Newton's Method in Complex Algebra, Classical Algebra #43

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Angewandte Mathematik für Ingenieure

Published on Jul 6, 2024
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Why can the Newton method also be used for complex functions with a complex variable and what does the Newton method for complex functions of one variable have to do with Newton fractals? Dipl. physicist Dietmar Haase shows in this video that the Newton method for real functions can be easily transferred to complex functions of a complex variable. Here, too, the starting point is the linear part of the Taylor series of a complex function that can be differentiated at least once. Just as in the real case, the zeros of the linearization of the complex function provide the iteration rule for the approximate calculation of the complex zeros. Among other things, it is shown that chaotic behavior is hidden in the iteration formula. If you choose a starting value for the iteration close to the boundary of the catchment area of ​​a complex zero, it becomes clear that the smallest variations in the starting value lead to completely different zeros of the complex function, which mathematically represents a chaotic system. It is also shown that the catchment areas of complex zeros are fractals, so-called Newton fractals. Website: https://www.ingmathe.de Youtube channel: / ingmathede Online calculator: https://www.wolframalpha.com/

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