00:00:00 ZAmir KhanshanAmir Khanshan