[複變] 兩題複變請教
以下兩題複變請教><
1.A Jordan curve in C can be considered as a subset of C homeomorphic to the
unit circle. Now let Γ⊂C be a jordan cuvre and let φ:Γ→∂D be a
homeomorphism. Show that, given ε>0, there exists δ>0 such that
if a,b∈Γ and |a-b|<δ, then
(a)One of the two circular arcs determined by φ(a) and φ(b) has strictly
shorter length;
(b)The diameter of σ_ab, the image of this shorter arc under φ^-1,
is less than ε.
2.Let Ω be a bounded domain an let f:H+→Ω be bi-holomorphic.
Let r>0 and ω_r(t)=f(re^it), 0<t<π and l(ω_r)=length of ω_r.
Show that
(a)ω_r(0+)=a and ω_r(π-)=b for some a,b∈Γ if l(ω_r)<∞.
(b)There exist r_n↘0 such that l(ω_r_n)→0 as n→0.
Hint: Consider the fact that Area(f(D_r(0)∩H+))≦Area(Ω) for all
r>0, and use the integral form of Cauchy-Schwarz inequality.
(c)Choose a sequence r_n↘0 such that l(ω_r_n)→0 and set
ω_r_n(0+)=a_n, ω_r_n(π-)=b_n. Let Ω_n be the inside of the Jordan
curve ω_r_n∪σ_anbn. Then diam(Ω)→0.
以上兩題(雖然有很多小題)@@
請教各位了，希望附上過程，拜託><
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