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Z/ D 3z. Try it out! 58. z/ defined on C. Try to formulate what it means for orbits to be drastically or significantly different. Try it out! g/ for the Julia set of g. A on page 78. These should be read and compared with your own definitions, but it is not necessary to know these formal definitions well to continue on in the text; an intuitive understanding is enough. 0; 1/. 0; 1/. To see this, consider z0 D e i written in polar form. z00 /j > jz0 z00 j. z00 / along the unit circle is 2ˇ. z00 / along the unit circle is 2n ˇ as long as 2n ˇ < .

A//. Try it out! 170 explore several examples of global conjugation. 30. There is also a very useful technique called local conjugation that can simplify calculations. z/ D a1 z C a2 z 2 C : : : , having a fixed point at 0, can there be found be an analytic map defined near 0 such that ı f ı 1 is simply z 7! a1 z. This is called linearizing the map f near 0. It can always be done when 0 < ja1 j ¤ 1, but only sometimes when ja1 j D 1. , [1, 3, 24]). 24. ˛Cˇ/ 1, respectively, and then analyze the simpler map obtained by conjugation.

Describe (as best as you can) which seed values will find which of the roots 1; e 2 i=3, and e 2 i=3 . e 2 i=3 /. Try it out! , how the orbits behave for all seed values. We employ the Graph basins of attraction feature of the Complex Newton Method Applet, which uses different colors to display which initial guesses will find which roots. 27. 26. Try it out! 25, it appears that the boundary between them is a straight line; points on one side look closer to the root of f on that side than to the other root.

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A Cauchy-Type Problem for the Diffusion-Wave Equation with Riemann-Liouville Partial Derivative by Voroshilov A. A., Kilbas A. A.


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