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Complex n-dimensional open ball is presented. Abbreviation is n open ball. The notation is B ( a, R )
R > 0
Open set { z∈Cn | | z-a | < R }
2
Open set of Cn Ω
Map fromΩ to open set of Cn, Ω’ F = (f1, f2, …, fn )
Element of F fj
When fj is 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 )
f ∈A (Ω ) ⇒ 1/ f ∈A (Ω ╲ f -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.
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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)
A (Ω ) = {f ∈ L2loc (Ω) | ∂f /∂
j = 0, j = 0, 1, …, n }
j = 0, j = 0, 1, …, n }6
n open ball B (a, R ) ⊆ Ω
Volume element of ∂B (a, R ) dS
Vol ( ∂B (a, R ) ) : = ∫∂B (a, R )dS = 2πnR2n-1/(n-1)!
A (Ω ) is closed subspace on topology of L2convergence .
A (Ω ) and A (Ω )2 is separable.
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Domain Ω
Point a
z ∈Ω
ζ∈(∂B)n
ζ∈(∂B)n
For arbitrary z∈Ω and ζ∈(∂B)n, when (a1+ζ1・(z1-a1), …, an+ζn・(zn-an) ) ∈Ω is satisfied, Ω is called Reinhardt domain centered by a.
For arbitrary z∈Ω 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 ball B (a, R ) is complete Reinhardt domain.
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n dimensional complex ball that has center 0 D = B n
D’s logarithm image log D is defined by the next.
D’s logarithm image log D is defined by the next.
log D = {x∈(R∪{-∞})n | ex : = (ex1, …, exn) ∈D }
When dialog image is convex, D is logarithm convex.
Outer point of D a
Monomial ma(z)
supz∈D | ma(z) | < ma(a) = 1
Word, meaning element and distance are defined by the next at simplified level.
Word : = B n ( = complete Reinhardt domain centered by 0 )
Meaning element : = a ( = Outer point of D)
Distance : = supz∈D | ma(z) | of monomial ma(z)
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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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