, ïzn+m-znï<e Þïan+m-anï, ïbn+m-bnï£ïzn+m-znï<e, {an} {bn}, . {zn}. n
1.3. , ,
,.
. "A>0 $N(A): ïznï>A "n>N(A), {zn} .
. ) zn=zn |z|>1; ) zn= i n.
, , $z¥=¥ zn=¥.
. . {zn} ,
{xn=1/ zn}0. : 1/¥=0,. 1/0=¥, z¥=¥, z¹0, z+¥=¥, z/¥=0, z¹¥. 0/0 ¥/¥ .
C, .
R.
2. .
E , "zÎE w: zw, , E f(z)=w. E- f(z); M w- f(z). f(z) () EM.
. ) w=az+b (, ),
)w=zn, ) w=1/z ( , ).
E M . , E M- .
, (x,y).
. g Z , :
1) zÎg g.
2) z1, z2 Îg , zÎg.
. ) |z|<1 - ; ) |z|£1- ; ) {z: |z|<1}È{z: |z-5i|<1} ;
.
. z0 g, $ e- z0 : ïz-z0ï<e g.
. ) z=0 - |z|<1; ) z=i - |z|£1.
,
1) , g- .
2) , g- .
, - .
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. z0 g, " e- zÎg, zÏg.
. ) z=0 - |z|>0; ) z=i - |z|£1.
g g. (: g, C, G, S ..)
, (, , |z|>0).
. g, g g `g=g+g.
|z|£1 - .
.
, , w=f(z) g g D w.
( .
z1, z2 Îg z1¹ z2: f(z1)=w1¹w2= f(z2), gD.
g f(z) f(z) g.
. ) w=const, w=az+b - . : k , a, b. , ;
) w=zn - , . w=z2
, z= 0 . 2 . Imz>0 w , Rez>0 w .
) w= - .
. z ε-, z , ε-, , z .
,
.
) w=1/z - , . : .
) ,
,
, :
w=1. .
) w=ez- , .
) w=Ln zº ln|z|+i Arg(z), .
) w=za
, . :
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.
)
)
)
. +, .. .
)
gD D $ z=j(w), Dg.
gD , , , .
.
z=x+iy, f(z)=w=u+iv=u(x,y)+iv(x,y). f(z) g z g (x,y). .
3. .
1. ( ) z0Îg.
1. ( ) w0 f(z), zÎg, z0Îg, "{zn}z0 {f(zn)}w0.
. , z0 ( g.
. z0Îg ( ) g, " e- z0 g, z0.
2. ( ) w0 f(z), zÎg, z0Îg, "e>0 $d(e,z0)>0: ½f(z)-w0½<e, 0<½z-z0½<d
: f(z)= w0.
. z0 w0 .
, .
. ()
1) 21 (). f(z) 2. "e>0 d(e)>0. {zn}z0 N[d(e)]=N(e): "n³ N(e) 0<½zn-z0½<d. 2. 0<½f(zn)-w0½<e "n³ N(e). .. e>0- {zn}z0-, , {f(zn)}w0, .. 1.
2) 12 (). : 1, 2- . , $e0>0, "dn>0 $znÎg, 0<½zn-z0½<dn, ½f(z)-w0½>e0. {dn}0 {zn}, . , ${zn}z0, {f(zn)} w0. .. 1.- . . . .. 12. n
f(z) z0. f(z), zÎg, z0Îg, $ :
f(z)= w0 w0= f(z0), .. f(z)= f(z0).
, d- z0 f(z) e- w0= f(z0).
e-d. f(z), zÎg, z0Îg, "e>0 $d(e,z0)>0: "z: ½z-z0½<d; ½f(z)-f(z0)½<e.
1. , .
. z0 g, $ e-, g.
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2. z0Îg.
3. f(z), zÎg, z0Îg z0=¥.
f(z) z¥ {f(zn)}, {zn}- " .
e-d f(z) z¥ ½z-z0½<d ½z½>R.
: ) w=az+b, w=z*, w=const, w=Re z, w=zn, w=|z| - .
) w=arg(z) , z=0, z=¥, , .
. f(z), zÎg, g, "zÎg.
: f(z)ÎC(g).
f(z)ÎC(`g), f(z)ÎC(g). zÎ`g zÎg {zn}, znÎ`g znÎg.
4. f(z)ÎC(g) e d (e,z) (d=d(e,z)), .. e- " w=f(z)ÎD d- z, d z- .
f (z)= u (x, y)+ iv (x, y), u (x, y), v (x, y)- .
3.1. f (z)Î C (g) , u (x, y) v(x, y) g (x, y) .
3.2. f(z) h(z) Î C (g). f(z)h(z) Î C (g), f(z)*h(z) Î C (g), f(z)/h(z) Î C (g), h(z)≠0