Complex Dynamics
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Home > Art, Film & Photography > Complex Dynamics: Classification of Fatou Components, Complex Quadratic Polynomial, Complex System, Filled Julia Set, Herman Ring, Mande
Complex Dynamics: Classification of Fatou Components, Complex Quadratic Polynomial, Complex System, Filled Julia Set, Herman Ring, Mande

Complex Dynamics: Classification of Fatou Components, Complex Quadratic Polynomial, Complex System, Filled Julia Set, Herman Ring, Mande


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About the Book

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 28. Chapters: Classification of Fatou components, Complex quadratic polynomial, Complex system, Filled Julia set, Herman ring, Mandelbrot set, No-wandering-domain theorem, Periodic points of complex quadratic mappings, Siegel disc. Excerpt: The Mandelbrot set is a mathematical set of points whose boundary is a distinctive and easily recognizable two-dimensional fractal shape. The set is closely related to Julia sets (which include similarly complex shapes), and is named after the mathematician Benoit Mandelbrot, who studied and popularized it. Images of the Mandelbrot set are made by sampling complex numbers and determining for each whether the result tends towards infinity when a particular mathematical operation is iterated on it. Treating the real and imaginary parts of each number as image coordinates, pixels are colored according to how rapidly the sequence diverges, if at all. More precisely, the Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn + c remains bounded. That is, a complex number c is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets. For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand, c = i (where i is defined as i = 1) gives the sequence 0, i, ( 1 + i), i, ( 1 + i), i, ..., which is bounded, and so i belongs to the Mandelbrot set. Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts. The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules, and is one of the best-known examples of mathematical v


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Product Details
  • ISBN-13: 9781156739020
  • Publisher: Books LLC, Wiki Series
  • Publisher Imprint: Books LLC, Wiki Series
  • Height: 246 mm
  • No of Pages: 30
  • Spine Width: 2 mm
  • Weight: 73 gr
  • ISBN-10: 1156739020
  • Publisher Date: 23 Apr 2013
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Sub Title: Classification of Fatou Components, Complex Quadratic Polynomial, Complex System, Filled Julia Set, Herman Ring, Mande
  • Width: 189 mm


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