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Some notes from | wikipedia
Z
2
{\displaystyle Z^{2}}
R
e
a
l
=
x
2
−
y
2
{\displaystyle Real=x^{2}-y^{2}}
I
m
a
g
=
2
∗
x
∗
y
{\displaystyle Imag=2*x*y}
Z
3
{\displaystyle Z^{3}}
R
e
a
l
=
x
3
−
3
∗
y
2
∗
x
{\displaystyle Real=x^{3}-3*y^{2}*x}
I
m
a
g
=
3
∗
x
2
∗
y
−
y
3
{\displaystyle Imag=3*x^{2}*y-y^{3}}
Z
4
{\displaystyle Z^{4}}
R
e
a
l
=
x
4
−
6
∗
x
2
∗
y
2
+
y
4
{\displaystyle Real=x^{4}-6*x^{2}*y^{2}+y^{4}}
I
m
a
g
=
4
∗
x
3
∗
y
−
4
∗
x
∗
y
3
{\displaystyle Imag=4*x^{3}*y-4*x*y^{3}}
Z
5
{\displaystyle Z^{5}}
R
e
a
l
=
x
5
−
10
∗
x
3
∗
y
2
+
5
∗
x
∗
y
4
{\displaystyle Real=x^{5}-10*x^{3}*y^{2}+5*x*y^{4}}
I
m
a
g
=
5
∗
x
4
∗
y
−
10
∗
x
2
∗
y
3
+
y
5
{\displaystyle Imag=5*x^{4}*y-10*x^{2}*y^{3}+y^{5}}
Z
6
{\displaystyle Z^{6}}
R
e
a
l
=
x
6
−
15
∗
x
4
∗
y
2
+
15
∗
x
2
∗
y
4
−
y
6
{\displaystyle Real=x^{6}-15*x^{4}*y^{2}+15*x^{2}*y^{4}-y^{6}}
I
m
a
g
=
6
∗
x
5
∗
y
−
20
∗
x
4
∗
y
2
+
6
∗
x
∗
y
5
{\displaystyle Imag=6*x^{5}*y-20*x^{4}*y^{2}+6*x*y^{5}}
Z
7
{\displaystyle Z^{7}}
R
e
a
l
=
x
7
−
21
∗
x
5
∗
y
2
+
35
∗
x
3
∗
y
4
−
7
∗
x
∗
y
6
{\displaystyle Real=x^{7}-21*x^{5}*y^{2}+35*x^{3}*y^{4}-7*x*y^{6}}
I
m
a
g
=
7
∗
x
6
∗
y
−
35
∗
x
4
∗
y
3
+
21
∗
x
2
∗
y
5
−
y
7
{\displaystyle Imag=7*x^{6}*y-35*x^{4}*y^{3}+21*x^{2}*y^{5}-y^{7}}
E
x
p
(
Z
)
{\displaystyle Exp(Z)}
R
e
a
l
=
E
x
p
(
x
)
∗
C
o
s
(
y
)
{\displaystyle Real=Exp(x)*Cos(y)}
I
m
a
g
=
E
x
p
(
x
)
∗
S
i
n
(
y
)
{\displaystyle Imag=Exp(x)*Sin(y)}
L
n
(
Z
)
{\displaystyle Ln(Z)}
R
e
a
l
=
0.5
∗
L
n
(
x
2
+
y
2
)
{\displaystyle Real=0.5*Ln(x^{2}+y^{2})}
I
m
a
g
=
A
r
c
t
a
n
(
y
/
x
)
{\displaystyle Imag=Arctan(y/x)}
S
i
n
(
Z
)
{\displaystyle Sin(Z)}
R
e
a
l
=
S
i
n
(
x
)
∗
(
(
E
x
p
(
y
)
+
E
x
p
(
−
y
)
)
/
2
)
{\displaystyle Real=Sin(x)*((Exp(y)+Exp(-y))/2)}
I
m
a
g
=
C
o
s
(
x
)
∗
(
(
E
x
p
(
y
)
−
E
x
p
(
−
y
)
)
/
2
)
{\displaystyle Imag=Cos(x)*((Exp(y)-Exp(-y))/2)}
C
o
s
(
Z
)
{\displaystyle Cos(Z)}
R
e
a
l
=
C
o
s
(
x
)
∗
(
(
E
x
p
(
y
)
+
E
x
p
(
−
y
)
)
/
2
)
{\displaystyle Real=Cos(x)*((Exp(y)+Exp(-y))/2)}
I
m
a
g
=
−
S
i
n
(
x
)
∗
(
(
E
x
p
(
y
)
−
E
x
p
(
−
y
)
)
/
2
)
{\displaystyle Imag=-Sin(x)*((Exp(y)-Exp(-y))/2)}
S
i
n
H
(
Z
)
{\displaystyle SinH(Z)}
R
e
a
l
=
C
o
s
(
y
)
∗
(
(
E
x
p
(
x
)
−
E
x
p
(
−
x
)
)
/
2
)
{\displaystyle Real=Cos(y)*((Exp(x)-Exp(-x))/2)}
I
m
a
g
=
S
i
n
(
y
)
∗
(
(
E
x
p
(
x
)
+
E
x
p
(
−
x
)
)
/
2
)
{\displaystyle Imag=Sin(y)*((Exp(x)+Exp(-x))/2)}
C
o
s
H
(
Z
)
{\displaystyle CosH(Z)}
R
e
a
l
=
C
o
s
(
y
)
∗
(
(
E
x
p
(
x
)
+
E
x
p
(
−
x
)
)
/
2
)
{\displaystyle Real=Cos(y)*((Exp(x)+Exp(-x))/2)}
I
m
a
g
=
S
i
n
(
y
)
∗
(
(
E
x
p
(
x
)
−
E
x
p
(
−
x
)
)
/
2
)
{\displaystyle Imag=Sin(y)*((Exp(x)-Exp(-x))/2)}
test
1
/
237142198758023568227473377297792835283496928595231875152809132048206089502588927
≈
4.21687917729220928973942962050800760308398455294740302003110521004325771638790385468222014406044810316697454753662...
∗
10
−
81
{\displaystyle 1/237142198758023568227473377297792835283496928595231875152809132048206089502588927\approx 4.21687917729220928973942962050800760308398455294740302003110521004325771638790385468222014406044810316697454753662...*10^{-}81}
1
2
267
−
1
=
1
237142198758023568227473377297792835283496928595231875152809132048206089502588927
≈
4.216879177292209
∗
10
−
81
{\displaystyle {\frac {1}{2^{267}-1}}={\frac {1}{237142198758023568227473377297792835283496928595231875152809132048206089502588927}}\approx 4.216879177292209*10^{-81}}
1
→
1
/
267
267
{\displaystyle 1{\xrightarrow {1/267}}267}
1
→
267
/
268
268
{\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
C
∖
0
{\displaystyle \mathbb {C} \setminus 0}
C
/
Z
{\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
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
pl-N
Polski jest językiem ojczystym tego użytkownika.
en-2
This user can read and write intermediate English .
.
↑ NEWTON’S METHOD ON THE COMPLEX EXPONENTIAL FUNCTION MAKO E. HARUTA
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