f(z)ÎC¥(g) 0 (zi¹zk, znÎg f(zn)=0), ( ) z*Îg ( zn=z*Îg). f(z)º0, "zÎg.
. f(z*)=0 => .. , f(z)= cn(z-z*)n, |z-z*|<r(z*) ( z* )=> c0=0, f(z)=(z-z*)f1(z); f1(z)= cn(z-z*)n-1; , f(zn)=0=>f1(zn)=0=> f1(z*)=0 => c1=0 => cn=0 "n. , .. 0, f(z)º0 |z-z*| <r(z*), r(z*) , z* g. f(z)º0 g .
, f(z**)=0, z**Îg - , |z-z*| <r(z*). z* z** L, g g d>0. " |z-z*|<r(z*), g f(z), z=z1 L |z-z*|=r(z*), , f(z)º0 |z-z1|<r(z1), r(z1)³d. , L , d, f(z)º0. z=z** => f(z**)º0. z** => f(z)º0 g.n
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1. f(z)ÎC¥(g) f(z)¹0 g - .
2. f(z)ÎC¥(g) f(z)¹0 g, " `g'Ìg f(z).
`g'- , , zÎ`g' =>f(z)º0 g, . n
3. f(z)- , " `g' f(z). , .
.3. .
. f1(z) f2(z)ÎC¥(g) $ {zn}z*Îg, zi¹zk f1(zn)=f2(zn), f1(z)ºf2(z) "zÎg.
, h(z)=f1(z)-f2(z) º0 g.
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g ,
a) {zn}z*Îg, zi¹zk
b) xÎCÌg, C- - .
c) zÎ`g'Ìg
: g a), b), c).
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. - . f(zn) f(C) f(`g')!!!