Saturday 28 April 2018

Reversion Analysis Theory



1
Complex n-dimensional open ball is presented. Abbreviation is n open ball. The notation is B aR )
> 0
Open set { zCn | | z-| < R }
2
Open set of Cn     Ω
Map fromΩ to open set of CnΩ’     F = (f1, f2, …, f)
Element of F     fj
When fis normal function over Ω, F is called holomorphic map.
Composition of holomorphic map is also holomorphic map.
3
Set of all the holomorphic functions over Ω     A (Ω )
(Ω  1/ f (Ω  -1(0) )
Holomorphic map that has holomorphic inverse map is called biholomorphic map
When there exists biholomorphic function from Ω to Ω is called biholomorphic equivalent.
Bijective holomorphic map is biholomorphic.
Biholomorphic map from Ω to Ω is called holomorphic automorphism that becomes group by product as composition.
The group is called holomorphic automorphism group. The notation is Aut Ω.
4
Each n open ball is holomorphic equivalent.
B ( (0,0, …, 0 ) is notated as B n.
5
All the locally 2 powered integrable functions     L2loc (Ω)
(Ω ) = {f  L2loc (Ω) | ∂f /∂ = 0, = 0, 1, …, n }
6
n open ball     B (aR ) ⊆ Ω
Volume element of B (aR )     dS
Vol ( B (aR ) ) : = B (aR )dS = 2πnR2n-1/(n-1)!
(Ω ) is closed subspace on topology of L2convergence .
(Ω ) and (Ω )is separable.
7
Domain     Ω
Point     a
Ω
ζ
(∂B)n
For arbitrary zΩ and ζ(∂B)nwhen (a1+ζ1(z1-a1), …, an+ζn(zn-an) ) Ω is satisfied, Ω is called Reinhardt domain centered by a.
For arbitrary zΩ and ζ∂Bn, when (a1+ζ1(z1-a1), …, an+ζn(zn-an) ) Ω is satisfied, Ω is complete Reinhardt domain centered by a.
n open balB (aR ) is complete Reinhardt domain.
8
n dimensional complex ball that has center 0    D = n   
D’s logarithm image log D is defined by the next.
log = {x(R{-∞})n ex : = (ex1, …, exn) D }
When dialog image is convex, D is logarithm convex.
Outer point of D     a
Monomial ma(z)  
supzD | ma(z) | < ma(a) = 1
Word, meaning element and distance are defined by the next at simplified level.
Word : = n ( = complete Reinhardt domain centered by 0 )  
Meaning element : = a ( = Outer point of D)
Distance : = supzD | ma(z) | of monomial ma(z)
9
Word, meaning element and distance are considered in connection with Cauchy-Riemann equation.

[References]
<Distance>


Tokyo June 8, 2008

[Postscript June 19]
On holomorphic, refer to the next.

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