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7 f(z) H f(z)dz = f(z), f (z)dz = 9 3

4 7 7.. 7. 7.. 3 f(z) D D z 7. f(z) = f () d (7.) z f(z) - =( z) = z = z Res(z) =lim!z ( z) z = f(z): (7.) 7. (7.) f(z)

7. 5 7.. f(z) z = a jz aj = r z = a r Ref(a) > Ref(a + re i ): (7.3) z a = re i ; dz =ire i d =i(za)d f (a) = Z df(a + re i ): (7.4) (7.3) f(z) f(z) z = a f(z) r Z jf(a)j > djf (a + re i )j (7.5) jf(a)j = Z df(a + re i ) Z djf(a + re i )j (7.6) jf (z)j m z = =f (z) j=f(z)j,

6 7 7.. 3 :f(z) jf(z)j 33 Liouville : f(z) z f(z) f (z) =c + c z + c z + (7.7) 7.3 6 z 6= z! jc n z n j <M (7.8) jc n j < M jz n j : (7.9) c n = (n 6= ) (7.) c

7. 7 z ;z ( 7.) f(z j )= f () z j d ; (j =; ): (7.) f (z ) f(z ) = = z z n o d z z f () ( z )( z ) d: (7.) R ( = Re i ; d =ire i d) jf(z ) f (z )j jz z j = jz z j Z Z M R jf(z)j <M: R! z ;z dire i f (Re i ) (Re i z )(Re i z ) d j(e i z =R)(e i z =R)j : (7.3) jf(z ) f (z )j!: (7.4) f (z )=f(z )= : (7.5) 34 d'alembert : f(z) =a z n + a z n + + a n z + a n =; (n ;a 6=) (7.6) n f(z) g(z) = f(z) (7.7)

8 7 7.3 49. g(z) f(z) z f(z) =(zz )h(z) (7.8) h(z) n z h(z) 49 Poisson u(r; ) (x = r cos ;y = r sin ) r<r u(r; ) = Z R r u(r; ) R Rr cos( ) +r d (7.9) f(z) f(z) = f () d (7.) z R z = re i ; = Re i (r <R) Z f(re i )= d f(rei )Re i : Re i (7.) rei

7. 9 z 7.3) z =(R =r)e i z = Z z d = (7.)(7.) d f(rei )Re i : Re i (7.) (R =r)ei Z f(re i ) = df (Re i Re i i )( Re i re re i re i Re i ) Z = df (Re i R r ) (7.3) R Rr cos( ) +r (7.)(7.) Z f(re i ) = = Z df(re i )( Re i Re i re i + f(z) =u(z) +iv(z) Z v(r; ) =v + f (a) = i re re i Re i ) df(re i R + r Rre i() ) (7.4) R Rr cos( ) +r Rr sin( ) u(r; ) R Rr cos( ) +r d (7.5) Z v = v(r; )d = v() (7.6) Z jzaj=r dz f (z) z a = Z df(a + Re i ) a =

3 7 7. 7.. f(z) n 35 Goursat : f (z) D D z z D f (n) (z) = n! ( z) d: (7.7) f f(z +z) f(z) (z) = lim z! z = lim z! = ( z f () ( z) f () z z z )d d: (7.8) n = n n f (n) (z f () +z) f (n) (z) (z) = lim z! z = n! = n! = lim z! (n +)! ( z) ( z z) f ()d z ( z z) ( z) (n + )( z) n f ()d ( z) () ( z) n+ d: (7.9) f(z) f(z)

7. 3 7.4. 4 f(z) D f(z) D D jzj r jzj = r jf ()j <M jf (n) (z)j = n! n! jzj=r jzj=r ( d z) jj jdj j zj n! Z M dr r = n!m r n (7.3) 5 : f(z) jz z jr jz z j = r jf(z)j <M jf (n) (z )j n!m r n (7.3) 36 Morera : f (z) D D 7.4 f(z) D f(z)dz = (7.3)

3 7 7.5 5 R. F R z (z) = z d 7.4 F (z) z F (z) =f(z): (7.33) F (z) f(z) =F (z) 7.. 5 Z = dx ; (n >) (7.34) ( + x ) R R R 7.5 R dz ( + z ) = dz (z i) (z +i) f(z) ==(z +i) R dz = ( + z ) n! f (n) (z =i)= (n)! : (7.35) ( n n!) : (7.36)

7.3 33 R! lim R! R dz ( + z ) = Z dx : (7.37) ( + x ) Z dx ( + x ) 3 5 (n ) = 4 6 (n) (n )!! = : (7.38) (n)!! 5 Z = cos n d: (7.39) e i = z; Z = ( ei +e i ) n d = i n jzj= ( + z ) n z dz (7.4) d = idz=z f (z) =(+z ) n = i f (n) n () = i (n)! (n )!! = (n)!! (n)! n (n!) (7.4) 7.3 7.3. f(z) D D z a z a D j aj > jz aj 7.6 =( z) z = ( a) (z a) = a n= z a ( a )n (7.4)

34 7 7.6. jz aj < j aj f(z) = = z d = n= f(z) = (z a) n n= f (n) (a) n! n= ( a) (z a) n f ()d ( a) d: (7.43) (z a) n (7.44) 7.43 7.44 D r z (Taylor) r 5 log( + z) log( + z) = n= n zn () n (7.45)

7.3 35 7.3. a k k D f(z) ; ; j aj > jz aj > j aj (7.46) f(z) f (z) = f( ) z d + f( ) z d (7.47) z z = = ( a) (z a) = a ( a) (z a) = z a n= z a ( a )n (7.48) n= a ( z a )n (7.49) f(z) = + n= n= (z a) n (z a) n ( a) d ( a) n d (7.5) D f (z) ; f (z) = = n= + (z a) n n= n= (z a) n ( a) d ( a) n d c n (z a) n (7.5) c n c n = d ( <n<) (7.5) ( a)

36 7 7.7. Laurent 53 54 n< 7.5 z = a z = a (z a) n z = a n 7.5 z = a z = a n! z = a z = a Resf (a) = f (z)dz = c (7.53) jzaj= 53 z = jzj < f (z) = z f (z) = ( z=) = jzj > f(z) = z( =z) = z n= (7.54) ( z )n (7.55) n= ( z )n (7.56)

7.3 37 jzj > z = 7.56 z = z = 54 e =z (7.57) z = e =z =+! z +! ( z ) + + n! ( z )n + (7.58) z =

38 7 7.4 7. Z sin n d. () sin z (z =) () (z =i) ( + z ) n t (3) exp (z o z ) (z =): (4) log( + z) (z =) (5) e z z (z =) (6) tan z (z =) 3. jzj R f (z) f(z) =u +iv;f(z) =a + a z + a z + a = Z u(re i )d 4. P n (z) a n = Z u(re i ) P n (z) = d n n n! dz n [(z ) n ] P n (z) = ( ) n n ( z) d z 5. L n (z) L n (z) =e z dn dz n (zn e z ) L n (z) = n! n e (z) d ( z) z (Re i ) n d (n )

7.4 7 39 6. (z )(z ) z () < jzj < () < jz j < (3) jzj < (4) < jzj

4 7

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