User:Adam majewski

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Some notes from | wikipedia

$Z^{2}$ $Real=x^{2}-y^{2}$ $Imag=2*x*y$

$Z^{3}$ $Real=x^{3}-3*y^{2}*x$ $Imag=3*x^{2}*y-y^{3}$

$Z^{4}$ $Real=x^{4}-6*x^{2}*y^{2}+y^{4}$ $Imag=4*x^{3}*y-4*x*y^{3}$

$Z^{5}$ $Real=x^{5}-10*x^{3}*y^{2}+5*x*y^{4}$ $Imag=5*x^{4}*y-10*x^{2}*y^{3}+y^{5}$

$Z^{6}$ $Real=x^{6}-15*x^{4}*y^{2}+15*x^{2}*y^{4}-y^{6}$ $Imag=6*x^{5}*y-20*x^{4}*y^{2}+6*x*y^{5}$

$Z^{7}$ $Real=x^{7}-21*x^{5}*y^{2}+35*x^{3}*y^{4}-7*x*y^{6}$ $Imag=7*x^{6}*y-35*x^{4}*y^{3}+21*x^{2}*y^{5}-y^{7}$,

$Exp(Z)$ $Real=Exp(x)*Cos(y)$   $Imag=Exp(x)*Sin(y)$

$Ln(Z)$ $Real=0.5*Ln(x^{2}+y^{2})$   $Imag=Arctan(y/x)$

$Sin(Z)$ $Real=Sin(x)*((Exp(y)+Exp(-y))/2)$   $Imag=Cos(x)*((Exp(y)-Exp(-y))/2)$

$Cos(Z)$ $Real=Cos(x)*((Exp(y)+Exp(-y))/2)$   $Imag=-Sin(x)*((Exp(y)-Exp(-y))/2)$

$SinH(Z)$ $Real=Cos(y)*((Exp(x)-Exp(-x))/2)$   $Imag=Sin(y)*((Exp(x)+Exp(-x))/2)$

$CosH(Z)$ $Real=Cos(y)*((Exp(x)+Exp(-x))/2)$   $Imag=Sin(y)*((Exp(x)-Exp(-x))/2)$

test

$1/237142198758023568227473377297792835283496928595231875152809132048206089502588927\approx 4.21687917729220928973942962050800760308398455294740302003110521004325771638790385468222014406044810316697454753662...*10^{-}81$ ${\frac {1}{2^{267}-1}}={\frac {1}{237142198758023568227473377297792835283496928595231875152809132048206089502588927}}\approx 4.216879177292209*10^{-81}$( land on the root point of period 267 component : c267 = 0.250137369683480-0.000003221184145 i with angled internal adress : $1{\xrightarrow {1/267}}267$

( land on the root point of period 268 component c268 = 0.250137369683480-0.000003221184145i period = 10000 i with angled internal adress : $1{\xrightarrow {267/268}}268$

mandelbrot set

the Mandelbrot set for the function 1/z - z∙(1 + 0.001∙z)/(1 - 0.002∙z + 0.001∙z2) = "1 -0.002 - 0.999 -0.001 0 1 -0.002 0.001":

 $f(z)={\frac {1}{z}}-{\frac {z*(1+0.001*z)}{(1-0.002*z+0.001*z^{2})}}$

http://www.juliasets.dk/UFP.htm

People

Marius-F. Danca

petal

"An attracting petal, P + , for a map M at zero is an open simply connected forward invariant region with 0 ∈ ∂P + , that shrinks down to the origin under iteration of M . More precisely, P + is an attracting petal if M (P + ) ⊂ P + ∪ {0} and n≥0 M n (P + ) = {0}. "

example

http://mathoverflow.net/questions/104482/parabolic-immediate-basins-always-simply-connected?rq=1 "An example is f(z)=z+1−1/zf(z)=z+1−1/z. There is one petal for the neutral point at infinity. Let AA be the dmain of attraction of ∞∞. Critical points are ±i±i. Everything is symmetric with respect to the real line, because the function is real. One critical point is in AA, so by symmetry the other one is also in AA. The map f:A→Af:A→A is 2-to-1 (because ff is of degree 22), so Riemann and Hurwitz tell us that AA is infinitely connected."

shareciteeditflag answered Aug 12 '12 at 13:38

Alexandre Eremenko

cylinder

What is the difference between the cylinder $\mathbb {C} \setminus 0$ and cylinder $\mathbb {C} /\mathbb {Z}$ ?

topology

"A topologist is someone who doesn't know the difference between a cup of coffee and a donut."

references

• Computer Methods and Borel Summability Applied to Feigenbaum's Equation By Jean Pierre Eckmann

Format Hardback | 297 pages, Publication date 01 May 1985, Publisher Springer , Publication City/Country United States , ISBN10 0387152156, ISBN13 9780387152158

• http://mathoverflow.net/questions/157309/power-series-expansion-of-the-koenigs-function?rq=1
• T. M. CHERRY, A singular case of iteration of analytic functions: A contribution to the small divisor problem. In: Nonlinear Problems of Engineering, W. F. AMES (Ed.), New York, 1964, 29–50.
• Cherry, T. M., "A Singular Case of Iteration of Analytic Functions: A Contribution to the Small-Divisor Problem," in Nonlinear Problems of Engineering (edited by W. F. Ames), Academic Press, New York, 1964, 29-50.
• MR178125 30.40 (57.48) Cherry, T. M. A singular case of iteration of analytic functions: A contribution to the small-divisor problem. 1964 Nonlinear Problems of Engineering pp. 29–50 Academic Press, New York

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