Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
The Mandelbrot Set M is a subset of the complex plane defined as follows: Given c E C, consider the function and define fe: C → C Z → 2² +c f(n)(z) = fe(fc(….. fc(z))); n times i.e. the function f(n) is the function fe composed with itself n times. Then M = {c ≤ C : |ƒ(¹)(0)| is bounded as n → ∞}. (We have not defined what bounded' means, but you can use your intuition here.) So, for example, if we look at c = 1, and want to determine whether 1 € M or not, we need to check the moduli of the numbers f₁ (0) = 0² + 1 = 1 ƒ(²) (0) = ƒ₁ (ƒ₁(0)) = f₁(1) = 1² + 1 = 2 f(0) = 2² + 1 = 5 ƒ(¹)(0) = 5² + 1 = 26 etc. You can see very quickly that the moduli of these numbers grow beyond any bounds as n→∞. So this tells us that 1 & M. Your job is to determine whether i E M. Show your work.
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Transcribed Image Text:The Mandelbrot Set M is a subset of the complex plane defined as follows: Given c E C, consider the function and define fe: C → C Z → 2² +c f(n)(z) = fe(fc(….. fc(z))); n times i.e. the function f(n) is the function fe composed with itself n times. Then M = {c ≤ C : |ƒ(¹)(0)| is bounded as n → ∞}. (We have not defined what bounded' means, but you can use your intuition here.) So, for example, if we look at c = 1, and want to determine whether 1 € M or not, we need to check the moduli of the numbers f₁ (0) = 0² + 1 = 1 ƒ(²) (0) = ƒ₁ (ƒ₁(0)) = f₁(1) = 1² + 1 = 2 f(0) = 2² + 1 = 5 ƒ(¹)(0) = 5² + 1 = 26 etc. You can see very quickly that the moduli of these numbers grow beyond any bounds as n→∞. So this tells us that 1 & M. Your job is to determine whether i E M. Show your work.
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