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Some notes from | wikipedia

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

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

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

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

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

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

${\displaystyle Exp(Z)}$ ${\displaystyle Real=Exp(x)*Cos(y)}$   ${\displaystyle Imag=Exp(x)*Sin(y)}$

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

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

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

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

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

## test

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

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

# 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}. "[1]

## cylinder

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

## 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

pl-N Polski jest językiem ojczystym tego użytkownika.
en-2 This user can read and write intermediate English.

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1. NEWTON’S METHOD ON THE COMPLEX EXPONENTIAL FUNCTION MAKO E. HARUTA