IntroCA Book

&

pi

22 2015

i

1 pipi 1

1.1 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 pi pipi 11

2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 ( ) 15

3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 CauchyRiemann . . . . . . . . . . . . . 16

4 37

4.1 - pi . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 pi Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.6 Cauchy Liouville . . . . . . . . . . . . . . . . . . . . . 84

4.7 - - . . . . . . . . . . 94

4.8 - - Schwarz . . . . . . . . . . . . . . . . . . 106

5 Laurent- - pipi 119

5.1 Laurent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2 . . . . . . . . . . . . . . . . . . 127

iii

iv

5.3 pipi . . . . . . . . . . . . . . . . . . . . . . . 137

5.4 pipi . . . . . . . . . . . . . . . . . . . . . . 137

5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

169

.1 pi . . . . . . . . . . . . . . . 169

.1.1 pi ( ) . . . . . . . . . . . . . . . . . . . . . . . 169

.2 Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

.3 Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

181

R pi

R+ pi

Z

N

Q

n N,n! = 1 2 3 n,(2n)!! = 2 4 6 (2n 2) (2n) (2n+ 1)!! = 1 3 5 (2n 1) (2n+ 1).

x R, [x], k Z k x < k + 1.

C pipi

C = C {} pi pipi

ii

D (z0, r) z0 C r > 0 ( D (z0, r) pi z0 C)

D (z0, r) z0 C r > 0

D(z0, r) := {z C : 0 < |z z0| < r} z0 C r > 0

C (z0, r) z0 C r > 0

C+ (z0, r) z0 C, r > 0

C (z0, r) z0 C, r > 0

A pi C,

z0 A A, pi pi D (z0, ) z0 D (z0, ) A,

z0 C (.) A, D (z0, ) (A \ {z0}) 6= , pi D (z0, ) z0.

f A C, A 6= , , pi M > 0 |f(z)| M z A.

f pi f

f (k) k-pi f

exp

log

Log pi

pi

pi pi

f (z) dz pipi f pi pi

iii

f (z) dz pipi f pi pi f (z) dz pipi f pi pi

f (z) dz pipi f pi pi

n (, z0) I (, z0) Ind (z0) pi pi z0 /

Res (f, z0) pipi f z0

Euler

Riemann

1

pipi

1.1 pi

1.1. n-

1, , 2, . . . , n1 , pi = e2pini = cos

2pi

n+ i sin

2pi

n.

n- pi {z C : |z| = 1} -

piPn.

pi || = 1, pi Pn

`n =

n1k=0

|k+1 k| =n1k=0

||k| 1| =n1k=0

|1 | = n|1 | .

|1 |2 =(1 cos 2pin

) i sin 2pi

n

2 = 2 2 cos 2pin = 4 sin2 pin1

2 1.

pi

`n = 2n sinpi

n.

,

limn `n = limn 2n sin

pi

n

= 2pi limn

sin(pi/n)

pi/n

= 2pi limx0

sinx

x= 2pi . ( )

1.2. z1, z2, . . . , zn , pi

|z1 + z2 + + zn| |z1|+ |z2|+ + |zn| . (1.1)

(1.1) z1, z2, . . . , zn pi

.

pi. pi pi z1, z2, . . . , zn .

z1 = |z1|ei1 z2 = |z2|ei2 , . . . , zn = |zn|ein pi 1, 2, . . . , n R. z1 + z2 + + zn C. z1 + z2 + + zn = 0, (1.1) pi . z1 + z2 + + zn 6= 0.

z1 + z2 + + zn = |z1 + z2 + + zn|ei , pi R .

pi

|z1 + z2 + + zn| = ei(z1 + z2 + + zn)= 1. f(0) = 0,

2pi0

u(cos t, sin t)4 dt 36 2pi

0v(cos t, sin t)4 dt

2pi0

v(cos t, sin t)4 dt 36 2pi

0u(cos t, sin t)4 dt .

pi. pi Cauchy f4 z0 = 0.

11. p N.

(a)1

2pii

|z|=1

(z 1

z

)p dzz

=1

2pii

|z|=1

(z2 1)pzp+1

dz =

(1)n(2nn ) p = 2n0 p = 2n 1

4.3. CAUCHY 65

(b)1

2pii

|z|=1

(z +

1

z

)p dzz

=1

2pii

|z|=1

(z2 + 1)p

zp+1dz =

(

2nn

) p = 2n

0 p = 2n 1 .

. pi (a) (b),

2pi0

sinp t dt =

2pi0

cosp t dt =

2pi4n

(2nn

) p = 2n

0 p = 2n 1 .

. pi pi Stirling

Lebesgue,

limn

2pi0

sin2n t dt = limn

2pi0

cos2n t dt = 0 .

12. f pi |z| = 1,

|z|=1z f(z) dz = pii f (0) .

13. f pi |z| = 1 a C.

1

2pii

|z|=1

f(z)

z a dz =

f(0) |a| < 1

f(0) f(1/a) |a| > 1 .

14. f pi pi

D(0, R) = {z C : |z| R}, z, |z| < R,

f(z) =1

2pi

2pi0

[Reit

Reit z +z

Reit z]f(Reit) dt

=1

2pi

2pi0

R2 |z|2|Reit z|2 f(Re

it) dt .

15. f D(0, 1). 0 < r < 1,

pi

1

2pii

|z|=r

f(z) f(z)z2

dz .

66 4.

d = supz,wD(0,1)

|f(z) f(w)|

pi f ,

|f (0)| d2.

16. f pi G pi,

pi : [, ] C pi pi z = a.

() n N f(a)n =

1

2pii

f(z)n

z a dz .

() M = max{|f(z)| : z = ([, ])}, ` pi d pi a pi pi ,

|f(a)|n `Mn

2pid.

() pi () |f(a)| M . |f | pi G pi pi (pipipi ,

4.64).

17. f pi G pi pi D(z0, R) =

{z C : |z z0| R}. |f(z)| M |z z0| = R, z1, z2

{z C : |z z0| R2

}

|f(z1) f(z2)| 4MR|z1 z2| .

18. ( Riemann) pi pi

f = u+ iv pi G pi . -

pi Green u dv =

(uvx dx+ uvy dy)

=

G

[(u

x

)2+

(u

y

)2]dxdy =

G

|f (z)|2 dxdy > 0 .

pi u dv f G.

4.4. 67

4.4

4.24. f D C. F f(z) D C,

(1) F D

(2) exp(F (z)) = f(z), eF (z) = f(z) z D.

pi F (z) = logD f(z) F (z) = log f(z)( -

D).

F f(z),

G(z) = F (z) + 2kpii , k Z ,

f(z).

F z D C,

(1) F D

(2) exp(F (z)) = z, eF (z) = z z D.

pi F (z) = logD z.

F z,

G(z) = F (z) + 2kpii , k Z ,

z.

pi pi F pi pi (2),

F (z) = ln |f(z)|+ i arg(f(z)) . (4.6)

arg(f(z)) .

arg(f(z)), pi (4.6) pi (

68 4.

) D. D pi pi f D,

pi f(z) D.

4.25 ( ). pi

f : D C pi pi D C f(z) 6= 0 z D. pi z0 D pi log f(z0)(, pi c0 C ec0 = f(z0)). F : D C

F (z) :=

zz0

f ()f()

d + log f(z0) . (4.7)

F f(z), F

D eF (z) = f(z) z D. pipi,

F (z) =f (z)f(z)

, z D .

pi. pi f /f pi pi D, pi -

4.19 pipi f /f

pi pi 4.18 F D

F (z) =f (z)f(z)

, z D .

G(z) := f(z)eF (z) , z D .

G D

G(z) = f (z)eF (z) f(z)F (z)eF (z) = f (z)eF (z) f (z)eF (z) = 0

z D. pi pi 3.17 G D. ,

G(z) = G(z0) = f(z0)eF (z0) = f(z0)e log f(z0) = 1 .

eF (z) = f(z) z D ,

F (z) = logD f(z).

4.4. 69

4.26. n N f : D C pi pi D C f(z) 6= 0 z D. pi g D

f(z) = g(z)n , z D .

pi n- f D.

pi. n N. F f(z),

g(z) := exp

(1

nF (z)

), z D .

g(z)n = exp(F (z)) = f(z) , z D .

4.27. pi f, g : C C pi

f2(z) + g2(z) = 1 , z C.

pi h

f(z) = cos(h(z)) g(z) = sin(h(z)) .

. pi pi

[f(z) + ig(z)] [f(z) ig(z)] = 1 , z C.

pi f+ig C, pi 4.25 pi

F f(z) + ig(z) = exp(F (z)). h := iF , h

f(z) + ig(z) = eih(z) .

f(z) ig(z) = 1f(z) + ig(z)

= eih(z)

70 4.

f(z) =eih(z) + eih(z)

2= cos(h(z)) g(z) = e

ih(z) eih(z)2i

= sin(h(z)) .

pi 4.25 pipi pi f(z) =

z, z D.

4.28 ( log z). pi D C pi pi 0 / D. pi z0 D pi log z0(,pi w0 C ew0 = z0). F : D C

F (z) :=

zz0

d

+ log z0 . (4.8)

F log z

F (z) =1

z, z D .

. pi pi D = C \ (, 0] z0 = 1 D. pi log 1 = 0 e0 = 1( e2kpii = 1, k Z).

F (z) :=

z1

d

+ log 1 =

z1

d

log z. C (0, |z|) |z| pi z.

4.4. 71

,

F (z) =

[1, |z|]

d

+

d

=

|z|1

dx

x+

0

i|z|eit|z|eit dt ((t) = |z|e

it)

= ln |z|+ i = ln |z|+ iArg z ,

pi pi < = Arg z < pi. pi

F (z) = ln |z|+ iArg z , z C \ (, 0] ,

F (z) = Log z. pi pi() .

, N0 = {rei0 : r 0}, 0 [pi, pi) pi pi D := C \N0 . pi 4.28

log z = ln |z|+ i arg z , 0 < arg z < 0 + 2pi ,

z D. pipi (i) 0 = 0 (ii) 0 6= 0.

4.29. pi ( C) f1, f2

. pi 1 2

f1(z)1(z) + f2(z)2(z) = 1 , z C .

pi. pi pi g := f1 f2 C. pipi 4.25 pi F C

g(z) = eF (z) f1(z) f2(z) = eF (z) z C .

,

f1(z)1(z) + f2(z)2(z) = 1 z C ,

pi 1(z) := eF (z) 2(z) := eF (z) .

4.30. pi a, b C , pi 2pi

0ln |a+ bei| d = 2pimax {ln |a|, ln |b|} . (4.9)

72 4.

pi. 1 pipi: |a| > |b|. f(z) := a + bz pi pi D pi pi C(0, 1)( a + bz = 0

z = a/b. |z| = | a/b| > 1). pi 4.25 pi f , log f(z). pi pi C+(0, 1)

z = ei, 0 2pi, 2pi

0ln |a+ bei| d = 0, N N pi z, > 0, N N z D.

pi fn f D pi fn f D.

fn f := sup{|fn(z) f(z)| : z D} ,

pi fn f D fn f 0.

4.32. 1. fn(z) = zn

D(0, 1). zn 0. pi

sup{|zn 0| : |z| < 1} = 1 9 0 n

4.5. 75

sup{|zn 0| : |z| 1 } = (1 )n n 0 , (0, 1) ,

zn 0 . zn 0 D(0, 1 ).

2. fn(z) = (1 + n2z2)1 D(0, 1). z, |z| < 1,

fn(z) =1/n2

1/n2 + z2 f(z) =

0 z 6= 0

1 z = 0 .

pi

fn f = sup{|fn(z) f(z)| : |z| < 1}

fn( 1n

) f

(1

n

)=

11 + n2 (1/n)2 0 = 12 ,

fn f = sup{

1

|1 + n2z2| : |z| < 1}9 0 n

pi fn 0 .

pi fn D(0, 1), f

0.

limz0

limn fn(z) = 0 limn limz0 fn(z) = 1 .

pi pi pi pipi pipi pi

fn f .

4.33 ( ). (fn) -

D C fn f D. fn, n N, z0 D, f z0

limn limzz0

fn(z) = limzz0

limn fn(z) = f(z0) .

78 4.

D(0, r) r < 1.

. pi

1 zn+1 = (1 z)(1 + z + z2 + + zn) 11 z (1 + z + z

2 + + zn) = zn+1

1 z pi 11 z

nk=0

zk

= |z|n+1|1 z| rn+11 r .pi 0 < r < 1,

limn

rn+1

1 r

1

1 z =n=0

zn .

pi z z, 1

1 + z=n=0

(1)nzn .

pi pi |z| < 1. Sn1(z) =n1k=0 z

k,

sup|z|

4.5. 79

4.37. : [a, b] G pi G C (fn) G.

(1) fn f = ([a, b]),

limn

fn(z) dz =

( limn fn(z)) dz =

f(z) dz .

(2)

n=1 fn(z) = ([a, b]),

n=1

fn(z) dz =

( n=1

fn(z)

)dz .

pi. pi (1). > 0. pi N N n N |fN (z) f(z)| < z = ([a, b]). pi

(fn(z) f(z)) dz

|fn(z) f(z)||dz|

|dz|

= `() ,

pi `() pi .

limn

(fn(z) f(z)) dz = 0 lim

n

fn(z) dz =

f(z) dz .

4.38 (CauchyTaylor). f pi G z0 G.pi D(z0, ) pi G(pi pi

pi G: =, D(z0, ) = G = C). z D(z0, )

f(z) =n=0

an(z z0)n =n=0

f (n)(z0)

n!(z z0)n . (4.10)

pipi, pipi r 0 < r < pi pi

an =f (n)(z0)

n!=

1

2pii

C

f()

( z0)n+1 d , (4.11)

80 4.

pi C = C(z0, r) z0 r.

(4.10) Taylor f pi z0. pi

D(z0, ) pi pi.

pi. z D(z0, ). pi |z z0| = < r < .

pi pi Cauchy

f(z) =1

2pii

C

f()

z d .

pi C z z0 z0 = r < 1 ,

1

z =1

( z0) (z z0) =1

( z0)(

1 zz0z0) = 1

z0n=0

(z z0 z0

)n( )

pi

f(z) =1

2pii

C

[ n=0

f()(z z0)n

( z0)n+1]d .

M = max{|f()| : C}, f() (z z0)n( z0)n+1 M 1r (r)n , C .

4.5. 81

pi (/r) < 1,

n=0(/r)n pi M - Weier-

strass n=0

f()(z z0)n

( z0)n+1 C .

, pi 4.37 pi Cauchy pi

f(z) =n=0

[1

2pii

C

f()

( z0)n+1 d]

(z z0)n =n=0

f (n)(z0)

n!(z z0)n .

. pi pi |z z0| = < r < . pipi pi |z z0| = r, r < , C1 = C1(z0, r1), pi r < r1 < pi C C1. pi

Cauchy

an =1

2pii

C1

f()

( z0)n+1 d =1

2pii

C

f()

( z0)n+1 d

pi pi pi (4.11).

4.39. pi Taylor

f(z) =z i

(z + 2)2

z0 = i. ;

.

f(z) =z i

((z i) + (2 + i))2 =z i

(2 + i)2(

1 + zi2+i)2 .

1/(1 + w) =

n=0(1)nwn, |w| < 1, pi

1(1 + w)2

=n=1

(1)nnwn1 1(1 + w)2

=n=1

(1)n+1nwn1 , |w| < 1 .

pi z i2 + i < 1 |z i| < |2 + i| = 22 + 12 = 5 ,

f(z) =z i

(2 + i)2

n=1

(1)n+1n(z i2 + i

)n1=n=1

(1)n+1 n(2 + i)n+1

(z i)n , |z i| 1 |z| R > 1.

4.

f(z) =cos z 1z 2 +

1

z + 3.

Taylor f i.

84 4.

5. pi Taylor f(z) = z2 cos2 3z pi z = 0.

;

6. pi Taylor

f(z) =z

(z + 1)2

z0 = 0. ;

7. pi Taylor

f(z) =z

z2 + 1

z0 = 1. ;

8. f(z) = 1/z D(0, 2) = {z C :0 < |z| < 2}. pi (pn) pi pi f pi pi D(0, 2).

pi. T pi pi pi D(0, 2).

9. f(z) =

n=0 anzn, z C, f(x) R

x R. f(z) = f(z), z C.

10. f(z) =

n=0 anzn, z C, f(R) R f(iR)

iR. f(z) = f(z), z C.

4.6 Cauchy Liouville

pi D(z0, R) z0 C, pipi pi z0

.

4.41 ( Cauchy1 pi). pi f :

D(z0, R) C D(z0, R) z0 C. |f(z)| M

4.6. CAUCHY LIOUVILLE 85

z D(z0, R),

|f (n)(z0)| n!MRn

, n N .

pi. 0 < r < R. pi pi Cauchy pi

f (n)(z0) =n!

2pii

|zz0|=r

f(z)

(z z0)n+1 dz .

pi pi |f(z)| M z C(z0, r) pi

|f (n)(z0)| = n!2pii

|zz0|=r

f(z)

(z z0)n+1 dz

n!2pi

|zz0|=r

|f(z)||z z0|n+1 |dz|

n!2pi

|zz0|=r

M

rn+1|dz|

=n!M

2pirn+1

|zz0|=r

|dz|

=n!M

2pirn+12pir =

n!M

rn.

|f (n)(z0)| n!Mrn

, 0 < r < R .

pi pipi pi r, pi

r R

|f (n)(z0)| n!MRn

, n N .

pi f pi C(z0, R)

z0 C |f(z)| M z C(z0, R). pi pi Cauchy pi C(z0, R) pi

pi 4.41, pi pipi Cauchy

|f (n)(z0)| n!MRn

, n N .

86 4.

4.42 ( Cauchy2 pi). f pi

C(z0, R) z0 C |f(z)| M z C(z0, R).

|f (n)(z0)| n!MRn

, n N .

4.43. f : C C

|f(z)| |z|2 + |z|3 , z C . (4.13)

f(z) = a2z2 + a3z

3

|a2| 1 |a3| 1.

. f : C C ( C) pi

f(z) =n=0

anzn =

n=0

f (n)(0)

n!zn = f(0) +

f (0)1!

z + + f(n)(0)

n!zn + , z C.

C(0, R) = {z C : |z| = R} 0 R > 0. M = max|z|=R |f(z)|,pi (4.13)

M R2 +R3

pi Cauchy n > 3

|an| =f (n)(0)

n! MRn R

2 +R3

RnR

0 .

pi an = 0, n > 3. f pi pi 3,

f(z) = a0 + a1z + a2z2 + a3z

3. pi (4.13) f(0) = 0 pi a0 = 0 pi

f(z) = a1z + a2z2 + a3z

3. pi

|a1| = |f (0)| MR R

2 +R3

R= R+R2

R00 ,

pi a1 = 0.

f(z) = a2z2 + a3z

3 .


pi pi Cauchy

|a2| =f (2)(0)

2! MR2 R

2 +R3

R2= 1 +R

R01

|a3| =f (3)(0)

3! MR3 R

2 +R3

R3=

1

R+ 1

R1 .

4.44 ( Liouville). pi f : C C pi A,B > 0 k 0,

|f(z)| A+B|z|k , |z| R0 > 0 . (4.14)

f pi pi k.

pi. f : C C ( C) pi

f(z) =n=0

anzn =

n=0

f (n)(0)

n!zn = f(0) +

f (0)1!

z + + f(n)(0)

n!zn + , z C.

C(0, R) = {z C : |z| = R} 0 R > R0. M = max|z|=R |f(z)|,pi (4.14)

M A+BRk .

pi Cauchy n > k

|an| = |f(n)(0)|n!

MRn A+BR

k

Rn=

A

Rn+

B

RnkR

0 .

pi an = 0, n > k. f pi pi k.

k = 0 pi , |f(z)| M |z| R0 ( z C), pi M = A+B, f . Liouville.

88 4.

4.45 (Liouville). f : C C . f C, pi M > 0 |f(z)| M z C, f C.

, |f(z)| M z, |z| R > 0, f C.

pi Cauchy, 4.13 pi

. pi

pi Liouville.

4.46 ( ). pi

n 1 C. pi, pi n 1 n C

pi. p(z) = anzn + an1zn1 + + a1z + a0, an 6= 0, pi n 1.pi p(z) 6= 0 z C. pi p pi 1/p pi . pi 1.5 pi R 1

|p(z)| 12|an||z|n , |z| R .

pi, 1p(z) 2|an||z|n 2|an|Rn , |z| R .

, 1/p z, |z| R 1 pi Liouville pi 1/p , 1/p(z) = .

pi p(z) = 1/ , pi. pi pi pi

pi p . , p C.

4.47. f : C C M > 0.

( i)


( ii)

90 4.

g |g(z)| < 1/r z C. pi Liouville g pi f , pi. f(C)

pi C.

1. f a C, pi

|f (n)(a)| n!nn , n N;

pi .

2. f : C C

|f(z)| C|z|2pi , |z| > 1, pi C > 0 .

f pi pi 6.

3. f, g : C C pi R


6. pi f : C C

f(z + 1) = f(z) f(z + i) = f(z) , z C .

f .

7. f : C C C / R.

f(z + 1) = f(z) f(z + ) = f(z) , z C ,

f C.

8. f : C C

lim|z|

f(z)

zn= 0 , pi n N .

f pi pi n 1.

9. f : C C pi 0 pi

lim|z|

f (z)z

= 0 .

10. f : C C z

|f (z)| |z| .

f(z) = a+ bz2 , pi a, b C |b| 12.

11. f : C C

|f(z)| 2|z|+ |z|4 , z C .

f(z) = a1z + a2z2 + a3z

3 + a4z4

|a1| 2, |a2| 3, |a3| 32 3

4 |a4| 1.

92 4.

12. f : C C |f(z)| Aeax z = x + iy C,pi a,A > 0.

f(z) = ceaz , pi c C .

pi |f(z)| Aea|z|, z C;

13. f = C \ (, 0]. |f(z)| |Log z|, z , pi w = Log z pi() , f C.

14. pi f pi pi

|f(z)| A+B ln |z| , |z| 1 ,

pi A B ;

15. f

|f (z)| A+B|z| ,

pi A B .

16. f

|f(z)| A(1 +|z + i|) ,

pi A > 0.

17. f

|f(z)| M(1 + |z i|) ,

pi M > 0.

18. f

|f (z)| < |f(z)| , z C .


19. f(z) =

n=0 anzn D(0, 1)

|f(z)| 1 + |z|1 |z| , |z| < 1 .

|an| (2n+ 1)(

1 +1

n

)n, n N .

20. f(z) =

n=0 anzn D(0, 1) a0 6= 0

z0 f . r |z0| < r < 1 M(r) := max|z|=r |f(z)|.

|z0| r|a0|M(r) + |a0| .

21. f(z) =

n=0 anzn D(0, 1)

|f (z)| 11 |z| , |z| < 1 .

|an| < e, n N.pi.

f (z) =n=1

nanzn1 , pi nan =

1

2pii

|z|=r

f (z)zn

dz, 0 < r < 1 .

22. f : C C ,

f(z) =n=0

anzn =

n=0

f (n)(0)

n!zn , z C .

pi pi M > 0 |f(z)| Me|z| z C. r > 0,pi pi Cauchy pi

|an| M er

rn, n N .

pi

|an| M en

nn, n N .

23. f : C C pipi|f(rei)| d r20/3 ,

r > 0. f(z) = 0, z C.

94 4.

4.7 - - -

4.49. f : G C G z0 G f(z0) = 0. z0 k 1 f , pi g : G C

f(z) = (z z0)kg(z) g(z0) 6= 0 . (4.15)

4.50. f : G C G z0 G. z0 k 1 f ,

0 = f(z0) = f(z0) = = f (k1)(z0) f (k)(z0) 6= 0 . (4.16)

pi. z0 k 1 f , (4.15). pi (4.16).

, pi (4.16). pi f z0, pi pi

D(z0, R) G

f(z) =n=0

an(z z0)n =n=0

f (n)(z0)

n!(z z0)n , z D(z0, R) .

pi (4.16)

f(z) =

n=k

an(z z0)n = (z z0)kn=0

ak+n(z z0)n , ak 6= 0, z D(z0, R) . (4.17)

g : G C

g(z) =

f(z)

(zz0)k z G \ {z0} ,

ak z = z0 .

g z G \ {z0} pi g(z) =

n=0 ak+n(z z0)n, z D(z0, R), g z0.

f(z) = (z z0)kg(z) g(z0) = ak 6= 0 ,

pi g G.

4.7. - - 95

4.51. f : D(0, 1) C

f(z) =

z7

1cos z z 6= 0

0 z = 0 .

f

f(z) =z7

1(

1 z22! + z4

4! z6

6! + )

=z7

z2

2! z4

4! +z6

6!

=z5

12! z

2

4! +z4

6! = z5g(z) ,

pi

g(z) =1

12! z

2

4! +z4

6! D(0, 1) g(0) = 1/2 6= 0. 0 5 f .

z0 G pi f ,

f (n)(z0) = 0 n N {0} .

: Zf Z(f)

f : G C G.

Zf = {z G : f(z) = 0} .

Zf G. , (zk) Zf limk zk =

z G. pi f G, pi

0 = limk

f(zk) = f(z)

pi z Zf . Zf G.

96 4.

4.52 ( ). f : G C piG C. pi pi :

(1) f 0, f(z) = 0 z G.

(2) pi f pi . pi a G f (n)(a) = 0 n N {0}.

(3) pi f G pi Zf ,

f .

(4) Zf , f , (.) G.

(5) pi pi pi K G K Zf pipi .

(6) Zf pi, f G

pipi .

pi. (1) (6): f 0, Zf = G pi Zf pi., G pi pi pi

pi pipi .

(6) (5): pi (Kn) pi pi G Kn Kn+1 G =

n=1Kn.

Zf = G Zf =n=1

(Kn Zf ) .

pi Zf pi, pi n N Kn Zf pipi- ( pipi

). , pi pi pi K G

K Zf pipi .(5) (4): pi K Zf pi pi K. K Zf . K. pi Zf . K G, f . G.

4.7. - - 97

(4) (3): Zf G. a . Zf G, pi (zn) Zf limn zn = a. pi Zf

a Zf , a f . a f (zn) f a.

(3) (2): f pipi Zf . z0 Zf pipi f . pi m N

f(z) = (z z0)mg(z) , pi g g(z0) 6= 0 .

pi g g(z0) 6= 0, pi pi D(z0, r) z0 g(z0) 6= 0 z D(z0, r). f(z) 6= 0 z D(z0, r) \ {z0} pi z0 f Zf . pi pi pi a f pi

Zf a pi f G.

(2) (1): E =

{z G : f (n)(z) = 0 , n N {0}

}.

pi pi E 6= . pi f (n) , E G. pi E G.

, a E. pi f a, pi pi D(a, r) a

f(z) =n=0

f (n)(a)

n!(z a)n = 0 , z D(a, r).

pi D(a, r) E pi E G. G pi E 6= G G. E = G pi f(z) = 0 z G.

pi pi pi .

4.53. f : G C piG C. pi pi :

(1) f 6 0, f 0 G.

98 4.

(2) f ( pi) pipi .

(3) f G Zf ,

f .

(4) Zf , f , (.) G.

(5) pi pi K G K Zf pipi . pi pi K G pi pipi pi f .

(6) Zf pi .

Zf f . pi G, pi

. G G. pi G Zf pi,

Zf . G(pi 5).

pi

. , f R

f(x) =

e1/x2 x > 0

0 x 0 .

pi f pi pi R f (n)(0) = 0,

n N. pi 0 pi f . pi Zf f R pi.

pi 4.52 .

4.54 ( ). f, g : G C pi G C.

(1) f(z) = g(z) z G,

Zfg := {z G : f(z) = g(z)}

(.) G.

4.7. - - 99

(2) (zn) G limn zn = a G. f(zn) = g(zn) n N, f(z) = g(z) z G.

(3) pi X G . G. f(z) = g(z) z X, f(z) = g(z) z G.

(4) f(z) = g(z) . pi G, f(z) = g(z)

z G.

(5) f(z) = g(z) z D(a, r) G, f(z) = g(z) z G.

pi. (1) h(z) := f(z) g(z).

(2) pi pipi pi (1).

(3) pi X Zfg, pi pipi pi (1).

(4) pi [, ] G. pi . [, ] G, pi pipipi (3).

(5) f(z) = g(z) z D(a, r) G, f(z) = g(z) . pi pi D(a, r). pi pi (4) pi

f(z) = g(z) z G.

4.55. 4.54 (2)

pi G. f(z) = e1/(1z) D(0, 1) .

zn = 1 1/2npii, f(zn) = e

1/(1zn) = e2npii = 1 .

limn zn = 1 / D(0, 1).

pi . pi f g

, f = g f = g pi

pi .

100 4.

4.56. pi pi -

. pi,

f(z) = sin2 z + cos2 z , z C .

f f(x) = 1 sin2 x+ cos2 x = 1, x R. pi, pi 4.54 (4) pi sin2 z + cos2 z = 1, z C.

. pi

(R,+, ) pi (integral domain), , (

x, y , x y = 0 x = 0 y = 0).

4.57. G C pipi pi, H(G) -() G pi. f, g : G C pi G

f(z)g(z) = 0 , z G ,

f 0 g 0 G.

pi. g(z0) 6= 0 pi z0 G. pi g z0, pi piD(z0, r) G g(z) 6= 0 z D(z0, r). f(z) = 0 z D(z0, r) pi 4.54 (5) f 0 G.

4.58. f D(0, 1),

pi n N f

(1

n

)6= 1n+ 1

.

. pi

f

(1

n

)=

1

n+ 1=

1/n

1 + 1/n, n N .

g(z) := z1+z , f(1/n) = g(1/n) n N limn0 1/n = 0 D(0, 1). pi {z D(0, 1) : f(z) = g(z)} (.) D(0, 1) pi

4.7. - - 101

f(z) = z1+z z D(0, 1). f 1 D(0, 1), pi. pi n N

f

(1

n

)6= 1n+ 1

.

4.59. f D(0, 1)

|f(1/n)| 2n , n = 2, 3, 4, . . . .

f .

. pi pi

|f(0)| = limn |f(1/n)| limn 2

n = 0 f(0) = 0 .

0 pi f , pi f D(0, 1).

pi f . 0 m f ,

f(z) = zmg(z) ,

pi g 0 g(0) 6= 0. pi |g(1/n)| = nm|f(1/n)| nm2n,

|g(0)| = limn |g(1/n)| limnn

m2n = 0 .

g(0) = 0, pi. f D(0, 1).

4.60. pi:

f {0 < |z| < 2} f(1/n) = 0, n =1,2,3, . . ., f .pi: . pi f(z) = sin(pi/z). f

{0 < |z| < 2}, f(1/n) = sin(pin) = 0, n =1,2,3, . . .. f D(0, 2) = {z C : |z| < 2}, pi0 D(0, 2), pi f .

102 4.

4.61. f pi G = {z : |z| > a}. f pipi (a,) R, f pi pi (,a)

. pi

G = {z : z G} = {z : |z| > a} = {z : |z| > a} = G ,

pi 3.16 f(z) := f(z) G. pi f(x) = f(x)

x (a,), pi f(z) = f(z) = f(z) z G. x (,a) f(x) = f(x). , f(x) R x (,a).

4.62. f : C C lim|z| f(z) = , lim|z| |f(z)| =, f pi.

pi. pi lim|z| |f(z)| =, piM > 0 |f(z)| > 1 |z| > M .pi D(0,M) pi , pi 4.53 f pipi pi

D(0,M). 1, 2, . . . , N f D(0,M).

g : C C

g(z) :=f(z)

(z 1)(z 2) (z N ) .

pi |f(z)| > 1 |z| > M , g(z) 6= 0 z C pi

h(z) :=1

g(z)=

(z 1)(z 2) (z N )f(z)

. pi h(z) 6= 0 z C. pi z 6= 0h(z)

zN=

(1 1/z)(1 2/z) (1 N/z)f(z)

pi pi lim|z| |f(z)| =,

lim|z|

h(z)zN = 0 .

= 1. pi R > 0 h(z)zN < 1 |h(z)| < |z|N , |z| > R .

4.7. - - 103

pi pi Liouville h pi pi

N . pi h(z) 6= 0 z C, h(z) = c

f(z) =1

c(z 1)(z 2) (z N ) .

1. f pi U 0, f(0) = f (0) = 0 f (0) 6= 0. pi pi V 0, f(z) = (z)2

z V .

2. f : G C G z0 G n 1 f . pi pi D(z0, ) z0, f(z) = (z)n z D(z0, ) G. f n- pi z0.

3. f U z0 U

f pi z0. pi

k, pi V U z0 h V ,

f(z) = (z z0)keh(z) , z V .

4. f 1, 2, . . . , n, pi j mj , 1 j n. pi g

f(z) = (z 1)m1(z 2)m2 (z n)mneg(z) , z C .

5. f D(0, 1) \ {0}

f(z) = sin

(1

z

)sin

(1

1 z), 0 < |z| < 1 .

f .

6. f pipi pi , pi pi

p g f(z) = p(z)g(z) z C.

104 4.

7. f z0 C. f(z) 0 pi z0 pi r > 0 f(z) 6= 0 z : 0 < |zz0| < r.

8. f D(0, 1)

f(z) = sin

(pi

1 z).

zn = 1 1/n, f(zn) = 0 n N. 4.54;

9. f D(0, 1),

f

(1

n

)6= 1n+ 2

, pi n = 2, 3, . . . .

10. pi f D(0, 1)

f

(1

2n

)= f

(1

2n 1)

=1

n, n = 2, 3, . . . ;

11. pi a C pi f D(0, 1),

f

(1

n

)=

1

n+ a, n 2 ;

12. pi f D(0, 1) pi

pi pipi pi pi .

(i) f

(1

n

)=

1

n2 1 , n = 2, 3, . . . .

(ii) f

(1

n

)= (1)n 1

n, n = 2, 3, . . . .

(iii) f

(1

2n+ 1

)=

1

2n, n N .

(iv) |f (n)(0)| (n!)2 , n N .

13. f, g D(0, 1) f(z) 6= 0 g(z) 6= 0, |z| < 1. pi

f (

1n

)f(

1n

) = g ( 1n)g(

1n

) , n = 2, 3, . . . . pi c f(z) = cg(z) |z| < 1.pi. h(z) = f(z)g(z) D(0, 1).

4.7. - - 105

14. f f ( 1ln(n+ 2)) 1n , n N .

f 0.

15. (i) f pi D(0, 1)

f

(1

n

)= n2f

(1

n

)3, n = 2, 3, 4, . . . .

(ii) g pi D(0, 1)

g

(1

n

)= n4g

(1

n

)5, n = 2, 3, 4, . . . .

16. (ak)

k=0 |ak| . k=0

aknk = 0 , n N ,

ak = 0, k N {0}.

17. f : C C , f 6 0, f(x) R x R f(0) = 0. f pi C,

pi .

pi. f(z) := f(z) C.

18. f : C C pi pi pi =z = 0, pi , =z = pi. f pi 2pii,

f(z + 2pii) = f(z) , z C .

pi. f(z) := f(z) C.

19. f pi D(0, R) = {z C : |z| < R} 0 f (0) 6= 0. pi g pi 0

f(zn) = f(0) + (g(z))n .

106 4.

20. f, g : C C

|f(z)| (1 + |z|)|g(z)| , z C .

pi a, b C |a| 1 |b| 1

f(z) = (az + b)g(z) , z C .

21. f, g : C C . pi pi k N |z|, |f(z)| |zkg(z)|. pi R > 0

|f(z)| |zkg(z)| , |z| > R .

f(z) = h(z)g(z), z C, pi h ( h pi pi).

pi. g 6 0. z1, . . . , zm g D(0, R),

G(z) := f(z)

mn=1(z zn)g(z)

.

4.8 - - Schwarz

4.63 ( 1 ). f pi

G C |f | pi z0 G, f G.

pi. |f | pi z0 G, pi pi D(z0, ) G |f(z)| |f(z0)| z D(z0, ). r > 0 r < pi Gauss , 4.14,

f(z0) =1

2pi

2pi0

f(z0 + reit) dt .

4.8. - - SCHWARZ 107

pi C(z0, r) D(z0, ), pi pi |f(z0 + reit)| |f(z0)| t [0, 2pi] pi

|f(z0)| = 12pi

2pi0

f(z0 + reit) dt

1

2pi

2pi0|f(z0 + reit)| dt

12pi

2pi0|f(z0)| dt = |f(z0)| .

|f(z0)| = 12pi

2pi0|f(z0 + reit)| dt

1

2pi

2pi0

(|f(z0)| |f(z0 + reit)|) dt = 0 , |f(z0)| |f(z0 + reit)| 0 .pi pi |f(z0)| |f(z0 + reit)| = 0 |f(z0 + reit)| = |f(z0)| t [0, 2pi]. pi r > 0, 0 < r < , pi |f(z)| = |f(z0)| z D(z0, ). pi 3.18 f D(z0, ) pipi , 4.54 (5), f G.

G C pi, G pipi . f G, pi -

|f | pi G. f G, pi pi |f | pi pi pi G. pi |f | pi G G. pi pi pi .

4.64 ( 2 ). pi G ,

pi C, G pi. f

G, G , |f | pi G G .

108 4.

pi f pi G C , f G. 4.63 g = 1/f , pipi

pi pi.

4.65 ( 1 ). pi f

pi G C G. |f | pi z0 G, f G.

f pi G, G

G, pi pi |f | pi G G.

4.66 ( 2 ). G ,

pi C, G pi. pi f

G. f G, G , |f | pi G G .

pi pi pi

, pipipi 10.

4.67 ( ). pi

u : G R piG. u pi pi G.

pi. pi u pi z0 = x0 + iy0 G, pi pi D(z0, ) G z0 u(x, y) u(x0, y0) z = x + iy D(z0, ). pi D(z0, ) pi pi, pi f

D(z0, ) u =

4.8. - - SCHWARZ 109

z = x+ iy D(z0, )

|g(z)| = |e 0, pi pi

|f(z)| = |ei(x+ix)2 | = e2x2 x .

pi f(z) = eiz2

pi A.

4.69. R

pi pi pi pi. -

pi Weierstrass, pipipi [12] [33].

110 4.

f : [a, b] C , > 0 pi pi

p(t) =nk=0

aktk (n N , ak C)

|f(t) p(t)| < , t [a, b] .

f(t) = eit [0, 2pi], pi pi

Weierstrass pi pi p

|eit p(t)| < , t [0, 2pi] .

pi pi pi eit t. ,

f(z) = 1z T = {z C : |z| = 1} pi pi( ) pi C. = 1 pi pi pi

p(z) =n

k=0 akzk, ak C, 1z p(z)

< 1 , z T . q(z) := zp(z), pi q C

|1 q(z)| < 1 , z T .

pi pi pipi

|1 q(z)| < 1 , |z| 1 .

z = 0 pi

1 = |1 q(0)| < 1

pi pi. , pi > 0 pi p

1z p(z) , z T

|eit p(eit)| , t [0, 2pi] .

4.8. - - SCHWARZ 111

pi f(z) = 1z pi T pi

pi. f(z) = 1z pi T pi

pi, > 0 pi pi P

|eit P (eit)| < , t [0, 2pi] .

pi pi pi pi pi

StoneWeierstrass(pi [30], [33]).

4.70. f : T C > 0. pi N N (cn)

Nn=N C pi

P (z) =

Nn=N

cnzn

( P (eit) =

Nn=N

cneint

)

pi

|f(z) P (z)| < , z T ( |f(eit) P (eit)| < , t [0, 2pi]) .

pi piG C pi pi pi pi pi G, pipi

Runge, pipipi [27, 13.9 Theorem].

f pi pi G, pi pi K G > 0, pi pi p

|f(z) p(z)| < , z K.

Runge pi f(z) = 1z pi

pipi. T pi pi C f

C \ {0}, f 0.

4.71. f(z) = ez

z :12 |z| 1.

pi |f | pi pi maxz |f(z)|, minz |f(z)|.

112 4.

. pi f C \ {0} , pi / |f | pi pi : |z| = 12 |z| = 1. pi |z| = R, pi R = 12 R = 1, z() = Rei =R(cos + i sin ), 0 2pi. z

|f(z) = eR(cos +i sin )R(cos + i sin )

=eR cos eiR sin

R=eR cos

R.

pi |f | pi cos = 1 = 0 cos = 1 = pi. = 0 R = 1/2,

|f(1/2)| = e1/2

1/2= 2e 3, 3 .

= 0 R = 1, |f(1)| = e

1= e 2, 7 .

pi maxz |f(z)| = |f(1/2)| = 2e 3, 3.

= pi R = 1/2,

|f(1/2)| = e1/2

1/2=

2e 1, 2 .

= pi R = 1,

|f(1)| = e1

1=

1

e 0, 4 .

pi minz |f(z)| = |f(1)| = 1e 0, 4.

4.72. pi f : D(a, r) C D(a, r) |f | C(a, r) pi , |f(z)| = c z C(a, r). f D(a, r) f D(a, r).

pi. pi f D(a, r). pi

|f | pi D(a, r), C(a, r). pi |f(z)| = c C(a, r), |f(z)| = c z D(a, r). pi 3.18 f D(a, r).

4.8. - - SCHWARZ 113

4.73. f D(0, 1)

D(0, 1).

|f(z)|

2 |z| = 1, =z 0

3 |z| = 1, =z < 0 ,

|f(0)| 6.

. g(z) := f(z)f(z). A = {z C : |z| 1,=z 0} K = {z C : |z| 1,=z 0}, pi z A z K z K z A. pi g D(0, 1) D(0, 1).

|g(z)| = |f(z)||f(z)|

2 3 = 6 |z| = 1, =z 0

3 2 = 6 |z| = 1, =z < 0 , |g(z)| 6 |z| = 1. pi pi |g(z)| 6 z D(0, 1). |g(0)| = |f(0)||f(0)| 6 pi |f(0)|2 6. |f(0)| 6.

4.74. f D(0, 1).

f(z) = 1 z + z() = ei, 0 pi, f(z) = 1 z D(0, 1).

. F (z) := (f(z) 1)(f(z) 1) pi D(0, 1). z +, F (z) = 0. pi z , pi z() = ei, pi 2pi, z + pi pi F (z) = 0. pi,

114 4.

z C(0, 1) F (z) = 0 pi F (z) = 0

z D(0, 1). ,

(f(z) 1)(f(z) 1) = 0 , D(0, 1).

, pi 4.57 f(z) 1 0 f(z) 1 0 D(0, 1). pi f(z) 1 0 D(0, 1) f(z) 1 0 D(0, 1), f(z) 1 D(0, 1).

4.75. f pi G pi pi

D(0, 3) = {z C : |z| 3}. f(1) = f(i) = 0,

|f(0)| 180

max|z|=3

|f(z)| . (4.18)

f pi (4.18).

. 1,i f pi

f(z) = (z 1)(z + 1)(z i)(z + i)g(z) = (z4 1)g(z) ,

pi g G. pi

|g(0)| max|z|=3

|g(z)|

pi

|f(0)| = |g(0)| max|z|=3

|g(z)|

= max|z|=3

|f(z)||z4 1|

max|z|=3

|f(z)||z4| 1

=1

34 1 max|z|=3 |f(z)| =1

80max|z|=3

|f(z)| .

(4.18) pi |g(0)| = max|z|=3 |g(z)| pi pi g D(0, 3), g(z) = c. pi pi

g(z) = c z G pi f(z) = c(z4 1), z G. , f pi (4.18)

f(z) = c(z4 1), pi c C.

4.8. - - SCHWARZ 115

4.76. pi p(z) = zn + an1zn1 + + a0. p(z) zn pi w C(0, 1) |p(w)| > 1.

. pi pi w C(0, 1) |p(w)| > 1. |p(z)| 1 |z| = 1. p(z) zn. pi

q(z) := znp

(1

z

)= 1 + an1z + + a0zn .

pi |p(z)| 1 |z| = 1, |p(ei)| 1 [pi, pi] pi

max|z|=1

|q(z)| = max|z|=1

|p(1/z)| = max[pi,pi]

|p(ei)| 1 .

q(0) = 1 pi pi q pipi

. pi q(z) 1 . pi q(z) 1 z C pi an1 = = a0 = 0. p(z) zn.

1. f(z) = ez2 , pi 1 |z| 2. pi |f | pi pi min |f(z)|, max |f(z)|.

2. S = [0, 2pi] [0, 2pi] pi | sin z| pi pi maxzS | sin z|..

| sin z| = | sin(x+ iy)| = (sin2 x+ sinh2 y)1/2 .3. R 0, pi, i pi + i.

f(z) =

sin zz z 6= 0

1 z = 0 .

R pi |f | pi pi minzR |f(z)|, maxzR |f(z)|.

4. pi g(x, y) = (1 + 3x2y y3)2 + (3xy2 x3)2. /

116 4.

D(0, 1) = {z C : |z| 1},

maxx2+y21

g(x, y) = g

(

3

2,1

2

)= g(0,1) = 4

minx2+y21

g(x, y) = g

(3

2,1

2

)= g

(

3

2,1

2

)= g(0, 1) = 0 .

pi. f(z) = f(x + iy) = (1 + 3x2y y3) + i(3xy2 x3) |f(x+ iy)|2 = g(x, y).

5. f

|f(0)| max|z|=1

|f(z)| .

pi f ; pi .

6. a C n N, max|z|1

|a+ zn| .

7. pi f : 1 < |z| < 2 . |f(z)| 1 |z| = 1 |f(z)| 4 |z| = 2, |f(z)| |z|2

z .

8. f : G C C pi G f (z) 6= 0 z G. z0 G pi f(z0) 6= 0. D(z0, ) G pi z0, pi z1, z2 D(z0, )

|f(z1)| > |f(z0)| |f(z2)| < |f(z0)| .

9. f pi G pi pi D(0, 2) = {z C : |z| 2}. f(2) = f(2i) = 0,

|f(0)| 13

max|z|=3

|f(z)| . ()

f pi ().

4.8. - - SCHWARZ 117

10. ( ) ,

pi

p(z) = anzn + an1zn1 + + a1z + a0 , an 6= 0 ,

n 1 C.pi. pi pi 1.5,

pi r > 0

|p(z)| > |p(0)| = |a0| , |z| = r .

118 4.

5

Laurent- -

pipi

5.1 Laurent

pi Taylor pi .

pi Laurent pi pi

. pi f(z) = e1/z2 , z 6= 0.pi

ew =n=0

wn

n!, w C ,

w 1/z2 pi

e1/z2

= 1 z2 + 12z4 1

6z6 + .

z pi f(z) = e1/z2

, z 6= 0., f z0 C, pi f pi (z z0). pi pi( ) Laurent pi

pi Laurent 1843 1. pi 1Pierre Alphonse Laurent[18131854] Laurent.

119

120 5. LAURENT- -

pi

pi pipi.

Laurent.

5.1 (Laurent). f

= {z C : R1 < |z z0| < R2} , pi 0 R1 < R2 + .

f pi

f(z) =+

n=an(z z0)n

an =1

2pii

|zz0|=r

f(z)

(z z0)n+1 dz , n Z , (5.1)

pi |z z0| = r z0 r, R1 < r < R2. pipi , pi Laurent, pi

pi pi .

5.2. z0

f , f

D(z0, ) := {z C : 0 < |z z0| < }

5.1. LAURENT 121

D(z0, ). f pi z0.

5.3.

f(z) =1

z2 + 1=

1

(z i)(z + i) .

( pi) Laurent f

z0 = i.

. i f pi i, 1 = {z C : 0 < |z i| < 2} 2 = {z C : |z i| > 2}. pi f Laurent pi.

1 pi. pi

1

1 + w=n=0

(1)nwn , |w| < 1 .

1 pipi: 1 : 0 < |z i| < 2. ,

f(z) =1

(z i)[2i+ (z i)]=

1

2i(z i) [1 + zi2i ]=

1

2i(z i)n=0

(1)n(z i

2i

)n( zi

2i

< 1 |z i| < |2i| = 2)=

n=0

(1)n 1(2i)n+1

(z i)n1

=

n=1(1)n+1 1

(2i)n+2(z i)n .

122 5. LAURENT- -

2 pipi: 2 : |z i| > 2. ,

f(z) =1

(z i)[2i+ (z i)]=

1

(z i)2[1 + 2izi

]=

1

(z i)2n=0

(1)n(

2i

z i)n

( 2izi < 1 |z i| > |2i| = 2)

=n=0

(1)n(2i)n 1(z i)n+2

=n=2

(1)n2(2i)n2 1(z i)n =

2n=

(1)n(2i)n2(z i)n .

2 pi. pi Laurent.

1 pipi: 1 : 0 < |z i| < 2. |z i| = r1, 0 < r1 < 2, 1, pi pi .

(i) n 1 n+ 2 1. pi Laurent f(z) = n= an(z i)n

5.1. LAURENT 123

pi pi

an =1

2pii

|zi|=r1

f(z)

(z i)n+1 dz

=1

2pii

|zi|=r1

1

(z + i)(z i)n+2 dz

=1

(n+ 1)!

{(n+ 1)!

2pii

|z|=r1

1z+i

(z i)n+2 dz}

=1

(n+ 1)!

(1

z + i

)(n+1)z=i

( pi Cauchy)

=1

(n+ 1)!(1)n+1(n+ 1)! 1

(z + i)n+2

z=i

= (1)n+1 1(2i)n+2

.

(ii) n 2 (n+ 2) 0. pi Cauchy

an =1

2pii

|zi|=r1

f(z)

(z i)n+1 dz =1

2pii

|zi|=r1

(z i)(n+2)(z + i)

dz = 0 .

pi,

f(z) =

n=1

(1)n+1 1(2i)n+2

(z i)n , 0 < |z i| < 2 .

2 pipi: 2 : |z i| > 2. |z i| = r2, r2 > 2, 2 pi |z i| = r1, |z + i| = r3 pi |z i| = r2. pi pi .

(i) n 1 n + 2 1. pipi pi Laurent

124 5. LAURENT- -

pi pi

an =1

2pii

|zi|=r2

f(z)

(z i)n+1 dz

=1

2pii

|zi|=r1

f(z)

(z i)n+1 dz +1

2pii

|z+i|=r3

f(z)

(z i)n+1 dz ( Cauchy)

= (1)n+1 1(2i)n+2

+1

2pii

|z+i|=r3

1(zi)n+2z + i

dz (pi 1 pipi)

= (1)n+1 1(2i)n+2

+1

(z i)n+2z=i

(pi Cauchy)

= (1)n+1 1(2i)n+2

+1

(2i)n+2 = 0 .

(ii) n 2 (n+ 2) 0. ,

an =1

2pii

|zi|=r2

f(z)

(z i)n+1 dz

=1

2pii

|zi|=r2

(z i)(n+2)z + i

dz

= (z i)(n+2)z=i

(pi Cauchy)

= (1)n(2i)(n+2) .

,

f(z) =2

n=(1)n(2i)n2(z i)n , |z i| > 2 ,

5.4.

f(z) =1

z(z 1)(z 2) =1

2 1z

+1

1 z 1

2 1

2 z .

( pi) Laurent f z0 = 0

.

. 0, 1 2 f pi

0, 1 = {z C : 0 < |z| < 1}, 2 = {z C : 1 < |z| < 2} 3 = {z C : |z| > 2}. pi

1

1 w =n=0

wn , |w| < 1 .

5.1. LAURENT 125

1 pipi: 1 : 0 < |z| < 1. ,

f(z) =1

2 1z

+1

1 z 1

4 1

1 z2=

1

2 1z

+

n=0

zn 14

n=0

(z2

)n( z

2

< 1| |z| < 2)=

1

2z1 +

n=0

(1 2n2)zn .

2 pipi: 2 : 1 < |z| < 2.

f(z) =1

2 1z 1z 1

1 1z 1

4 1

1 z2=

1

2 1z 1z

n=0

(1

z

)n 1

4

n=0

(z2

)n(1z

< 1| |z| > 1 z2 < 1 |z| < 2)= 1

2 1zn=1

zn1 n=0

2n2zn = 2

n=zn 1

2z1

n=0

2n2zn .

3 pipi: 3 : |z| > 2. pipi

f(z) =1

2 1z 1z 1

1 1z+

1

2z 1

1 2z=

1

2 1z 1z

n=0

(1

z

)n+

1

2z

n=0

(2

z

)n(pi |z| > 2, 1z < 12 < 1. pi, 2z < 1 |z| > 2)

= n=2

1

zn+1+n=2

2n11

zn+1

=n=2

(2n1 1)zn1 =3

n=(2n2 1)zn .

5.5. pi Laurent

f(z) =1

1 z +1

(z2 + 4)2

z0 = 0 pi pi 1 i.

. f : 1 2i. pi piLaurent f pi 1 = {z C : 0 |z| < 1} (pi Taylor),

126 5. LAURENT- -

2 = {z C : 1 < |z| < 2} 3 = {z C : 2 < |z| < }. pi 1 i 2, pi f 2 = {z C : 1 < |z| < 2}. pi

1

1 w =n=0

wn 11 + w

=

n=0

(1)nwn , |w| < 1 .

, pi

1(1 + w)2

=n=1

(1)nnwn1 , |w| < 1

1

(1 + w)2=n=1

(1)n+1nwn1 =n=0

(1)n(n+ 1)wn , |w| < 1 .

pi,

f(z) =1

1 z +1

(z2 + 4)2

= 1z 1

1 1z+

1

42 1(

1 + z2

4

)2= 1

z

n=0

(1

z

)n+

1

42

n=0

(1)n(n+ 1)(z2

4

)n(1z

< 1 |z| > 1 z24 < 1 |z| < 2)=

n=0

1

zn+1+n=0

n+ 1

4n+2z2n =

1n=

zn +n=0

n+ 1

4n+2z2n .

pipi pi 2 = {z C : 1 < |z| < 2} pi - pi pi 1 i.

5.6. pi, pi 5.3, -

pi (5.1) pi pi an

pi Laurent f . pi pi

. pi Laurent pi

, pi. pi pi Lau-

rent f pi, pi pi f(z)

pi , pi Laurent f . pi pi

pi, pi pi

an pipi pi pi (5.1).

5.2. 127

5.2

pipi Laurent, 5.1, R1 = 0

R2 = R > 0. pipi f

D(z0, R) : 0 < |z z0| < R

( z0 f ) D(z0, R) pi

Laurent

f(z) =+

n=an(z z0)n (5.2)

an =1

2pii

|zz0|=r

f(z)

(z z0)n+1 dz , n Z , (5.3)

pi |z z0| = r z0 r, 0 < r < R.

pi (5.2)

f(z) =

1n=

an(z z0)n ++n=0

an(z z0)n .

D(z0, R). pi

1

n=an(z z0)n

pi Laurent.

z0 f pi pi

an n < 0 pi (5.2). z0

128 5. LAURENT- -

pi pi f an = 0 n < 0 ,

pi k(k 1) f ak 6= 0 an = 0 n < k ,

f an 6= 0 pi pi n .

pi 1, 2, 3, . . . pi, pi, pi, . . . pi.

5.7. f D(z0, R) : 0 < |z z0| < R( z0 f ). pipi f z0,

Res(f, z0) Resz=z0

f(z), a1 pi Laurent (5.2) f

D(z0, R). pi (5.3) pipi f z0 pi

pi

Res(f, z0) = a1 =1

2pii

|zz0|=r

f(z) dz , (5.4)

pi |z z0| = r z0 r, 0 < r < R.

5.8. U C , z0 U f U \ {z0}. pi :

(1) z0 pi f .

(2) limzz0 f(z) pi.

(3) pi M > 0 > 0 |f(z)| < M 0 < |z z0| < . f pi z0.

(4) limzz0(z z0)f(z) = 0.

pi. pi (1) (2) (3) (4).(4) (1):

f(z) =

+n=

an(z z0)n , 0 < |z z0| < R ,

5.2. 129

pi Laurent f , pi

an =1

2pii

|zz0|=r

f(z)

(z z0)n+1 dz , n Z

|z z0| = r z0 r, 0 < r < R, pi ( z0) U .

> 0. pi limzz0(z z0)f(z) = 0, pi > 0 z U 0 < |z z0| < |(z z0)f(z)| < . pi pi r > 0 r < min{1, } |z z0| = r

|f(z)| < |z z0| =

r.

n = 1, 2, . . .

|an| = 12pii

|zz0|=r

f(z)

(z z0)n+1 dz

12pi

|zz0|=r

|f(z)||z z0|n1 dz

12pi r rn1

|zz0|=r

|dz|

=1

2pirn22pir = rn1 .

pi > 0, pi an = 0 n = 1, 2, . . . . , z0

pi f .

f D(z0, R) : 0 < |z z0| < R ( z0 f )

f(z) =

n=

anzn

pi Laurent f D(z0, R). z0 pi f ,

f(z) =

n=0

anzn , 0 < |z z0| < R .

g D(z0, R)

g(z) =

f(z) z 6= z0a0 z = z0 .

130 5. LAURENT- -

g z0

g(z) =

n=0

anzn , z D(z0, R) .

pi, z0 pi f , pi f

pi z0 f z0.

f z0.

5.9. (1)

f(z) =cos z 1

z2, z 6= 0 .

f(z) =

(1 z22! + z

4

4! z6

6! + ) 1

z2= 1

2!+z2

4! z

4

6!+

pi z = 0 pi f .

(2)

g(z) =sin5 z

z5+ cos z , z 6= 0 .

limz0 sin zz = 1 pi

limz0

g(z) = 1 + cos 0 = 2 .

pi limz0 g(z) pi, pi pi pi pi z = 0 pi-

g. pi Laurent g

pi z = 0 pi 0 pi

g.

5.10. U C , z0 U f U \ {z0}. pi :

(1) z0 pi k.

(2) pi M > 0 > 0

|f(z)| < M|z z0|k 0 < |z z0| < .

5.2. 131

(3) limzz0(z z0)k+1f(z) = 0.

(4) limzz0(z z0)kf(z) pi.

pi. 5.8 g(z) := (z z0)kf(z).

5.11. f D(z0, R) : 0 < |z z0| < R.

() z0 pi k f

limzz0

(z z0)kf(z) = 6= 0 .

() z0 pi k f

f(z) =g(z)

(z z0)k ,

pi g D(z0, R) g(z0) 6= 0.

pi. () z0 pi k f ,

f(z) =

+n=k

an(z z0)n , 0 < |z z0| < R ,

pi ak 6= 0. pi

(z z0)kf(z) = ak + + a1(z z0)k1 ++n=0

an(z z0)n+k , 0 < |z z0| < R

pi

limzz0

(z z0)kf(z) = ak 6= 0 .

, limzz0(z z0)kf(z) = 6= 0. g(z) := (z z0)kf(z), 0 < |z z0| < R, pi 5.8 z0 pi g. pi

g(z) =

+n=0

an(z z0)n , |z z0| < R a0 = 6= 0 .

132 5. LAURENT- -

f(z) =a0

(z z0)k + +ak1z z0 +

+n=0

an+k(z z0)n , 0 < |z z0| < R ,

pi a0 = 6= 0. , z0 pi k f .

() pi pi .

5.12. f D(z0, R) : 0 < |zz0| < R, z0 pi f

limzz0

|f(z)| = + .

pi. z0 pi k f . g(z) := (z z0)kf(z), 0 < |z z0| < R, pi 5.11() limzz0 g(z) = 6= 0 pi

limzz0

|f(z)| = limzz0

g(z)(z z0)k = || limzz0 1|z z0|k = + .

, limzz0 |f(z)| = +. limzz0 1f(z) = 0 pi z0 pi- 1/f . pi limzz0 |f(z)| = +, pi > 0, R, f D(z0, ) : 0 < |z z0| < 1/f D(z0, ). h D(z0, )

h(z) =

1

f(z) z 6= z00 z = z0 .

h D(z0, ) h(z0) = 0. z0 k 1 h,

h(z) = (z z0)kh1(z) ,

pi h1 D(z0, ) h1(z0) 6= 0.

limzz0

(z z0)kf(z) = limzz0

1

h1(z)=

1

h1(z0)6= 0

pi 5.11() z0 pi k f .

5.2. 133

5.13. f D(z0, R) : 0 < |zz0| < R, z0 f limzz0 f(z) pi

+( limzz0 |f(z)| 6= +).

pi. z0 f pi

pi f . pi pi pi 5.8 5.12.

5.14. f, g U C. pi z0 U m f n g.

(1) m n, z0 pi f/g.

(2) m < n, z0 pi nm f/g.

pi. pi pi

f(z) = (z z0)mf1(z) g(z) = (z z0)ng1(z0) ,

pi f1, g1 U f1(z0) 6= 0 g1(z0) 6= 0. f1 g1 pi D(z0, ) U z0. h(z) := f1(z)/g1(z) D(z0, ). pi,

z D(z0, ) : 0 < |z z0| < f(z)

g(z)=

(z z0)mf1(z)(z z0)ng1(z0) = (z z0)

mnh(z) .

(1) m n, limzz0 f(z)g(z) pi pi pi 5.8 z0 pi f/g.

(2) m < n,

f(z)

g(z)=

h(z)

(z z0)nm h(z0) 6= 0 , 0 < |z z0| <

pi pi 5.11() z0 pi nm f/g.

134 5. LAURENT- -

5.15. (1)

f(z) =z(z 1)2sin2 piz

, z / Z .

pi 0 2 pi pi f , z = 0

pi pi f .

pi 1 2 pi f , z = 1 pi

f .

pi z, z 6= 0, 1, 2 pi f , pi 2 f .

(2)

g(z) =1 cos(z + 1)

(z + 1)2, z 6= 1 .

pi 1 2 pi g, z = 1 pi g. z 6= 1

g(z) =1

(1 (z+1)22! + (z+1)

4

4! (z+1)6

6! + )

(z + 1)2=

1

2! (z + 1)

2

4!+

(z + 1)4

6!

g :

g(z) =

1cos(z+1)

(z+1)2 z 6= 1

12 z = 1 ,

g ( C).

. pi limz1 g(z) pi pi

LHpital. ,

limz1

1 cos(z + 1)(z + 1)2

= limz1

sin(z + 1)

2(z + 1)= lim

z1cos(z + 1)

2=

1

2.

(3)

h(z) = cos(e1/z) , z 6= 0 .

zn =1

ln(npi), n N .

5.2. 135

pi e1/zn = npi cos(e1/zn) = cos(npi) = (1)n, limn zn = 0 limn h(zn) = limn(1)n pi. pi limz0 h(z) pi +( limz0 |h(z)| 6= +). , z = 0 h.

5.16. z0 C R > 0, pi f, g D(z0, R) : 0 < |z z0| < R. z0 f pi g, z0 fg, f/g f + g.

. z0 fg. z0 pi

k N g. ,

g(z) =h(z)

(z z0)k , pi h |z z0| < R h(z0) 6= 0 .

pi pi R h(z) 6= 0 |z z0| < .(i) pi z0 pi fg. pi

fg z0 fg |z z0| < R. pi

f(z) =f(z)g(z)

g(z)=f(z)g(z)

h(z)(z z0)k , 0 < |z z0| < ,

limzz0 f(z) = 0 pi z0 pi f (pi).

(ii) pi z0 pi m N fg.

f(z)g(z) =H(z)

(z z0)m , pi H |z z0| < R H(z0) 6= 0 .

pi

f(z) =f(z)g(z)

g(z)=

H(z)/h(z)

(z z0)mk , 0 < |z z0| < ,

pi w = H(z)/h(z) |z z0| < H(z0)/h(z0) 6= 0. m > k, z0 pi m k f (pi). m k, limzz0 f(z) pi pi z0 pi f (pi).

z0 fg.

pi z0 f/g f+g

( 2).

136 5. LAURENT- -

5.17. f C \ {0}, 0 pi pi f f(T) R, pi T ,

f(z) = az +a

z+ b ,

pi a C \ {0} pi b R.

. pi 0 pi pi f , pi Laurent f :

0 < |z| < + f(z) =

+n=1

an(z z0)n , a1 6= 0 ,

pi an pi pi

an =1

2pii

|z|=1

f(z)

zn+1dz , n 1 . (5.5)

pi |z| = 1 z() =ei, [0, 2pi]. pi pi pi f(ei) R, [0, 2pi], pi pi (5.5) n = 0

a0 =1

2pii

2pi0

f(ei)

eiiei d =

1

2pi

2pi0

f(ei) d R .

pi n N pi pi (5.5)

an =1

2pii

2pi0

f(ei)

ei(n+1)iei d

=1

2pi

2pi0

f(ei)

eind

=1

2pi

2pi0

(f(ei)

ein

)d (pi f(ei) R)

=1

2pi

( 2pi0

f(ei)

eind

)= an .

n = 1 a1 = a1 a1 = a1. pi a1 6= 0, a1 C \ {0}. n 2. an = an pi an = 0 n 2, pi

an = 0 , n = 2,3,4, . . . .

,

f(z) = a1z +a1z

+ a0 ,

pi a0 R a1 C \ {0}.

5.3. 137

5.3 pipi

5.4 pipi

1. f D(z0, R) : 0 < |z z0| < R. z0 pi k f

f(z) =g(z)

(z z0)k ,

pi g D(z0, R) g(z0) 6= 0.

2. z0 C R > 0, pi f, g D(z0, R) : 0 < |z z0| < R. z0 f pi g, z0 f/g f+g.

3. z = 0 f , 0

f2.

4. f U C z0 U n 1 f . pi f1 U pi D(z0, ) U z0,

f(z) = (z z0)nf1(z) , f1(z) 6= 0 z D(z0, ) .

f (D(z0, )) D(0, R), pi g D(0, R) : 0 < |w| < R 0 pi m 1 g. z0 pi mn h := g f .

5. f f(n) = 0 n Z. f(z)/ sin(piz) pi.

6. f f(z + 1) = f(z) z C f(0) = 0.

|f(z)| epi|=z| , z C ,

f(z) = c sin(piz) pi c.

pi. Liouville pi

138 5. LAURENT- -

7.

g(z) = exp

(z + 1/z

2

)= e(z+1/z)/2 .

g(z) =

n= cnzn pi Laurent g : 0 < |z| 1}. |1/(z + 1)| < 1 pi

f(z) =1

z2(z + 1)=

1

[1 1/(z + 1)]2 (z + 1)3

=1

(z + 1)3

n=1

n1

(z + 1)n1

=

n=1

n1

(z + 1)n+2

= 3

n=(n+ 2)(z + 1)n .

5.20.

f(z) =z2 + 4z + 4 + 4i

(z2 + 4)(z + i).

pi Laurent f z0 = 0 pi pi

1 i. pi pi Laurent f ;

. i,2i f .

f(z) =z2 + 4

(z2 + 4)(z + i)+

4z + 4i

(z2 + 4)(z + i)=

1

z + i+

4

z2 + 4.

140 5. LAURENT- -

, 1/(1w) = n=0wn 1/(1 +w) = n=0(1)nwn, |w| < 1 ( ). = {z C : 1 < |z| < 2}, 1 i pi Laurent f

f(z) =1

z + i+

4

z2 + 4

=1

z

1

1 + i/z+

1

1 + (z/2)2

=1

z

n=0

(1)n(i

z

)n+n=0

(1)n(z

2

)2n=n=0

(1)n in

zn+1+n=0

(1)n4n

z2n

=

n=1

(1)n1 in1

zn+

n=0

(1)n4n

z2n .

. f

f(z) =1

z + i+

4

z2 + 4=

1

z + i+

i

z + 2i iz 2i .

5.21. pi pi Laurent

f(z) = cot z z0 = 0 : 0 < |z| < pi.

. pi

sin z = 0 z = npi , n Z ,

zn = npi, n Z, pi pi f(z) = cot z = cos zsin z . 0 pipi f(z) = cot z pi pi Laurent f(z) = cot z

0 < |z| < pi

cot z =a1z

+a0+a1z+a2z2+a3z

3+ cos z = sin z(a1z

+ a0 + a1z + a2z2 + a3z

3 + ).

cos z =n=0

(1)n z2n

(2n)!= 1 z

2

2!+z4

4! sin z =

n=0

(1)n z2n+1

(2n+ 1)!= z z

3

3!+z5

5!+ ,

z C. pi

1 z2

2!+z4

4! =

(z z

3

3!+z5

5!+

)(a1z

+ a0 + a1z + a2z2 + a3z

3 + )

5.5. 141

1 z2

2!+z4

4! = a1 + a0z +

(a1 a1

3!

)z2 +

(a2 a0

3!

)z3 +

(a3 a1

3!+a15!

)z4 + .

, a1 = 1, a0 = 0, a1a1/3! = 1/2! a1 = 1/3, a2 = 0, a3a1/3! +a1/5! = 1/4!a3 = 1/45.

cot z =1

z 1

3z 1

45z3 + , 0 < |z| < pi .

5.22. 1e2piz1 =

n= anz

n pi Laurent

f(z) = 1e2piz1 = {z C : 1 < |z| < 2} z0 = 0. pi

an, n < 0.

. pi

e2piz 1 = 0 e2piz = 1 2piz = 2npii z = ni , n Z ,

zn = ni, n Z, pi pi f(z) = 1e2piz1 . pi Laurent an pi pi

an =1

2pii

C+(0, r)

1/(e2piz 1)zn+1

dz =1

2pii

C+(0, r)

1

(e2piz 1) zn+1 dz ,

pi C+ (0, r) 0, r, 1 < r < 2

.

142 5. LAURENT- -

(i) n = 1: pipi i, 0 i f(z) = 1e2piz1

C+ (0, r) pi pi. pi

pipi

a1 =1

2pii

C+(0, r)

1

e2piz 1 dz

= Res

(1

e2piz 1 , i)

+ Res

(1

e2piz 1 , 0)

+ Res

(1

e2piz 1 , i)

=1

(e2piz 1)z=i

+1

(e2piz 1)z=0

+1

(e2piz 1)z=i

=1

2pie2pii+

1

2pi+

1

2pie2pii=

3

2pi.

(ii) n 2: pipi

an =

C+(0, r)

1

(e2piz 1) zn+1 dz =C+(0, r)

zn1

e2piz 1 dz ,

n 1 1. g(z) =

zn1

e2piz 1 ,

i i g C+(0, r) pi pi. pi 0 g 1 pi pi g, 0 pi . , pi

pipi

an =

C+(0, r)

zn1

e2piz 1 dz = Res(zn1

e2piz 1 , i)

+ Res

(zn1

e2piz 1 , i)

=zn1

(e2piz 1)z=i

+zn1

(e2piz 1)z=i

=(i)n12pie2pii

+in1

2pie2pii

=1

2pi

((i)n1 + in1

)=

0 n = 2k ,

(1)k /pi n = 2k + 1 .

5.23. pi C+(0, 2)

e1/z

1 + z2dz ,

pi C+(0, 2) 0 2 .

5.5. 143

. 0 i f(z) = e1/z/(1 + z2) C+(0, 2). i pi pi f ,

Res

(e1/z

1 + z2, i

)=

e1/z

(1 + z2)

z=i

= e1/i

2i.

0 f . pi

ew =

n=0

wn

n!, w C 1

1 + w=

n=0

(1)nwn , |w| < 1 ,

pi Laurent f z0 = 0 : 0 < |z| < 1

f(z) = e1/z 11 + z2

=

(1 +

1

z+

1

2!z2+

1

3!z3+

1

4!z4+

1

5!z5+

)(1 z2 + z4 z6 + )

= (

1 13!

+1

5!+

)1

z+ .

pi,

Res

(e1/z

1 + z2, 0

)= a1 = 1 1

3!+

1

5!+ = sin 1 .

sin z =n=0

(1)n z2n+1

(2n+ 1)!= z z

3

3!+z5

5!+ , z C .

pi, pi pipi

C+(0, 2)

e1/z

1 + z2dz = 2pii

{Res

(e1/z

1 + z2, i

)+ Res

(e1/z

1 + z2, 0

)+ Res

(e1/z

1 + z2, i

)}

= 2pii

{e1/i

2i+ sin 1 +

e1/i

2i

}

= 2pii

{sin 1 e

i ei2i

}= 2pii {sin 1 sin 1} = 0 .

5.24. pi pi

2pi0

1

1 + 8 cos2 d .

144 5. LAURENT- -

. z = ei, 0 2pi. pi dz = ieid = izd cos = 12

(z + z1

),

2pi0

1

1 + 8 cos2 d =

C+(0, 1)

1

iz

1

(1 + 2(z2 + 2 + z2))dz

=1

i

C+(0, 1)

z

2z4 + 5z2 + 2dz

=1

i

C+(0, 1)

z

(2z2 + 1)(z2 + 2)dz .

i/2 2i pi pi f(z) = z(2z2+1)(z2+2)

. pi

i/2 C+(0, 1), pi - pipi 2pi

0

1

1 + 8 cos2 d =

1

i2pii

{Res

(z

2z4 + 5z2 + 2,i2

)+ Res

(z

2z4 + 5z2 + 2, i

2

)}= 2pi

{z

(2z4 + 5z2 + 2)

z=i/

2

+z

(2z4 + 5z2 + 2)

z=i/2

}

= 2pi

{z

8z3 + 10z

z=i/

2

+z

8z3 + 10z

z=i/2

}

= pi

{1

4(i/

2)2 + 5+

1

4(i/2)2 + 5

}=

2pi

3.

5.25. R, z() = Rei, 0 pi, pipi 0 R > 0. pi

limR

R

1

(z2 + 1)2(z2 + 4)dz = 0

pi

0

1

(x2 + 1)2(x2 + 4)dx .

. z R 1(z2 + 1)2(z2 + 4) = 1|z2 + 1|2|z2 + 4| 1(|z|2 1)2(|z|2 4) = 1(R2 1)2(R2 4) .

R Rpi piR

1

(z2 + 1)2(z2 + 4)dz

Rpi(R2 1)2(R2 4) R 0 .

5.5. 145

,

limR

R

1

(z2 + 1)2(z2 + 4)dz = 0 .

i pi 2 2i pi pi f(z) = 1(z2+1)2(z2+4)

.

f pi pi pi pi pi

pipi R, z() = Rei, 0 pi [R,R]. R i 2i f

R.

pi pipi

RR

1

(x2 + 1)2(x2 + 4)dx+

R

1

(z2 + 1)2(z2 + 4)dz = 2pii (Res (f, i) + Res (f, 2i)) . (5.6)

Res (f, i) = Res

(1

(z2 + 1)2(z2 + 4), i

)= lim

zi

((z i)2 1

(z2 + 1)2(z2 + 4)

)= lim

zi

(1

(z + i)2(z2 + 4)

)= lim

zi2z(z + i) + 2(z2 + 4)

(z + i)3(z2 + 4)2

= i36

146 5. LAURENT- -

Res (f, 2i) = Res

(1

(z2 + 1)2(z2 + 4), 2i

)= lim

z2i(z 2i) 1

(z2 + 1)2(z2 + 4)

= limz2i

1

(z2 + 1)2(z + 2i)

= i36.

pi, pi (5.6) pipi

1

(x2 + 1)2(x2 + 4)dx = lim

R

RR

1

(x2 + 1)2(x2 + 4)dx =

pi

9.

, 0

1

(x2 + 1)2(x2 + 4)dx =

1

2

1

(x2 + 1)2(x2 + 4)dx =

pi

18.

5.26. f : C C limz f(z) = , pi f pi.

pi. f(z) =

k=0 akzk, z C.

g(z) = f

(1

z

)=k=0

akzk, |z| > 0 .

pi, pi f , 0 pi, pi g(z) = f(1/z). 0

pi g limz0 g(z) pi, pi g

limz0 |g(z)| = g limz0 |g(z)| pi . pi pi f limz f(z) pi, pi f

limz |f(z)| = f limz |f(z)| pi . pi pi pipi :(i) 0 pi g pi ak = 0, k = 1, 2, . . .. pi f(z) = a0,

f .

5.5. 147

(ii) 0 pi n g.

g(z) =nk=0

akzk

, an 6= 0 pi f(z) = a0 + a1z + + anzn , an 6= 0 .

pipi limz |f(z)| =.(iii) 0 g,

g(z) =k=0

akzk

, |z| > 0 , pi ak 6= 0 pi pi k .

pipi limz0 |g(z)|, pi pi limz |f(z)|, pi .pi pi limz |f(z)| =, f pi.

5.27. g : C C 1 1, pi g g(z) = az + b, a, b C, a 6= 0.

pi. pi g(z) = az + b, a, b C, a 6= 0, 1 1 pi. pi pi 1 1 pi pipi . .

pi () g, limz g(z) pi, lim|z| |g(z)| = c. , > 0 piM > 0 |g(z)| < c+ , |z| > M . pi K > 0 |g(z)| < K, |z| M . |g| C pi Liouville g . pi, pi() g.

g, limz g(z) pi . pi g , pi pipi > 0 > 0

w C |w g(0)| < , pi z C , |z| < , w = g(z) . (5.7)

g U pi , pi CasoratiWeierstrass R = {|g(z) : z U |} pi C. pi, pi|z1| > |g(z1) g(0)| < . pi (5.7) pi |z2| <

148 5. LAURENT- -

g(z2) = g(z1). pi, pi g 1 1., pi g pi g pi. pi 2 pi pi pi pi 1 1.pi pi g : C C 1 1, g(z) = az + b, a, b C, a 6= 0.

5.28. pi f : C pi C . a r > 0, D(a, r) = {z C : |z a| r} . |f | D(a, r), pi f D(a, r).

. f pi C |f(z)| = c, |za| = r,pi D(a, r) , f D(a, r) f pi.

pi. 1 pi. pi f |f(z)| = c, |z a| = r, pi z0 D(a, r) f(z0) D(0, c). z0

g(z) := f(z) f(z0) .

,

|g(z) f(z)| = |f(z0)| < c = |f(z)| , z C(a, r) .

pi g D(a, r), z0, pi Rouch

f D(a, r).

2 pi. pi f D(a, r), pi

|f | pi D(a, r). pi |f(z)| = c D(a, r), |f(z)| = c, z D(a, r). pi 3.18 f D(a, r) pi pi .

5.29. f : C C . c > 0, pi

{z C : |f(z)| < c} = {z C : |f(z)| c} .

5.5. 149

pi. pi f , G := {z C : |f(z)| < c} F := {z C : |f(z)| c} . pi G F , G F = F .pi pi pi F G. pi z0 F |f(z0)| = c, z0 G. ,

D(z0, ) G 6= , > 0 .

f pi D(z0, ), pi G. pi f D(z0, ). 0 < < . pi f , pi

f pi pi D(z0, ),

C(z0, ) D(z0, ). |f()| < |f(z0)| = c pi G. ,D(z0, ) G 6= .

5.30. C pi f, g : C . pi

|f(z)|+ |g(z)| sup {|f(w)|+ |g(w)| : w } , z .

pi. z0 . f(z0) = |f(z0)|ei g(z0) = |g(z0)|ei,

h(z) := f(z)ei + g(z)ei , z .

h pi . pi, pi

|h(z0)| = |f(z0)|+ |g(z0)| sup {|h(w)| : w } sup {|f(w)|+ |g(w)| : w } .

5.31. pi pi P (z) = anzn +an1zn1 + +a1z+a0 |P (z)| = 1, z C |z| 1. pi

1. |ak| 1, k = 0, 1, . . . , n,

2. |P (z)| |z|n, z C |z| 1.

pi. 1. pi Cauchy

|ak| =P (k)(0)k!

max|z|=1 |P (z)|1k = 1 , k = 0, 1, . . . , n .

150 5. LAURENT- -

2. D = {z C : 1 |z| R}, R > 1,

f(z) :=P (z)

zn

D. |z| = 1 |f(z)| = |P (z)| = 1, |z| = R

|f(z)| = |an + an1z

+ + a0zn| |an|+ |an1|

R+ + |a0|

Rn 1 + (R) ,

pi (R) = |an1|/R+ + |a0|/Rn. pi, |z| = R

|f(z)| 1 + (R) , limR

(R) = 0 .

, pi D

|f(z)| = |P (z)||z|n 1 |P (z)| |z|n , z C |z| 1 .

5.32. pi f : C C |f(z)| = 1, z C |z| = 1. pi

f(z) = czn , pi c |c| = 1 n N .

, p pi n |p(z)| = 1, z C |z| = 1, p(z) = czn, pi c C |c| = 1.

pi. 1 pi. f(z) =

n=0 anzn, z C.

g(z) := f

(1

z

)=

n=0

an1

zn,

g pi C \ {0}. |z| = 1, z = ei, pi pi f(ei)f(ei) = 1. pi,

g(z) := f(ei) =1

f(ei)=

1

f(z) f(z)g(z) = 1 , z C |z| = 1 .

fg pi C \ {0} f(z)g(z) = 1, z C |z| = 1. , pi

f(z)g(z) = 1 , z C \ {0} .

5.5. 151

pi f(z) 6= 0, z C \ {0} pi

f(z) = znh(z) , pi n N ,

pi h h(z) 6= 0, z C. pi |h(z)| = 1, z C |z| = 1 h(z) 6= 0, z C, pi 5.28 pipi h(z) = c, |c| = 1.,

f(z) = czn , pi c |c| = 1 n N .

2 pi. f pi pi pi D(0, 1). -

, pi f(n) = 0, pi (n)

, pi pi (kn) limn kn = D(0, 1) pi f = 0.

1, 2, . . . , n f D(0, 1).

g : C C, g(z) := f(z)/

(nk=1

z k1 kz

).

, Mbius

wk =z k1 kz , k = 1, 2, . . . , n ,

pi D(0, 1) D(0, 1)

C(0, 1) C(0, 1). pi, |z| = 1 |g(z)| = |f(z)| = 1 g D(0, 1). pi |g| pi D(0, 1),

|z| = 1. |g(z)| = 1, |z| = 1, pi |g(z)| = 1 z D(0, 1). g pi D(0, 1) pi

C. g(z) = c, pi c C |c| = 1. ,

f(z) = c

nk=1

z k1 kz = c

z 11 1z

z 21 2z

z n1 nz .

pi f , pipi 1 = 2 = = n = 0 pi

f(z) = czn , pi c |c| = 1 n N .

152 5. LAURENT- -

5.33. pi , pi -

.

pi. p(z) = anzn + an1zn1 + + a1z + a0, an 6= 0, pi n q(z) = anz

n. R > 0 ,

|an| > |an1|R

+ + |a1|Rn1

+|a0|Rn

.

, |z| = R

|p(z) q(z)| = |an1zn1 + + a1z + a0| |an1||z|n1 + + |a1||z|+ |a0|= |an1|Rn1 + + |a1|R+ |a0|< |an|Rn = |q(z)| .

|p(z) q(z)| < |q(z)|, |z| = R. pi q(z) = zn n ( ) |z| = R, pi Rouch pi p n .

5.34. pi pi f : (0, 1, 2) C, pi(0, 1, 2) = {z C : 1 < |z| < 2},

(f(z))2 = z , z (0, 1, 2) .

pi. g(z) = z, g 1/g (0, 1, 2). pi

pi f , (f(z))2 = g(z), pi [;, 13.11 Theorem]

(0, 1, 2) pipi pi pi, pi. , pi

f (0, 1, 2).

5.35. w = f(z) D(0, 1), f(0) = 0,

w = f(z) = eiz , R .

pi. pi f : D (0, 1) D (0, 1) 1 1 pi, f (0) = 0, pi Schwarz

|f (z)| |z| , |z| < 1 . (5.8)

5.5. 153

f1 : D (0, 1) D (0, 1) 1 1 pi, f1 (0) = 0, pi pi Schwarz f1 (z) |z| , |z| < 1 . (5.9)pi (5.8) (5.9) pi |f (z)| = |z| |z| < 1, pi pi pi Schwarz pi

w = f (z) = eiz , R .

5.36. w = f (z) D (0, 1), f (a) = 0, |a| < 1,

w = f (z) = eiz a1 az , R .

pi. , Mbius g (z) = (z a) / (1 az) 1 1 pi D (0, 1) pi D (0, 1), g (a) = 0. f

D (0, 1), f (a) = 0, h = f g1 D (0, 1), h (0) = 0.pi 5.35

h (z) = eiz , f (z) = ei z a1 az , R .

5.37. pi pi pi

D1 = {z C : |z| < 1,=z > 0}

D(0, 1) = {z C : |z| < 1}.

. w = f1(z) = i1z1+z pi D1 D2 = {z C : 0,=z > 0} w = f2(z) = z

2 pi D2 D3 = {z C : =z > 0}. , w = f3(z) = ziz+ipi D3 D(0, 1). , f = f3 f2 f1 pi D1 .

w = f(z) = f3 (f2 (f1(z))) =[i(1 z)/(1 + z)]2 i[i(1 z)/(1 + z)]2 + i =

(1 z)2 + i(1 + z)2(1 z)2 i(1 + z)2 .

154 5. LAURENT- -

5.38. a, b D(0, 1) a 6= b. D (0, 1) (a) = b pi .

. , w = f(z) D(0, 1), f(a) = 0, |a| < 1,

w = f(z) = eiz a1 az , R .

pi,

w = f1(z) =z + eia

ei + az, R ,

w = g(z) D(0, 1), g(0) = b,

w = g(z) =z + eib

ei + bz, R .

,

w = (z) := g (f(z)) =ei(z a)/ (1 az) + eibei + bei(z a)/ (1 az) =

(1 ba) z + b a(b a) z + 1 ab

D(0, 1), (a) = b.

5.39. C pi pi 0, 1 : [0, 1] pi a pi b. pi pi : [0, 1][0, 1],

(0, t) = 0 (t) , (1, t) = 1 (t) , t [0, 1]

(s, 0) = a , (s, 1) = b , s [0, 1] .

. C pi pi, pi 1 1 pi : D (0, 1) pi 1 pi (pipipi [;, 13.11 Theorem]). : [0, 1] [0, 1]

(s, t) := 1 ((1 s) (0(t)) + s (1(t))) .

,

(0, t) = 1 ( (0 (t))) = 0 (t) , (1, t) = 1 ( (1 (t))) = 1 (t) , t [0, 1]

5.5. 155

(s, 0) = 1 ((1 s) (a) + s (a)) = a , (s, 1) = 1 ((1 s) (b) + s (b)) = b ,

s [0, 1].

5.40. pi f D (0, 1) f (0) = 0 f (0) > 0

.

. pi 5.35, w = f (z) D (0, 1), f (0) = 0,

w = f (z) = eiz , R .

pi f (0) = ei > 0, pipi = 2npi, n Z. , w = f (z) = z.

5.41. C, 6= C, pi pi z0 . pi pi pi f pi D (0, 1), f(z0) = 0 f (z0) > 0.

pi. pi pi Riemann, pi pi f

pipi . g pi pi pi ,

h = f g1 D (0, 1) h (0) = 0 h (0) > 0. pi pi h (z) = z pi f = g.

5.42. z1, z2, z3, z4 C . pi

pi (z1, z2, z3, z4) pi z1, z2, z3, z4

.

pi. z2, z3, z4 C w2, w3, w4 C , pi C = C {}, pi Mbius w = f (z), wi = f (zi), i = 2, 3, 4. pipi pi pi

(w,w2, w3, w4) = (z, z2, z3, z4) (w w2) (w3 w4)(w w4) (w3 w2) =

(z z2) (z3 z4)(z z4) (z3 z2) .

, Mbius pi Cz

Cw. , Mbius

(w, 0, 1,) = (z, z2, z3, z4) w = f (z) = (z z2) (z3 z4)(z z4) (z3 z2)

156 5. LAURENT- -

pi Cz pi ( 0, 1, pi). pi, (z1, z2, z3, z4) = f(z1) R, z1, z2, z3, z4 pi Cz.

5.43. f D (0, 1) =

{z C : |z| 1} pi z0, |z0| = 1, pi pi pi(pi 1) f . f(z) =

n=0 anz

n, |z| < 1, pi

limn

anan+1

= z0 .

pi. pi z0 pi pi f , z 0 < |z z0| < R

f(z) =c

z z0 + g(z) ,

pi c = Res (f, z0) g D (z0, R) = {z C : |z z0| < R}.pi g , g(z) =

n=0 bnz

n.

|z| < 1

f(z) =c

z z0 + g(z)

= cz0

1

1 zz0+n=0

bnzn

= cz0

n=0

(z

z0

)n+

n=0

bnzn

=n=0

( czn+10

+ bn

)zn

an = c/zn+10 + bn anzn0 = c/z0 + bnzn0 . pi g(z) =

n=0 bnzn

z = z0, limn bnzn0 = 0. ,

limn

anan+1

= z0 limn

anzn0

an+1zn+10

= z0 limn

c/z0 + bnzn0c/z0 + bn+1zn+10

= z0 .

5.44. f : C C

f (z + 1) = f (z) f (z + i) = f (z) , z C .

f pi pi pi 1 i. pi f .

5.5. 157

pi. f S = {z = x+ iy C : 0 x 1, 0 y 1} pi pi . pi piM > 0, |f (z)| M , z S.pi pi pi

f(z + k) = f(z) f(z +mi) = f(z) , k,m Z .

t R, [t], [t] t < [t] + 1. pi 0 t [t] < 1. z = x+ iy C. ,

f(z) = f(x+ iy) = f(x [x] + iy) = f (x [x] + i(y [y])) = f(w) ,

pi w = x [x] + i (y [y]) S. , |f(z)| M z C pi Liouville f .

5.45. pi (

pi):

1. f pi pi D, pi F

pi D F (z) = f(z), z D.

2. f {0 < |z| < 2} f(1/n) = 0, n =1,2,3, . . ., f .

3. pi f D (0, 2) = {z C : |z| < 2},

|f(x+ iy)|2 = 4 x2 y2 , z = x+ iy D(0, 2) .

.

1. pipi pi f pi pi D.

f , pi, .

f(z) = z pi pi pi C

pi pi. pi F pi C F (z) = z,

w = F (z), w = z pi C, pi.

158 5. LAURENT- -

2. . {0 < |z| < 2} pi, pi . f(z) = sin(pi/z) pi {0 < |z| < 2} . f(1/n) = sin(pin) = 0, n = 1,2,3, . . .. f D(0, 2) = {z C : |z| < 2},pi 0 D(0, 2), pi f .

3. pi f , |f(0)|2 = 4 |f(0)| = 2, D(0, r) = {z C : |z| r}, 0 < r < 2, |z| = r |f(z)|2 = 4 r2 < 4 |f(z)| < 2. pi pi. , pi f D(0, 2).

5.46. pi, pi C

. pi f pi

z1, z2, . . . , zn .

Q(z) = (z z1)(z z2) (z zn) ,

pi

P (z) :=1

2pii

f(w)

Q(w)

Q(w)Q(z)w z dw

pi n1 pi f z1, z2, . . . , zn, P (zk) = f(zk), k = 1, 2, . . . , n.

. pi Q(zk) = 0, k = 1, 2, . . . , n, pi pi Cauchy

P (zk) =1

2pii

f(w)

w zk dw = f(zk) , k = 1, 2, . . . , n .

pi P pi n1. pi Q(w)Q(z) pi n pi z, pi ak(w)

Q(w)Q(z) = (w z)n1k=0

ak(w)zk .

,

P (z) =

n1k=0

(1

2pii

f(w)

Q(w)ak(w)

)zk .

5.5. 159

5.47. pi

sin z = z

pi pi z = 0 pi pi .

.

f(z) := sin z z

pi. pi f . pi Picard w C, pi , f(z) = w pi pi pi .pi sin z = z f(z) = 0 pipi pi

. pi Picard w = 2pi f(z) = 2pi pi pi .

(wn) pi f(wn) = 2pi sinwn wn = 2pi.

f(2pi + wn) = sin(2pi + wn) (2pi + wn) = sinwn 2pi wn = 0 .

pi, pi pi f(z) = 0 pipi pi .

sin z = z pi pi .

5.48. f, g : C C |f(z)| |g(z)|, z C. pi f(z) = cg(z), pi c C., |f(z)| | sin2 z|, z C, f(z) = c sin2 z, pi c C.

pi. pi pi pipi pi g . pi-

g . g

pi f/g pi. f/g pi pi-

|f(z)/g(z)| 1, z C. pi Liouville f/g .

5.49. f : C C, |f(z)| = | sin z|, z C. pi f(z) = c sin z, pi c C, |c| = 1.

160 5. LAURENT- -

pi. 1 pi. pi pi pi .

2 pi.

g(z) :=f(z)

sin z, g pi C \ piZ .

pi |g(z)| = 1 z C \ piZ, pi 3.18 g piC \piZ. g(z) = c, |c| = 1. f(z) = c sin z |c| = 1, z C\piZ. pi f w = sin z C, pi f(z) = c sin z z C, pic |c| = 1.

5.50. R > 0. n N pi

fn(z) = 1 +1

z+

1

2!z2+ + 1

n!zn

D (0, R) = {z C : |z| < R}.

pi.

gn(z) = fn(1/z) = 1 + z +z2

2!+ + z

n

n!.

pi gn(0) 6= 0 n N, gn |z| 1/R fn |z| R. gn n-

k=0 zk/k! = ez, pi pi pi C.

m = min|z|=1/R

|ez| ,

pi n0 N n n0

|gn(z) ez| < m |ez| , |z| = 1/R .

pi w = ez C, pi Rouch gn

|z| 1/R. , fn |z| R.

5.6

1. pi pi :

5.6. 161

() C+(0, 1/2)

cos z

1 + z + z2 + z3 + z4dz ,

pi C+ (0, 1/2) 0, 1/2 .

() C+(i, 1)

e2z

(z i)4 dz ,

pi C+ (i, 1) i, 1 .

2. pi f : C C lim|z| |f(z)| = , 0. pi f .

3. pi f G G, pi

G = {z C : |

162 5. LAURENT- -

7.

f(z) =1

z4 + 13z2 + 36.

pi Laurent f z0 = 0

pi pi 2 + i.

8. pi pi Laurent f(z) = ezsin z

z0 = 0 : 0 < |z| < pi.

9. 1sinpiz =

n= anzn pi Laurent f(z) = 1sinpiz

= {z C : 1 < |z| < 2} z0 = 0. pi an, n 1.

10. tan z =

n= anzn pi Laurent f (z) = tan z

= {z C : pi/2 < |z| < 3pi/2} z0 = 0. pi an, n 1.

11. pi pi :

() C+(0, 2)

epiz

z (z2 + 1)dz ,

pi C+ (0, 2) 0, 2 .

() C+(0, 1)

zne1/z dz ,

pi n N C+ (0, 1) .

() C+(0, 5)

cot z

z4 + z2dz ,

pi C+ (0, 5) 0, 5 .

() C+(0, 2)

z4e1/z

1 z4 dz ,

pi C+ (0, 2) 0, 2 .

5.6. 163

12. pi C+(0, 1)

1

z2 sin zdz =

pii

3,

pi C+ (0, 1) .

13. pi

1

z4 + 1dz = 2piiRes

(1

z4 + 1, e5pii/4

)=

2pi

4(1 + i) ,

pi : x2 xy+ y2 + x+ y = 0 .

14. pi pi

(i)

2pi0

sin2

5 + 4 cos d =

pi

4(ii)

2pi0

1

(1 + 2a cos + a2)2d =

2pi(1 + a2

)(1 a2)3 , 1 < a < 1 .

15. pi , pi n N 2pi

0sin2n d =

2pi

4n

(2n

n

).

16. pi

0

1

(x2 + a2) (x2 + b2)dx =

pi

ab (a+ b), a, b > 0 , a 6= b

1

(x2 + x+ 1)2dx =

4pi

3

3.

17. pi

(i)

cosx

x2 + a2dx =

pi

aea , a > 0 (ii)

cospix

x2 2x+ 2 dx = piepi .

18. () pi f, g -

D (0, 1) = {z C : |z| < 1}.

f (1/n)f (1/n)

=g (1/n)g (1/n)

, n = 2, 3, . . . ,

pi f/g D (0, 1).

164 5. LAURENT- -

() f C \ {0}

|f (z)| |z|1/2 + |z|1/2 ,

pi f .

19. pi G C f, g : G C pi :

f (z) g (z) = 0 , z G,

f 0 g 0 G.

20. pi f, g D (0, 1) =

{z C : |z| 1}, g D (0, 1) f pi n1 pi D (0, 1) = {z C : |z| < 1}. pi n N

max|z|1

zn f (z)g (z) 1 .

21. f, g D (0, R) = {z C : |z| R} pi f (z) 6= 0 z C (0, R) = {z C : |z| = R}. pi pi > 0 f f + g

C (0, R). , pi pi f

f C (0, R).

22. , 1, = e2pii/3 2 z3 = 1. pi

f C pi 1, 2 pi pi pi

f

Res (f , 1) = 1 , Res (f , ) = a 6= 0 Res (f , 2) = a1 .pi pi pi R0 > 0 pi M > 0 z2f (z) M , |z| > R0.() pi pipi pi

1 + a+ a1 = 0 .

pi a = a = 2.

5.6. 165

()

g (z) := f (z) 1z 1

a

z a1

z 2 ,

pi g C.

() pi

f (z) = 3(z3 1)1 f (z) = 3z (z3 1)1 .

23.

exp(az + bz1

)=

n=

cnzn , a, b C ,

pi Laurent f (z) = exp(az + bz1

) -

= {z C : 0 < |z| 0 ,

pi

12pii

C+(0, r)

f n (z)fn (z)

dz

pi fn = {z C : |z| > r}. (pi- fn pi C (0, r).)

()

1

2pii

C+(0, r)

f n (z)fn (z)

dz

n r > 0;

166 5. LAURENT- -

() n r > 0 pi fn

D (0, r) = {z C : |z| < r}.

pi. (Fn) C

Fn (z) := fn (1/z) = 1 + z +z2

2!+ + z

n

n!.

25. pi f D (0, 1) =

{z C : |z| 1} pi pi C (0, 1) = {z C : |z| = 1}, f .

pi. a = + i C, 6= 0, pi f (z) 6= a |z| < 1.pi f (z) a.

26. > 0

sin2 x+ i = exp(

12 Log

(sin2 x+ i

)), pi w = Log z -

pi C \ (, 0]. x [0, pi/2], pi

|1 cosx| sin2 x sin2 x+ i .

() pi pi

0

1sin2 x+ i

dx = 2

pi/20

1sin2 x+ i

dx

= 2

pi/20

cosxsin2 x+ i

dx+ 2

pi/20

1 cosxsin2 x+ i

dx .

() pi Lebesgue, pi

lim0+

2

pi/20

1 cosxsin2 x+ i

dx = 2 ln 2 .

() pi

2

pi/20

cosxsin2 x+ i

dx

pi

lim0+

(ln+

pi0

1sin2 x+ i

dx

)= 4 ln 2 pi

2i .

5.6. 167

27. pi (fn) , fn f pi pi pi D C. fn 1 1 n N, pi f f 1 1 D.

168 5. LAURENT- -

.1 pi

pi pi

. pi pi Karl Weierstrass 1860. pi

, pipipi [15, Theorem 3.1.8] [;, 10.28 Theorem ], pi

Morera Cauchy.

.1 ( ). 1. pi (fn)

C, fn f pi pi . , f f (k)n f (k),k N, pi pi .

2. (gn)

C g(z) = n=1 gn(z) pi pi , g g(k)(z) =

n=1 g

(k)n (z), k N,

pi pi .

.1.1 pi ( )

f D C (f,D). (f1, D1), (f2, D2) ,

(f1, D1) = (f2, D2) D1 = D2 f1 = f2. (f2, D2)

169

170 .

pi (f1, D1)

D1 D2 6= f1(z) = f2(z) , z D1 D2 .

pi (f,D) pi. pi D1 D f1 = f |D1 , f1 pi f D1, (f1, D1) pi (f,D). pi .

pi . pi pi.

(f,D) D = D (0, 1)

f(z) =n=0

zn , z D(0, 1) .

f D(0, 1). D1 = C \ {1}

f1(z) =1

1 z , z C \ {1} ,

(f1, D1) pi (f,D).

.2. (f1, D1), (f2, D2) pi (f,D), f1 = f2.

pi. pi pi D D1 6=

f1(z) = f(z) = f2(z) , z D D1 .

, pi pi f1 = f2.

(f,D) , f pi ; pi pi

pi .

.3.

f(z) :=

k=0

z2k

= z + z2 + z4 + z8 + , z D(0, 1) .

(f,D (0, 1)) pi.

pi. pi (f,D(0, 1)) pi. f

pi pi C(0, 1) pi

D (0, 1). pi

ei , pi = 2mpi2n

m,n N .

.2. EULER 171

limr1

f (rei) = + . pi (pi pi f )

pi (f,D(0, 1)) pi pi.

z = rei,

f (z) =n1k=0

z2k

+k=n

r2k

exp(

2kn+1mpii)

=n1k=0

z2k

+k=n

r2k (exp

(2kn+1mpii

)= 1)

= g (z) + h (r) ,

pi g (z) =n1

k=0 z2k D (0, 1)

h (r) :=k=n

r2k. (0 < r < 1)

pi limr1 h (r) = a. , N N n+Nk=n

r2k a (0 < r < 1)

pi

N + 1 = limr1

n+Nk=n

r2k a . (pi)

, limr1 h (r) = + pi

limr1

f (rei) = + .

.2 Euler

pi

0ettx1 dt

172 .

x > 0. : (0,) R

(x) :=

0

ettx1 dt .

pi (pipipi [35, Section 5.3]) pi ,

limx0+ (x) = limx (x) = . log-, x, y > 0 , 0 + = 1

(x+ y) (x) + (y) .

pi pi

(x+ 1) = x (x) x > 0, (1) = 1 (n+ 1) = n! .

pipi

pipi D = {z C : 0}. pi t > 0

tz1 = exp {(z 1) ln t} |tz1| = tx1 , pi x = 0 .

pi 0

ettz1 dt = 0

ettx1 dt = (x) , z D ,

0 ettz1 dt pi pi pi

D

(z) :=

0

ettz1 dt , z D .

D R

(z) dz =

R

( 0

ettz1 dt)dz

pi R pi D. pi R

( 0

ettz1 dt) dz = R

(x) dz

, pi . pi, pi

ettz1 z, R

(z) dz =

0

Rettz1 dz dt

=

0

0 dt ( Cauchy)

= 0 .

.2. EULER 173

, pi Morera pi D.

pi pi pi

(z + 1) = z (z) . ( 0)

pi C,

pi. 1 1

1 (z) := (z + 1)

z. ( 1)

1 pi pipi 1 pi 0. pi (1) = 1 6= 0, 0 pi pi 1. 1 = D = {z C : 0}.

D1 = {z C : 1} \ {0} ,

(1, D1) pi (, D). pi

(z + 2) = (z + 1) (z + 1) = z (z + 1) (z) . ( 0)

pi 2 2

2 (z) := (z + 2)

z (z + 1). ( 2)

pi pi pi pi (2, D2) (, D), pi

D2 = {z C : 2} \ {0,1} .

1 = 2 1. pi C. pi ,

C pi n (n = 0, 1, 2, . . .) pi pi pi Euler. , n N

(z + n) = z (z + 1) (z + n 1) (z) . (z C)

pi

(z) = (z + n)

z (z + 1) (z + n 1)

174 .

pi pi pi n (n =0, 1, 2, . . .)

Res (,n) = limzn (z + n) (z)

= limzn

(z + n+ 1)

z (z + 1) (z + n 1)=

(1)

(n) (n+ 1) (n+ n 1)=

(1)nn!

.

pi .

.4. C pi

(z + 1) = z (z) . (z C)

pi n (n = 0, 1, 2, . . .), pi pi

Res (,n) = (1)n

n!.

. pi

pi ( pi pipipi [3, 267, 268]).

.5.

(z) (1 z) = pisin (piz)

. (z C)

IntroCA Book

Documents

Transcript of IntroCA Book