Polar form of Complex Numbers: De Moiver’s Theorem | MATH 144, Study notes of Trigonometry

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SECTIONE.3 Polar Form of Complex Numbers; DaMoivra's Theorem 8 : Fractals iT hia as Fractals are geometric objects that exhibit more and more detail the more we PROJECT magnify them (see Mathematics in the Modern Wierld om page 600), Many frac- tals can be described by iterating functions of complex numbers. The most famous such fractal is illustrated in Figures | and 2 called the Mandelbrot set, named after Benoit Mandelbrot, the mathematician who discovered it in the 1950s. Image oot available due to copyright restrictions Figure 2 Detail from the Mandelbrot set Here is how the Mandelbrot set is defined. Choose a complex number c, and define the complex quadratic Function faj=e24e Starting with zo = 0, we form the iterates of f as follows: = = f(0) = IU) = fle) =F +e AALNON) = He* + ) = fe? + cf +e As we continue calculating the iterates, one of two things will happen, depend- ing on the valu of c. Either the iterates z., .. form a bounded set (that Soe page S07 for the definition of is, the moduli of the iterates are all hess than some fixed number X°), or else they reondialies (phural mocdudi), eventually grow larger and larger without bound. The calculations in the table on page 606 show that for c = 0.1 + 0.2i, the iterates eventually stabilize at about 0.05 + 0.22i, whereas forc = 1 = i, the iterates quickly become so large that a calculator can’t handle them.