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DESCRIPTION
αναλυση 1 πολυτεχνειοTRANSCRIPT
-
&
pi
13 2013
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i
1 1
1.1 201112 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 200910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3 20089 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.4 20078 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.5 200607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2 89
2.1 201112 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.2 200910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
2.3 20089 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.4 20078 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.5 20067 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
-
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
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 f (z) dz pipi f pi pi
-
iii
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
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iv
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1
1.1 201112
&
1
1. : z4 = 1 +
3i.
. pi 1 +3i = 2e2pii/3, z4 = 1 +3i
zk = 21/4e(2kpii+2pii/3)/4 , k = 0, 1, 2, 3 .
pi z4 = 1 +3i
z0 = 21/4epii/6 , z1 = 21/4e2pii/3 , z2 = 21/4e7pii/6 z3 = 21/4e5pii/3 .
2. n N,
zn1 = z , z C .
pipi (i) n = 1, (ii) n = 2 (iii) n 3.
. (i) n = 1: z = 1 z = 1.
1
-
2 1.
(ii) n = 2: z = z z = x R, pi.
(iii) n 3: zn1 = z z = 0. z 6= 0, zn1 = zpi
|z|n1 = |z| |z|n2 = 1 |z| = 1 .
pi z 6= 0zn1 = z zn = 1
zn = 1 n- ,
zk = e2kpii/n , k = 0, 1, 2, . . . , n 1 .
n 3 zn1 = z {0, 1, e2pii/n, e4pii/n, . . . , e2(n1)pii/n
}={
0, 1, , 2, . . . , n1}, pi = e2pii/n .
3. f : C C
f(z) =
z2
z z 6= 0,
0 z = 0 .
pi pi CauchyRiemann (0, 0) pi
pi f (0).
. x R, x 6= 0,
f(x, 0) f(0, 0)x 0 =
x2/x
x= 1
pif
x(0, 0) = lim
x0f(x, 0) f(0, 0)
x 0 = 1 .
, y R, y 6= 0,
f(0, y) f(0, 0)y 0 =
(iy)2/iy
y= i
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1.1. 201112 3
pif
y(0, 0) = lim
y0f(0, y) f(0, 0)
y 0 = i .
fx
(0, 0) = ify
(0, 0) = 1, pi CauchyRiemann
(0, 0).
pi f pi 0. z C,z 6= 0,
f(z) f(0)z 0 =
f(z)
z=
(z
z
)2.
z = x+ ix C R x 6= 0. ,
f(z) f(0)z 0 =
(x+ ix
x+ ix
)2=
(1 i1 + i
)2=
(1 2 2i
1 + 2
)2 pi
limz0
z=x+ix
f(z) f(0)z 0 =
(1 2 2i
1 + 2
)2 pi . , pi f (0) pi.
. pi pi CauchyRiemann
(0, 0) pi : z = x+ iy 6= 0,
z2
z=
z3
|z|2 =(x iy)3x2 + y2
=x3 3xy2x2 + y2
+ iy3 3x2yx2 + y2
.
pi f = u+ iv
u(x, y) =
x33xy2x2+y2
(x, y) 6= (0, 0),
0 (x, y) = (0, 0) v(x, y) =
y33x2yx2+y2
(x, y) 6= (0, 0),
0 (x, y) = (0, 0) .
ux(0, 0) = vy(0, 0) = 1 uy(0, 0) = vx(0, 0) = 0, pi - CauchyRiemann (0, 0).
4. pi
f(z) = (3x2y y3 + e2y cos 2x) + i(3xy2 x3 e2y sin 2x+ 3)
C pi pi f (z). f
z, pi pi pi f (z).
-
4 1.
. f (z) = u (x, y) + iv (x, y), u (x, y) = 3x2y y3 + e2y cos 2x v (x, y) =3xy2 x3 e2y sin 2x + 3. u, v pi R2. f C pipi pi CauchyRiemann.
, (x, y) R2 ux = vy = 6xy 2e2y sin 2xuy = vx = 3x2 3y2 + 2e2y cos 2x
. pi pi pi
f (z) =f
x(x, y) = (6xy 2e2y sin 2x) + i(3y2 3x2 2e2y cos 2x) .
f (z) = u (z, 0) + iv (z, 0) = cos 2z + i(z3 sin 2z + 3) = e2iz iz3 + 3i .
pi,
f (z) = 2ie2iz 3iz2 .
5. pi f : G C pi(, )G C. z G f(z) = 0 f (z) = 0, pi f G.
pi. f2.
. g := f2 G. pi
pi g(z) = 2f(z)f (z) = 0 z G. pi, pi pi g G pi |g| = |f |2 G. |f | G pi pi pi pi f G.
6. pi
u(x, y) = y3 3x2y xx2 + y2
, (x, y) R2 \ {(0, 0)} ,
R2\{(0, 0)} f = u+iv C \ {0} f (1) = 1.
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1.1. 201112 5
. u pi 2 R2 \ {(0, 0)}
ux = 6xy + x2 y2
(x2 + y2)2, uxx = 6y + 2x(3y
2 x2)(x2 + y2)3
uy = 3y2 3x2 + 2xy
(x2 + y2)2, uyy = 6y +
2x(x2 3y2)(x2 + y2)3
.
uxx + uyy = 0 pi u R2 \ {(0, 0)}. v u, f = u+iv C\{0}. pi CauchyRiemann uy = vx
vx = 3y2 + 3x2 2xy(x2 + y2)2
.
pi,
v(x, y) = 3xy2 + x3 + yx2 + y2
+ c(y) .
pi CauchyRiemann ux = vy pipi
6xy + x2 y2
(x2 + y2)2= 6xy + x
2 y2(x2 + y2)2
+ c(y) c(y) = 0 c(y) = c .
v(x, y) = 3xy2 + x3 + yx2+y2
+ c C \ {0}
f(z) = u(z, 0) + iv(z, 0) = 1z
+ i(z3 + c) , c R .
f(1) = 1, pi c = 1. ,
f (z) = 1z
+ iz3 i .
7. pi w = cos z = cos(x + iy) = cosx cosh y i sinx sinh ypi
={z = x+ iy : 0 x pi
2, y 0
} z-pipi 4 w-pipi.
.
w = cos z = u(x, y) + iv(x, y) u(x, y) = cosx cosh y v(x, y) = sinx sinh y .
pi w =
cos z.
-
6 1.
x = 0, y 0, u = cosh y, y 0 v = 0. x = 0,y 0 ( ) . [1,) pi Ou.
x = pi2 , y 0, u = 0 v = sinh y, y 0. x = pi2 , y 0 .
y = 0 0 x pi2 , u = cosx v = 0. . y = 0, 0 x pi2 , . [0, 1] pi Ou.
pi pi pi
.
y = y0 > 0, 0 x pi2 , u = cosx cosh y0 0 v = sinx sinh y0 0. . y = y0 > 0, 0 x pi2 ,
u2
cosh2 y0+
v2
sinh2 y0= 1
4 . 0 < y0 < , - y = y0, 0 x pi2 , pi 4 , pi 4 . , w = cos z
4 .
8. pi
[R,R]R
|z|z dz = R3pii ,
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1.1. 201112 7
pi R pipi 0, R > 0
.
. pi R z(t) = Reit, 0 t pi pi
[R,R]R
|z|z dz =[R,R]
|z|z dz +R
|z|z dz
=
RR|x|x dx+
pi0R2eit iReit dt
= 0R
x2 dx+
R0x2 dx+ i
pi0R3 dt
= R3pii .
9. z1, z2 C
ez2 ez1 =[z1,z2]
ez dz ,
pi [z1, z2] z1 pi z2.
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8 1.
. pi
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1.1. 201112 9
(iv) C pi 0 pi. C1 C2 pi
C, C1 pi
0 C2 pi pi. ,
I =1
2pii
C+1
cos z
z(z pi)3 dz +1
2pii
C+2
cos z
z(z pi)3 dz ( Cauchy)
= 1pi3
+pi2 2
2pi3(pipi (ii) (iii))
=pi2 4
2pi3.
2. f D(0, R), pi 2pi0
f(rei)
reid = 2pif (0) ,
pi 0 < r < R.
. z = rei, 0 2pi, dz = ireid = izd, pi d = dz/iz. pi, 2pi0
f(rei)
reid =
|z|=r
f(z)
iz2dz
= 2pi
{1
2pii
|z|=r
f(z)
(z 0)2 dz}
= 2pif (0) . ( pi Cauchy pi)
3. pi |z|=1
ez
zn+1dz , n N
pi 2pi0
ecos cos(n sin ) d = 2pin!.
. pi pi Cauchy pi |z|=1
ez
zn+1dz =
2pii
n!{n!
2pii
|z|=1
ez
zn+1dz
}=
2pii
n! (ez)(n)
z=0
=2pii
n! e0 = 2pi
n!i .
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10 1.
z = ei, 0 2pi. pi dz = ieid = izd,
|z|=1
ez
zn+1dz = i
2pi0
eei
eind
= i
2pi0
ecos +i sin
eind
= i
2pi0
ecos ei(sin n) d
=
2pi0
ecos sin (n sin ) d + i 2pi0
ecos cos (n sin ) d .
, 2pi0
ecos cos(n sin ) d = 2pin!
2pi0
ecos sin (n sin ) d = 0 .
4. () f1(z) =z2 + 6z
(2 z)(z + 2)2 =1
2 z 2
(z + 2)2. pi-
f1 Taylor z0 = 0, pi
.
() pi f2(z) =1
z2 + 4z 3i z0 = 2. ;
. pi
1
1 w =n=0
wn 11 + w
=n=0
(1)nwn , |w| < 1 .
()
1
2 z =1
2(1 z/2) =1
2
n=0
(z2
)n=n=0
zn
2n+1. (|z/2| < 1 |z| < 2)
11+w =
n=0(1)nwn, |w| < 1, pi
1(1 + w)2
=
n=1
(1)nnwn1
1
(1 + w)2=
n=1
(1)n1nwn1 =n=0
(1)n(n+ 1)wn , |w| < 1 .
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1.1. 201112 11
pi
2
(z + 2)2=
1
2(1 + z/2)2=
1
2
n=0
(1)n(n+ 1)(z
2
)n=
n=0
(1)n(n+ 1) zn
2n+1.
(|z/2| < 1 |z| < 2) |z| < 2
f1(z) =n=0
1
2n+1[1 + (1)n+1 (n+ 1)] zn .
R = 2.
()
f2(z) =1
(z + 2)2 (4 + 3i)= 1
4 + 3i 1
1 (z+2)24+3i= 1
4 + 3i
n=0
((z + 2)2
4 + 3i
)n( (z+2)24+3i < 1 |z + 2|
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12 1.
6. f(z) = z23i, D(0, 2) = {z C : |z| 2} pi |f | pi .
. pi f C, pi |f | pi D(0, 2) pi |z| = 2 z() = 2ei, pi pi. |z| = 2
|f(z)| = |z2 3i| = |4e2i 3i| = |4 cos 2 + i(4 sin 2 3)|=
(16 cos2 2 + (4 sin 2 3)2
=
25 24 sin 2 .
|f | pi sin 2 = 1 = pi/4, z = 2eipi/4 =
2(1 i). pi,max|z|2
|f(z)| = |f(
2(1 i))| = 7 .
f D(0, 2) = {z C : |z| < 2}. f(z) = 0
z2 = 3i (x+ iy)2 = 3i x2 y2 + 2ixy = 3i {x2 y2 = 0, 2xy = 3} . x = y =
3/2 x = y = 3/2.
2 pi. pi z2 = 3i = 3epii/2,
zk =
3e(2kpii+pii/2)/2, k = 0, 1, z0 =
3epii/4 z1 =
3e5pii/4 .
,
min|z|2|f(z)| =
f(
3
2(1 + i)
) =f(
3
2(1 + i)
) = 0 .
7. f D(0, 1) = {z C : |z| 1} |f(z)| > 1 |z| = 1. f(0) = i, pi f D(0, 1) = {z C : |z| < 1}.
. pi f D(0, 1) = {z C : |z| < 1}., pi f pi pi -
|z| = 1. pi pi
|f(z)| > 1 |z| = 1 |f(0)| = |i| = 1 . (pi)
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1.1. 201112 13
f D(0, 1) = {z C : |z| < 1}.
8. pi f D(0, R) = {z C : |z| < R} |f(z)| M < z D(0, R). 0 2 f , f(0) = f (0) = 0 f (0) 6= 0, pi
|f(z)| MR2|z|2 z D(0, R) (1)
|f (0)| 2MR2
. (2)
pi.
g(z) =
f(z)z2
0 < |z| < R,f (0)2! z = 0 .
.
f(z) =f (0)
2!z2 +
f (0)3!
z3 + + f(n)(0)
n!zn +
= z2
(f (0)
2!+f (0)
3!z + + f
(n)(0)
n!zn2 +
), |z| < R .
g(z) := f(0)2!
+f (0)
3!z + + f
(n)(0)
n!zn2 + , g
D(0, R)
g(z) =
f(z)z2
0 < |z| < R,f (0)2! z = 0 .
z D(0, R), z . pipi r |z| < r < R. pi
|g(z)| max|w|=r
|g(w)| = max|w|=r
|f(w)||w|2
M
r2.
pi r R pipi
|g(z)| MR2
z D(0, R)
pi (1) (2).
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14 1.
. pi (2) pi pi
Cauchy. ,
|f (0)| M2!R2
.
&
3
1. pi f : C C
lim|z|
|f(z)|1 + |z|1/2 pi .
pi f .
pi. Liouville.
. lim|z||f(z)|
1+|z|1/2 = a 0, > 0 pi M > 0 |z| > M |f(z)|1 + |z|1/2 a
< .pi
|f(z)| < a+ + (a+ )|z|1/2 |z| > M .
, pi Liouville f pi 0. , f
.
2. 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 pi
|an| M er
rn n N .
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1.1. 201112 15
pi
|an| M en
nn n N .
. pi pi Cauchy pi
an =f (n)(0)
n!=
1
2pii
C+(0, r)
f(z)
zn+1dz ,
pi C+ (0, r) 0 r > 0. ,
|an| = 12pi
C+(0, r)
f(z)
zn+1dz
1
2pi
C+(0, r)
|f(z)||z|n+1 |dz|
12pi
Mer
rn+1
C+(0, r)
|dz| (|f(z)| Me|z|)
=1
2pi
Mer
rn+12pir = M
er
rn.
pi
|an| M infr>0
er
rn.
r 7 errn pi r = n. ,
|an| M en
nn n N .
3.
f (z) =1
z2 (z + 1).
pi Laurent f
z0 = 1.
. 1/ (1 w) = n=0wn, |w| < 1 ( ).
1
(1 w)2 =n=1
nwn1 , |w| < 1 .
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16 1.
1 pipi: 1 = {z C : 0 < |z + 1| < 1}. ,
f (z) =1
z2 (z + 1)=
1
[1 (z + 1)]2 (z + 1)
=1
z + 1
n=1
n (z + 1)n1
=n=1
n (z + 1)n2
=
n=1(n+ 2) (z + 1)n .
2 pipi: 2 = {z C : |z + 1| > 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 .
4. pi Laurent
f(z) =z2 + 4iz
(z 1)(z2 + 4i) z0 = 0 pi pi 1 2i. pi pi Laurent f ;
.
f(z) =z2 + 4i
(z 1)(z2 + 4i) +4iz 4i
(z 1)(z2 + 4i) =1
z 1 +4i
z2 + 4i.
f : 1,
2(1 i) 2(1 + i). pi pi Laurent pi 1 = {z C : 0 |z| < 1}(pi Taylor), 2 = {z C : 1 < |z| < 2} 3 = {z C : 2 < |z| 2.
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1.1. 201112 17
, 1/ (1 w) = n=0wn 1/ (1 + w) = n=0 (1)nwn, |w| < 1 ().
f (z) =1
z 1 +4i
z2 + 4i
=1
z
1
1 1/z +4i
z21
1 + 4i/z2
=1
z
n=0
(1
z
)n+
4i
z2
n=0
(1)n(
4i
z2
)n(|z| > 2)
=n=0
1
zn+1+n=0
(1)n (4i)n+1
z2n+2.
pipi pi 3 = {z C : 2 < |z| 0
|h(z)| k|z|1/2 z .
h z = 0;
.
() z + 2 = w z 3 = w 5,
(w 5) sin 1w
= (w 5)(
1
w 1
3! 1w3
+1
5! 1w5
)= 1 5
w 1
3! 1w2
+
pi
f(z) = 1 5z + 2
13! 1
(z + 2)2+ .
z = 2 f Res(f,2) = 5.
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18 1.
() 1(z) := sin z z cos z, 1(0) = 1(0) = 1(0) = 0 1 (0) = 2 6= 0.pi, 0 3 g 4 pi
g. , 0 pi pi g. pi |z| > 0
g(z) =1
z4
[(z z
3
3!+z5
5!
) z
(1 z
2
2!+z4
4!
)]=
(1
2! 1
3!
)1
z(
1
4! 1
5!
)z + ,
Res(g, 0) =1
2! 1
3!=
1
3.
. pi 0 pi pi g, Res(g, 0) pi pi
pi
Res(g, 0) = limz0
zsin z z cos z
z4
= limz0
sin z z cos zz3
= limz0
z sin z
3z2( LHpital)
=1
3limz0
sin z
z=
1
3.
() |zh(z)| k|z|1/2 z pi limz0 zh(z) = 0. , 0 pi (pi) h.
6. pi
I =
C+(1i, 2)
[epiz
z2 + 1+ sin
1
z+
1
ez2
]dz ,
pi C+(1 i, 2) 1 i, 2 .
. i pi pi w = epiz/(z2 + 1). i C+(1 i, 2).
Res
(epiz
z2 + 1, i
)=
epiz
(z2 + 1)
z=i
=epii
2i = 1
2i .
pi
sin1
z=
1
z 1
3!z3+
1
5!z5 ,
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1.1. 201112 19
Res(sin 1z , 0) = 1. , pi pipi
I = 2pii
[Res
(epiz
z2 + 1, i
)+ Res
(sin
1
z, 0
)]= 2pii
[1
2i+ 1
]= pi + 2pii .
w = 1/ez2 C pi pi
Cauchy C+(1i, 2)
1
ez2dz = 0 .
7. z2e2iz1 =
n= anz
n pi Laurent f(z) = z2e2iz1 -
= {z C : pi < |z| < 2pi} z0 = 0 .
pi Laurent pi pi-
pi pi an, n 1.
. pi
e2iz 1 = 0 e2iz = 1 2iz = 2kpii z = kpi , k Z ,
zk = kpi, k Z \ {0}, pi pi f(z) = z2/(eiz 1). pi 0 2 pi pi f , 0 pi
(pi) . pi Laurent an pi pi
an =1
2pii
C+(0, r)
z2/(e2iz 1)zn+1
dz =1
2pii
C+(0, r)
zn+1
e2iz 1 dz ,
pi C+(0, r) 0, r, pi < r < 2pi
.
g(z) :=zn+1
e2iz 1 ,
pi pi g C+(0, r) pi pi.
(i) n = 1: pipi pi, 0 pi g(z) = 1eiz1
C+(0, r) pi pi. pi
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20 1.
pipi
a1 =1
2pii
C+(0, r)
1
e2iz 1 dz
= Res
(1
e2iz 1 , pi)
+ Res
(1
e2iz 1 , 0)
+ Res
(1
e2iz 1 , pi)
=1
(e2iz 1)z=pi
+1
(e2iz 1)z=0
+1
(e2iz 1)z=pi
=1
2ie2ipi+
1
2i+
1
2ie2ipi=
3
2i= 3
2i .
(ii) n 0: pipi
g(z) =zn+1
e2iz 1 n+ 1 1
pi pi pi g C+(0, r) pi pi. pi 0 1 pi pi g(z) = zn+1/(e2iz 1), 0 pi (pi) g. , pi pipi
an =1
2pii
C+(0, r)
zn+1
e2iz 1 dz = Res(zn+1
e2iz 1 , pi)
+ Res
(zn+1
e2iz 1 , pi)
=zn+1
(e2iz 1)z=pi
+zn+1
(e2iz 1)z=pi
=(pi)n+1
2ie2ipi+pin+1
2ie2ipi
=pin+1
2i
[(1)n+1 + 1
]=
0 n = 0,2,4, . . . ,
ipin+1 n = 1,3,5, . . . .
8. a > 0, pi pi0
1
a2 + cos2 d =
2pi0
1
2a2 + 1 + cosd =
pi
a
1 + a2.
. pi0
1
a2 + cos2 d =
pi0
2
2a2 + 1 + cos 2d =
2pi0
1
2a2 + 1 + cosd .
( = 2)
-
1.1. 201112 21
z = ei, cos = ei + ei
2=
1
2
(z +
1
z
) dz = ieid = izd d = dz
iz. -
pi, pi0
1
a2 + cos2 d =
|z|=1
1
2a2 + 1 + 12(z + 1z
) dziz
=2
i
|z|=1
1
z2 + 2 (1 + 2a2) z + 1dz .
pi z1,2 = 1 2a2 2a
1 + a2 pi z2 + 2(1 + 2a2)z+ 1 = 0,
1 2a2 2a1 + a2 pi pi g(z) = 1/(z2 + 2(1 +2a2)z + 1). 1 2a2 + 2a1 + a2 |z| = 1, 1 2a2 2a1 + a2 |z| = 1. , pi pipi
pi0
1
a2 + cos2 d =
2
i
|z|=1
1
z2 + 2 (1 + 2a2) z + 1dz
=2
i2piiRes
(1
z2 + 2(1 + 2a2)z + 1, 1 2a2 + 2a
1 + a2
)= 4pi
1
(z2 + 2(1 + 2a2)z + 1)
z=12a2+2a1+a2
= 4pi1
2(1 2a2 + 2a1 + a2) + 2(1 + 2a2) =pi
a
1 + a2.
9. R z () = Rei, 0 pi, pipi 0 R > 0 a, b > 0, a 6= b. pi
limR
R
1
(z2 + a2)(z2 + b2)dz = 0 .
pi pipi pi
0
1
(x2 + a2)(x2 + b2)dx .
. z R 1(z2 + a2)(z2 + b2) 1(|z|2 a2)(|z|2 b2) = 1(R2 a2)(R2 b2) , R > a, b .
R piR piR
1
(z2 + a2)(z2 + b2)dz
piR(R2 a2)(R2 b2) R 0 .
-
22 1.
,
limR
R
1
(z2 + a2)(z2 + b2)dz = 0 .
ai,bi pi pi f(z) = 1/(z2+a2)(z2+b2). f pi pi pi pi pi pipi
R z () = Rei, 0 pi [R,R]. R ai bi f
R.
pi pipi RR
1
(x2 + a2)(x2 + b2)dx+
R
1
(z2 + a2)(z2 + b2)dz = 2pii [Res (f, ai) + Res (f, bi)] .
(1.1)
Res (f, ai) = limzai
(z ai) 1(z ai)(z + ai)(z2 + b2) =
1
2ai(b2 a2)
Res (f, bi) = limzbi
(z bi) 1(x2 + a2)(z bi)(z + bi) =
1
2bi(a2 b2) .
pi, pi (1.1) pipi
1
(x2 + a2)(x2 + b2)dx = lim
R
RR
1
(x2 + a2)(x2 + b2)dx
= 2pii1
2i
(1
a(b2 a2) +1
b(a2 b2))
=pi
ab(a+ b).
, 0
1
(x2 + a2)(x2 + b2)dx =
1
2
1
(x2 + a2)(x2 + b2)dx =
pi
2ab(a+ b).
-
1.2. 200910 23
1.2 200910
1
1. pi f = u+iv piG. u = v2, pi
f .
. pi CauchyRiemann vy = ux
vx = uy
vy = 2vvx
vx = 2vvy
vy = 2vvx
vx = 4v2vx
vy = 2vvx(1 + 4v2
)vx = 0
.pi, vx = vy = ux = uy = 0 pi f (z) = ux (x, y) + ivx (x, y) = 0
pi G. , pi pi f pi G.
2. f (z) = u (x, y) + iv (x, y) D (0, r) =
{z C : |z| r}. pi C(0, r)
u
ydx u
xdy = 0 ,
pi C (0, r) 0 r.
. , u pi pi
Laplace2u
x2+2u
y2= 0 .
pi u pi, pi Green C+(0, r)
u
ydx u
xdy =
D(0, r)
(2u
x2+2u
y2
)dxdy = 0 .
, C(0, r)
u
ydx u
xdy = 0 .
3. f = u+ iv C \ {0}, f (1) = 2 + 2i,
u (x, y) =x(x2 + y2 + 1
)x2 + y2
.
-
24 1.
f z = x+ iy.
.
u (x, y) = x+x
x2 + y2
pi C \ {0} pi Laplace
uxx + uyy = 2xx2 3y2
(x2 + y2)3+ 2x
3y2 x2(x2 + y2)3
= 0 ,
u C \ {0}. v u, pi CauchyRiemann ux = vy
vy = 1 +y2 x2
(x2 + y2)2= 1 1
x2 + y2+
2y2
(x2 + y2)2.
pi y
v (x, y) = y yx2 + y2
+ c (x) .
pi CauchyRiemann uy = vx pipi
2xy(x2 + y2)2
= 2xy(x2 + y2)2
c (x) c (x) = 0 c (x) = c .
v (x, y) = y yx2 + y2
+ c , c R .
pi,
f (z) = u (z, 0) + iv (z, 0) = z +1
z+ ic , c R .
f (1) = 2 + 2i, pi c = 2. ,
f (z) = z +1
z+ 2i .
4. , w = tan z,
D = {z C : pi/4
-
1.2. 200910 25
y = 0, pi/4 x 0. , w = tan (x+ iy) = tanx.pi u = tanx [pi/4, 0], y = 0, pi/4 x 0 z-pipi pi v = 0,1 u 0 w-pipi.
x = 0, y 0. ,
w = tan (iy) = i tanh y , (tanh y) = 1/ cosh2 y > 0.
pi t = tanh y
limy tanh y = limy
ey eyey + ey
= limy
1 e2y1 + e2y
= 1 ,
x = 0, y 0 z-pipi pi u = 0,0 v 1 w-pipi.
x = pi/4, y 0. ,
w = u (x, y) + iu (x, y) = tan (pi/4 + iy)
=tan (pi/4) + tan (iy)
1 tan (pi/4) tan (iy)=1 + i tanh y1 + i tanh y
=(1 + i tanh y) (1 i tanh y)
1 + tanh2 y
=1 + tanh2 y1 + tanh2 y
+ i2 tanh y
1 + tanh2 y.
pi,
u2 (x, y) + v2 (x, y) =
(1 + tanh2 y)2(1 + tanh2 y
)2 + 4 tanh2 y(1 + tanh2 y
)2 =(1 + tanh2 y
)2(1 + tanh2 y
)2 = 1 .pi, pi y 0
1 u(x, y) = 1 2 11 + tanh2 y
< 0 limyu(x, y) = 0
0 v(x, y) = 2 tanh y1 + tanh2 y
1 .
, x = pi/4, y 0 z-pipi pi pi 2 w-pipi.
-
26 1.
, z = pi/6 + i ln 2 D pi w = tan (pi/6 + i ln 2) =(43 + 15i) /21., D z-pipi pi w = tan z
D ={w = u+ iv C : u2 + v2 1 , 1 u 0, 0 v 1} .
5. f pi
C (0, 1) a C. pi
I =1
2pii
C+(0, 1)
f (z)
z a dz ,
(i) |a| < 1 (ii) |a| > 1.pi. pi C+ (0, 1) -
z () = ei , [0, 2pi] .
-
1.2. 200910 27
.
I =1
2pii
2pi0
f (ei)
ei a iei d
=1
2pii
2pi0
(f (ei)
ei a (iei)
)d
=1
2pii
2pi0
f (ei)
ei a iei d
=1
2pii
2pi0
f (ei)
ei ae2i iei d
=1
2pii
C+(0, 1)
f (z)
z az2 dz
=1
2pii
C+(0, 1)
f (z)
zdz 1
2pii
C+(0, 1)
f (z)
z 1/a dz
pi, pi pi Cauchy pi Cauchy
I =1
2pii
C+(0, 1)
f (z)
z a dz =
f (0) |a| < 1,
f (0) f (1/a) |a| > 1 .
6. f pi
C (0, 1). |f (z) z| < 1 , pi f (1/2) 8 .
. 1 pi. pi pi Cauchy pi
f (1/2) =1
2pii
C+(0, 1)
f (z)
(z 1/2)2 dz ,
pi C+ (0, 1) . |z| = 1, z C (0, 1)
|z 1/2| |z| 1/2 = 1 1/2 = 1/2
|f (z)| = |(f (z) z) + z| |f (z) z|+ |z| < 1 + 1 = 2 .
-
28 1.
pi,
f (1/2) = 12pi
C+(0, 1)
f (z)
(z 1/2)2 dz
12pi
C+(0, 1)
|f (z)||z 1/2|2 |dz|
12pi
C+(0, 1)
2
(1/2)2|dz|
=8
2pi
C+(0, 1)
|dz|
=8
2pi2pi = 8 .
2 pi. pi Cauchy
f (1/2) M1/2
= 2M ,
piM = max {|f (z)| : z C (1/2, 1/2)}. pi z C (0, 1) |f (z)| < 2 C (1/2, 1/2) C (0, 1), pi pi
M < 2. , f (1/2) < 4 .
7.
f (z) =
n=0
anzn =
n=0
f (n) (0)
n!zn , z C .
pi R > 0
n=0
|an|Rn 2M (2R) , pi M (r) = max {|f (z)| : |z| = r}.
. pi Cauchy
|an| =f (n) (0)
n! M (2R)
(2R)n.
pi,n=0
|an|Rn n=0
M (2R)
(2R)nRn = M (2R)
n=0
1
2n= 2M (2R) .
-
1.2. 200910 29
8. (pi ). f (z) = eizez2/2, R,pi pi pipi,
pi
eixex2/2 dx =
2pie
2/2 pi
ex2/2 cosx dx =
2pie
2/2 .
2
1. pi f : C C =f (z) 0, z C. pi f .
pi. g (z) = eif(z).
. pi =f (z) 0, z C,
|g (z)| =eif(z) = ei(
-
30 1.
z D (0, 1). ,
(f(z) 1) (f(z) 1) = 0 , D (0, 1).
, pi pi ( ) pi 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).
. f, g : G C pi G C
f(z)g(z) = 0 , z G,
f 0 g 0 G.
pi.
g (z0) 6= 0 pi z0 G. , pi g pi piD (z0, r) G g (z) 6= 0 z D (z0, r). pi pi pi f (z) = 0 z D (z0, r) pi f (n) (z) = 0 z D (z0, r) n N.
A ={z G : f (n) (z) = 0 , n N
}.
A 6= . zk A limk zk = z, pi f (n)
f (n) (z) = limk
f (n) (zk) = 0 .
z A pi A . pi A . , a A. pi G , pi pi D (a, R) G. ,
f (z) =n=0
f (n) (a)
n!(z a)n = 0 , z D (a, R).
pi, f (n)(z) = 0 z D (a, R) n N. D (a, R) A pi A . pi pi G
G = A (G \A), pi A(A 6= ) G \A , pi G = A., f 0 G.
-
1.2. 200910 31
3.
f(z) =1
z4 + 13z2 + 36.
pi Laurent f z0 = 0 pi pi
2 + i. pi pi Laurent
f ;
. 2i, 3i f .
f (z) =1
(z2 + 4) (z2 + 9)=
1
5
[1
z2 + 4 1z2 + 9
].
, 1/ (1 + w) =
n=0 (1)nwn, |w| < 1 ( ). = {z C : 2 < |z| < 3}, 2 + i pi Laurent f
f (z) =1
5
[1
z2 + 4 1z2 + 9
]=
1
5
[1
z21
1 + 4/z2 1
9
1
1 + z2/9
]=
1
5
[1
z2
n=0
(1)n(
4
z2
)n 1
9
n=0
(1)n(z2
9
)n]
=1
5
[ n=0
(1)n 4n
z2n+2n=0
(1)n z2n
9n+1
].
4. f (z) =
n= an (z z0)n pi Laurent () f = {z C : 0 < |z z0| < R} z0 C. z
|f (z)| C |z z0|p , 0 < p < 1 ,
pi Laurent pi z0 pi()
f .
. pi Laurent an pi pi
an =1
2pii
C+(z0, r)
f (z)
(z z0)n+1dz ,
pi C+ (z0, r) z0 r, 0 < r < R. pi
-
32 1.
|an| = 12pi
C+(z0, r)
f (z)
(z z0)n+1dz
1
2pi
C+(z0, r)
|f (z)||z z0|n+1
|dz|
C2pirn+1+p
C+(0, r)
|dz| (|f (z)| C |z z0|p)
=C
2pirn+1+p2pir =
C
rn+p.
pi p (0, 1), n 1 n 1
|an| Crnp r0+
0 .
pi an = 0 n = 1,2,3, . . . . , z0 pi() f .
5. () ez1/z =
n= cnzn pi Laurent f (z) = ez1/z
z0 = 0 = {z C : 0 < |z|
-
1.2. 200910 33
pi Laurent f(z) = eze1/z z0 = 0 : 0 b > 0, pi pi pi0
sin2
a+ b cos d =
pi
b2
(a
a2 b2
).
. pi0
sin2
a+ b cos d =
1
2
2pi0
sin2
a+ b cos d =
1
4
2pi0
1 cos 2a+ b cos
d = 0}
= {w C : 0 < =w < pi} .
[1 + t (1 + i) , i], 0 < t < 1, 2 1 + t (1 + i) pi i. pi
(Log (Log z)) =1
z Log z, z A,
pi [1+t(1+i), i]
1
z Log zdz = Log (Log i) Log (Log (1 + t (1 + i)))
= Log(ipi
2
) Log (Log (1 + t (1 + i)))
= lnpi
2+ i
pi
2 Log (Log (1 + t (1 + i))) .
,
1
z Log zdz = ln
pi
2+ i
pi
2 limt0+
Log (Log (1 + t (1 + i)))
= lnpi
2+ i
pi
2 Log (ipi)
= lnpi
2+ i
pi
2(
lnpi + ipi
2
)= ln 2 .
6. f : G C pi pi G, f(z) 6= 0 z G. pi pi h G
f (z) = eh(z) .
-
42 1.
. h f G.
pi. , z0 G
F (z) :=
zz0
f (z)f (z)
dz , z G ,
F (z) = f (z) /f (z). H (z) := eF (z)/f (z).
pi.
H (z) :=eF (z)
f (z),
pi F G, F (z) = f (z) /f (z). H
G,
H (z) = F (z)eF (z)
f (z) eF (z) f
(z)f2 (z)
= 0 , z G.
pi, pi pi H pi
pi G. H(z) = C, C 6= 0. pi pi C C \ {0}, pi c C ec = C. ,
eF (z)
f(z)= C = ec f(z) = eF (z)c .
h(z) := F (z) c, h G
f(z) = eh(z) .
7. f, g : C C
f2(z) + g2(z) = 1 , z C.
pi pi h
f(z) = cos(h(z)) g(z) = sin(h(z)) .
pi. pi .
-
1.3. 20089 43
pi. pi pi, z C
[f(z) + ig(z)] [f(z) ig(z)] = 1 .
pi f + ig C, pi pi
pi h C
f(z) + ig(z) = eih(z) .
pi
f(z) ig(z) = 1f(z) + ig(z)
= eih(z) .
,
f(z) =eih(z) + eih(z)
2= cos(h(z)) g(z) = e
ih(z) eih(z)2i
= sin(h(z)) .
8. pi
I =1
2pii
C+
z + 1
z2(z 2) dz ,
pi C+ pi pi 0 2.
pipi.
.
() C+ pi 0 2: pi Cauchy I = 0.
() C+ pi 0: pi pi Cauchy
pi
I =1!
2pii
C+
(z + 1) / (z 2)z2
dz =
(z + 1
z 2)
z=0
=3
(z 2)2z=0
= 34.
() C+ pi 2: pi pi Cauchy -
I =1
2pii
C+
(z + 1) /z2
z 2 dz =z + 1
z2
z=2
=3
4.
-
44 1.
() C+ pi 0 2: C+1 C+2 pi -
C+, C+1 pi
0 C+2 pi 2. pi
Cauchy
I =1
2pii
C+
z + 1
z2 (z 2) dz =1
2pii
C+1
z + 1
z2 (z 2) dz +1
2pii
C+2
z + 1
z2 (z 2) dz
= 34
+3
4= 0 . (pipi () ())
2
1. () f D(0, R) = {z C : |z| R} M > 0. |f(z)| > M |z| = R |f(0)| < M , pi f D(0, R) = {z C : |z| < R}.
() pi () pi :
pi
p(z) = anzn + an1zn1 + + a1z + a0 , a0, a1, . . . , an1, an C , an 6= 0
n C.
pi. pi p C. lim|z| |p(z)| =.
.
() pi f D (0, R), pi -
pi ( f ). pi f
D (0, R).
-
1.3. 20089 45
() M > |a0|. pi lim|z| |p(z)| =, pi C (0, R) = {z C : |z| = R} |p(z)| > M z C (0, R), |z| = R. pi|p(0)| = |a0| < M , pi () pi p D (0, R) = {z C : |z| < R}.
2.
f (z) =z2 + 2z + 2 + 2i
(z + 1) (z2 + 2i).
pi Laurent f z0 = 0 pi pi
3/2. pi pi Laurent f
;
. 1, 2e3ipi/4 2e7ipi/4 f .
f (z) =1
z + 1+
2
z2 + 2i.
, 1/ (1 w) = n=0wn 1/ (1 + w) = n=0 (1)nwn, |w| < 1 ().
={z C : |z| > 2}, 3/2 pi Laurent f
f (z) =1
z + 1+
2
z2 + 2i
=1
z
1
1 + 1/z+
1
z22
1 + 2i/z2
=1
z
n=0
(1)n(
1
z
)n+
2
z2
n=0
(1)n(
2i
z2
)n=n=0
(1)n 1zn+1
+n=0
(1)n 2 (2i)n
z2n+2.
3. pi pi, , pi
z0 = 0 . n Z, pi Laurent pi
1
2pii
wn sin (1/w) dw =
(1)k(2k+1)! n = 2k, k = 0, 1, 2, . . . ,
0 .
-
46 1.
. pi z C
sin z =k=0
(1)k z2k+1
(2k + 1)!,
w sin (1/w) =
k=0
(1)k w2k
(2k + 1)!, 0 < |w|
-
1.3. 20089 47
pi Laurent f (z) = e1/ze2z z0 = 0 : 0 0, , pi pi
Cauchy C+(0, 1)
eaz
zdz =
C+(0, 1)
eaz
z 0 dz = 2pii eaz|z=0 = 2pii .
|z| = 1 z() = ei, pi pi pi
2pii =
C+(0, 1)
eaz
zdz =
pipi
eaei
eiiei d
= i
pipiea cos +ia sin d
= i
pipiea cos [cos (a sin ) + i sin (a sin )] d
= pipiea cos sin (a sin ) d + i
pipiea cos cos (a sin ) d .
, pipiea cos cos (a sin ) d = 2pi
pi0ea cos cos (a sin ) d = pi .
5. 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 n 1 pi f z1, z2, . . . , zn, P (zk) = f(zk), k = 1, 2, . . . , n.
-
60 1.
. 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 n 1. 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 .
6. pi p(z) =n
k=0 akzk, ak C, an 6= 0, 2,
n 2. pi pi pi, pi , pi
1
p(z)dz = 0 .
pi. pi R > 0,
|p(z)| 12|an||z|n , |z| R .
. R pi pi
|z| = R
|p (z)| 12|an| |z|n , |z| R .
,1
|p (z)| 2
|an| |z|n , |z| R .
pi Cauchy
1
p (z)dz =
|z|=R
1
p (z)dz .
-
1.4. 20078 61
pi
1
p (z)dz
=|z|=R
1
p (z)dz
|z|=R
1
|p (z)| |dz|
|z|=R
2
|an| |z|n |dz|
=2
|an|Rn|z|=R
|dz|
=2
|an|Rn 2piR =4pi
|an|Rn1 R 0 .
,
1
p (z)dz = 0 .
7. pi f : C C |f(z)| Aeax z = x+ iy C, pi a,A > 0. pi
f(z) = ceaz , pi c C .
pi |f(z)| Aea|z|, z C;
. g(z) = f(z)/eaz
|g(z)| = |f(z)||eaz| =|f(z)||eax+iay| =
|f(z)|eax
A , z = x+ iy C .
pi Liouville g , g(z) = c. pi f(z) = ceaz
pi c C. |f(z)| Aea|z|, z C, pi f(z) = ceaz pi c C. , f(z) = A sin az, a,A > 0, pi f(z) = ceaz, pi c C.
|f(z)| = Aeiaz eiaz2i
A2 (e|iaz| + e|iaz|) = Aea|z| .
8. pi:
-
62 1.
() f pi pi D, pi
F pi D F (z) = f(z), z D.
() pi f D (0, 2) = {z C : |z| < 2},
|f(x+ iy)|2 = 4 x2 y2 , z = x+ iy D (0, 2) .
.
() 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.
() 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).
9. R pi pi-
pi pi. pi
Weierstrass.
a < b R f : [a, b] R , > 0 pipi
p(x) =
nk=0
akxk (n N , ak R)
|f(x) p(x)| < , x [a, b] .
Weierstrass pipi. -
, f(z) = 1/z C (0, 1) =
{z C : |z| = 1} pi pi ( ) pi C.
-
1.4. 20078 63
= 1 pi pi pi
p(z) =nk=0
akzk (n N , ak C)
1z p(z) < 1 , |z| = 1 ,
pi pi.
. pi 1z p(z) < 1 |1 zp(z)| < 1 , |z| = 1 .
q(z) := zp(z). q pi pi
C,
|1 q(z)| < 1 , |z| = 1 .
pi pipi
|1 q(z)| < 1 , |z| 1 .
pi pipi z = 0
1 = |1 q(0)| < 1 ,
pi pi.
-
64 1.
1.5 200607
1
1. pi z1, z2, . . . , zn pi
p (z) = a (z z1) (z z2) (z zn) ,
a C, pipi =z > 0.
() pi x R
=p (x)p (x)
==z1
|x z1|2+
=z2|x z2|2
+ + =zn|x zn|2> 0 .
() pi =p (x)p (x)
dx .
.
() pi
p(z)p(z)
=1
z z1 +1
z z2 + +1
z zn =z z1|z z1|2 +
z z2|z z2|2 + +
z zn|z zn|2 ,
x R
=p(x)p(x)
= = x z1|x z1|2 + =x z2|x z2|2 + + =
x zn|x zn|2
==z1
|x z1|2 +=z2
|x z2|2 + +=zn
|x zn|2 > 0 .
() zk = xk + iyk, k = 1, 2, . . . , n, yk > 0
=zk|x zk|2 dx =
yk(x xk)2 + y2k
dx
= limR+r
Rr
yk(x xk)2 + y2k
dx
= limR+r
arctan
(x xkyk
)x=Rx=r
= limR+
arctan
(R xkyk
) limr arctan
(r xkyk
)=pi
2(pi
2
)= pi .
-
1.5. 200607 65
pi, =p(x)p(x)
dx =
=z1|x z1|2 dx+
=z2|x z2|2 dx+ +
=zn|x zn|2 dx
= pin .
2. f : C C,
f(z) =
e1/z4 z 6= 0,
0 z = 0 .
pi pi CauchyRiemann (0, 0)
pi pi f (0).
.
limx0+
f (x, 0) f (0, 0)x 0 = limx0+
e1/x4
x
= limt
t1/4
et( t = x4)
=1
4limt
t3/4
et(LHpital)
= 0
pi
limx0
f (x, 0) f (0, 0)x 0 = limx0
e1/x4
x= 0 .
pi fx (0, 0) = 0. pi pi
fy (0, 0) = limy0
f (0, y) f (0, 0)y 0 = 0 .
fx (0, 0) = ify (0, 0) = 0, pi CauchyRiemann (0, 0).
pi f pi 0.
f (0 + h) f (0)h
=f (h)
h=
e1/h4/h h R,
e1/4x4/ (1 + i)x h = x+ ix, x R .
-
66 1.
pi,
limh0hR
f(0 + h) f(0)h
= limh0
e1/h4
h= 0 lim
h0h=x+ix
f(0 + h) f(0)h
=1
1 + ilimx0
e1/4x4
x= .
, pi f (0) pi.
. f pi 0 pipi pi f
0. ,
limr0+
f(reipi/4
)= lim
r0+e1/r
4=
3. f : A C R, pi A . f pipi . f pi z0 A, pi f (z0) = 0.
pi.
limh0
f (z0 + h) f (z0)h
= f (z0) .
pi,
limh0hR
f (z0 + h) f (z0)h
= f (z0)
pi pi pi pipi pi f (z0)
pi . h = ik, k R, pi
limk0
f (z0 + ik) f (z0)ik
= f (z0) .
pi if (z0) R. pipi f (z0) R if (z0) R pi f (z0) = 0.
4. f : G C C pi G. pi pi L C f (z) L, z G. pi f pi G.
pi. L C
L = {z C : z = z0 + tw, t R}
z0 C w C, w 6= 0.
-
1.5. 200607 67
pi. f (z) L,
f (z) z0w
R , z G .
(f(z) z0) /w piG pi pi .pi pi (pi pi pi f (z) = 0,
z G), f pi G.
5. pi
f(z) = f(x+ iy) = e2xy[cos(y2 x2)+ i sin (y2 x2)]
C pi pi f (z).
. f = u+ iv u(x, y) = e2xy cos(y2x2) v(x, y) = e2xy sin(y2x2). u, v pi R2 pi CauchyRiemann.
, ux = vy = 2e
2xy[y cos(y2 x2) + x sin(y2 x2)]
uy = vx = 2e2xy[x cos(y2 x2) y sin(y2 x2)]
.pi f C. pi pi f (z):
1 pi.
f (z) = fx (x, y) = ux (x, y) + ivx (x, y)
= 2e2xy[(y cos
(y2 x2)+ x sin (y2 x2))
+i(y sin
(y2 x2) x cos (y2 x2))] .
2 pi. pi u, v ,
f(z) = u(z, 0) + iv(z, 0) = cos z2 i sin z2 = eiz2 .
,
f (z) = 2izeiz2 .
-
68 1.
6. pi u(x, y) = x3 + 6x2y 3xy2 2y3 R2. v u f(z) = u(x, y) +
iv(x, y).
. pi uxx + uyy = 0, u R2. pi
CauchyRiemann vy = ux
vy = 3x2 + 12xy 3y2 , pi v(x, y) = 3x2y + 6xy2 y3 + c(x) .
pi CauchyRiemann uy = vx pipi
6x2 6xy 6y2 = 6xy 6y2 c(x) c(x) = 6x2 .pi c(x) = 2x3 + c .
v(x, y) = 3x2y + 6xy2 y3 2x3 + c
f(z) = u(z, 0) + iv(z, 0) = z3 + i(2z3 + c) = (1 2i)z3 + ic ,
c R.
7. pi : R R pi. u (x, y) = (xy) R2, u.
v u pi f (z) = u (x, y) + iv (x, y).
. t = xy, ux = y (t), uxx = y2 (t) pi uyy = x2 (t). pi
u R2,
uxx + uyy =(x2 + y2
) (t) = 0 (t) = 0 , t R .
pi (t) = at+b, pi u (x, y) = axy+b, a, b R. pi CauchyRiemannvy = ux
vy = ay , pi v (x, y) =a
2y2 + c (x) .
pi CauchyRiemann uy = vx pipi
ax = c (x) c (x) = ax .pi , c (x) = a2x2 + c .
, v (x, y) = a(y2 x2) /2 + c, c R
f (z) = u (z, 0) + iv (z, 0) = b+ i(a
2z2 + c
)= ai
2z2 + b+ ic ,
c R.
-
1.5. 200607 69
8. , f (z) = ez,
D = {z = x+ iy : 1 x 2, 0 y pi} .
. x = 1, 0 y pi, pi w = eeiy,0 y pi, pipi e. , x = 2,0 y pi, pi w = e2eiy, 0 y pi, pipi e2. [1, 2] w = exei0 = ex, 1 x 2,
[e, e2
], [1 + ipi, 2 + ipi] pi
w = exeipi = ex, 1 x 2, pi [e2,e]. , D pi D .
9. pi
| sin z| coshR | cos z| coshR ,
z D(0, R) = {z C : |z| R}.
. |z| = |x+ iy| R, |y| R pi
| sin z| =eiz eiz2i
12 (eiz+ eiz) = 12 (eixy+ eix+y)=ey + ey
2= cosh y coshR .
-
70 1.
,
| cos z| =eiz + eiz2
12 (eiz+ eiz) = 12 (eixy+ eix+y)=ey + ey
2= cosh y coshR .
10. w = log z pi D = C \ {iy : y 0}, log i = 5pii/2. h (z) = z = e log z, C, pi pi h3/2 (1).
. log z = ln |z|+ i (z) + 2kpii, k Z, pi (z) (pi/2, 3pi/2).
pi
log i = ln |i|+ i (i) + 2kpii = ln 1 + ipi2
+ 2kpii =pii
2+ 2kpii
log i = 5pii/2, pi k = 1. ,
log z = ln |z|+ i [ (z) + 2pi] .
-
1.5. 200607 71
log (1) = 3pii
h (z) = e log z
z= e log z
elog z= e(1) log z = z1 = h1 (z) .
pi,
h3/2 (1) =3
2h1/2 (1) =
3
2elog(1)/2 =
3
2e3pii/2 = 3i
2.
2
1. () pi ez=z dz
(2 +2) e .pi 0, 1 1 + i.
() C, z() = Rei, 0 pi/4, 0 R > 0. pi
Ceiz
2dz
pi4R (1 eR2) .pi. sin 2/pi, [0, pi/2].
.
() z = x+ iy , 0 x, y 1 piez=z = e(xiy)y = exy e , z .
pi 2 +
2, ez=z dz
e |dz| = e ( ) = (2 +
2)e .
-
72 1.
() z() = Rei, 0 pi/4, eiz()2 = eiR2(cos 2+i sin 2)= eR
2 sin 2 eR24/pi . (pi sin 2 4/pi)
pi, Ceiz
2dz
C
eiz2 |dz|=
pi/40
eiz()2 z () d pi/40
eR24/piRd
= pi4R
eR24/pi
=pi/4=0
=pi
4R
(1 eR2
).
2. pi
1
zdz =
1
2ln 5 + i arctan
(1
2
),
pi : [0, 1] C pi pipi (t) = 1 + t+ i sin (52pit).. 1 pi. pi pi pi C \ (, 0] pi
Log z = ln |z|+ iArg z , pi Arg z < pi ,
-
1.5. 200607 73
pi 1/z, (Log z) = 1/z. pi -
1
zdz = Log (1) Log (0)
= Log(2 + i) Log(1)= Log(2 + i)
=1
2ln 5 + i arctan
(1
2
).
2 pi. pi pi C\(, 0] f(z) = 1/z . pi pi
pi pi [1, 2] pi
[2, 2 + i], z(y) = 2 + iy, 0 y 1. pi,
1
zdz =
[1,2]
1
zdz +
[2,2+i]
1
zdz
=
21
1
xdx+ i
10
1
2 + iydy
= ln 2 + i
10
2 iy22 + y2
dy
= ln 2 + i
10
2
4 + y2dy +
10
y
4 + y2dy
= ln 2 + i arctan(y
2
)y=1y=0
+1
2ln(4 + y2
)y=1y=0
= ln 2 + i arctan
(1
2
)+
1
2ln 5 1
2ln 4
=1
2ln 5 + i arctan
(1
2
).
3. pi Cauchy,
pi Cauchy, pi
1
2pii
C+(0, 2)
cos z
z(z 1)2 dz = 1 sin 1 cos 1 ,
pi C+ (0, 2) 0, 2 .
-
74 1.
. C1 C2 pi
C (0, 2), C1 pi 0 C2 pi
1.
pi Cauchy
1
2pii
C+(0, 2)
cos z
z(z 1)2 dz =1
2pii
C+1
cos z
z(z 1)2 dz +1
2pii
C+2
cos z
z(z 1)2 dz .
, pi pi Cauchy
1
2pii
C+1
cos z
z(z 1)2 dz =1
2pii
C+1
cos z/(z 1)2z
dz =cos z
(z 1)2z=0
= 1
pi pi Cauchy pi
1
2pii
C+2
cos z
z(z 1)2 dz =1
2pii
C+2
cos z/z
(z 1)2 dz
=(cos z
z
)z=1
=z sin z cos z
z2
z=1
= sin 1 cos 1 .
,1
2pii
C+(0, 2)
cos z
z(z 1)2 dz = 1 sin 1 cos 1 .
4. D pi pipi, pi D pi
pi pipi pi pi, pi, pi
-
1.5. 200607 75
pi. f : D C,f(z) = u(x, y) + iv(x, y), pi ( f
pi ), pi Green pipi-
- pi pi
Green
D
f
zdxdy = 1
2i
D
f dz , (1.2)
D
f
zdxdy =
1
2i
D
f dz . (1.3)
pi
f
z=
1
2
(f
x if
y
)=
1
2
[u
x+v
y+ i
(v
x uy
)]
f
z=
1
2
(f
x+ i
f
y
)=
1
2
[u
x vy
+ i
(v
x+u
y
)].
.
12i
D
f dz = 12i
D
(u+ iv) (dx idy)
=1
2
D
(vdx+ udy) + i2
D
udx+ vdy
=1
2
D
(u
x+v
y
)dxdy +
i
2
D
(v
x uy
)dxdy
(pi Green)
=
D
1
2
[u
x+v
y+ i
(v
x uy
)]dxdy
=
D
f
zdxdy .
pi (1.3) pi.
5. pi (1.2) (1.3) Green,
pi, pi
|z|1cos z dxdy = pi .
-
76 1.
. pi pi (1.2)
|z|1cos z dxdy =
|z|1
z(sin z) dxdy
= 12i
|z|=1
sin z dz (pi (1.2))
= 12i
|z|=1
sin z d
(1
z
)(pi |z| = 1 zz = 1 z = 1z )
=1
2i
|z|=1
sin z
z2dz
= pi
(1
2pii
|z|=1
sin z
z2dz
)= pi (sin z)
z=0
( pi Cauchy pi)
= pi cos 0 = pi .
pi pi (1.3),
|z|1cos z dxdy =
|z|1
z(z cos z) dxdy =
1
2i
|z|=1
z cos z dz
=1
2i
|z|=1
cos z
zdz
(pi |z| = 1 zz = 1 z = 1z )
= pi
(1
2pii
|z|=1
cos z
zdz
)= pi cos 0 = pi . ( pi Cauchy)
6. pi f : C C pi 2pi0
f (rei) d r5/2 , r > 0 . pi f (n)(0) = 0, n = 0, 1, 2, . . . . pi f ;
. z() = rei, 0 2pi, |z| = r 0 r . pi pi Cauchy pi
f (n)(0) =n!
2pii
|z|=r
f(z)
zn+1dz =
n!
2pii
2pi0
f(rei
)rn+1ei(n+1)
irei d =n!
2pirn
2pi0
f(rei
)ein
d
-
1.5. 200607 77
pi, r > 0 pi pi f (n) (0)n!
=1
2pirn
2pi0
f(rei
)ein
d
12pirn 2pi0
f (rei) d 12pir5/2n . (1.4)
n 3, n 5/2 3 5/2 = 1/2 > 0 pi (1.4) f (n) (0)n!
12pir5/2n =
1
2pi 1rn5/2
r 0 , f
(n) (0) = 0 .
n 2, 5/2 n 5/2 2 = 1/2 > 0 pi (1.4) f (n) (0)n!
12pir5/2n
r0+0 , f (n) (0) = 0 .
, f (n) (0) = 0, n = 0, 1, 2, . . . . pi
f (z) =n=0
f (n) (0)
nzn = 0 , z C ,
pi f 0.
7. a R (pi pi a > 0). w = ez2 pi R, R, R+ia R+ia, R > 0, pi
limR
RR
e(x+ia)2dx =
ex2dx =
pi .
. pi e(x+ia)2 = ea2 ex2
limR
RR
ex2dx =
pi ,
pi
limR
RR
e(x+ia)2dx pi .
w = ez2 C. pi pi
= 1 2 3 4, pi Cauchy 1
ez2dz +
2
ez2dz +
3
ez2dz +
4
ez2dz = 0 .
-
78 1.
1, 2, 3 4 z (x) = x, 1 x 1, z (y) = R+ iy,0 y a, z (x) = x+ ia, R x R z (y) = R+ iy, 0 y a. pi, R
Rex
2dx+
a0e(R+iy)
2idy
RR
e(x+ia)2dx
a0e(R+iy)
2idy = 0
pi pi RR
ex2dx+ ieR
2
a0ey
22Ryi dy RR
e(x+ia)2dx ieR2
a0ey
2+2Ryi dy = 0 . (1.5)
pi ieR2 a0ey
22Ryi dy eR2 a
0
ey22Ryi dy = eR2 a0ey
2dy
R0 ,
limR
ieR2
a0ey
22Ryi dy = 0 .
, pi (1.5), pi R, pipi
limR
RR
e(x+ia)2dx =
ex2dx =
pi .
8. pi f : C C |f (z)| |z|, z C. pi pi ( Liouville), f pi pi 1 pi f pi pi 2. pi
f(z) = a+ bz2 , a, b C , |b| 1/2 .
-
1.5. 200607 79
. pi |f (z)| |z|, z C, pi f (0) = 0. pi,
f(z) = f(0) + f (0)z +f (0)
2!z2 = a+ bz2 , a, b C .
, pi pi Cauchy f (0) = 12pii|z|=1
f (z)z2
dz
12pi|z|=1
|f (z)||z|2 |dz|
12pi
|z|=1
|z||z|2 |dz| (pi |f
(z) | |z|, z C)
=1
2pi
|z|=1
|dz|
=1
2pi2pi = 1 .
,
|b| = |f (0)|2
12.
3
1.
|sin z| = |sin (x+ iy)| = (sin2 x+ sinh2 y)1/2 S = [0, 2pi] [0, 2pi].. f (z) = sin z , C. pi
|f (z)| = |sin z| pi S.
-
80 1.
pipi :
(i) 0 x 2pi, y = 0. |sin z| = |sinx| |sin (pi/2)| =|sin (3pi/2)| = 1.
(ii) 0 y 2pi, x = 2pi. |sin z| = sinh y u = sinh y . pi, sinh 2pi.
(iii) 0 x 2pi, y = 2pi. |sin z| = (sin2 x+ sinh2 2pi)1/2 sin(pi2
+ 2pii) = sin(3pi2 + 2pii
) = (1 + sinh2 2pi)1/2 = cosh 2pi .(iv) 0 y 2pi, x = 0. |sin z| = sinh y. pi, pi pipi,
sinh 2pi.
, |sin z| sin(pi2
+ 2pii) = sin(3pi2 + 2pii
) = cosh 2pi .
2.
f (z) =z2 3z
(z 1)2 (z + 1) =1
z + 1 1
(z 1)2 .
pi Laurent f z0 = 1 .
. , pi 11w =
n=0wn, |w| < 1,
1
(1 w)2 =n=1
nwn1 =n=0
(n+ 1)wn , |w| < 1 .
i. 0 < |z + 1| < 2:
f (z) =1
z + 1 1
[(z + 1) 2]2 =1
z + 1 1
4[1 z+12
]2=
1
z + 1 1
4
n=0
(n+ 1)
(z + 1
2
)n=
1
z + 1n=0
n+ 1
2n+2(z + 1)n .
-
1.5. 200607 81
ii. |z + 1| > 2:
f (z) =1
z + 1 1
[(z + 1) 2]2
=1
z + 1 1
(z + 1)2[1 2z+1
]2=
1
z + 1 1
(z + 1)2
n=0
(n+ 1)
(2
z + 1
)n=
1
z + 1n=0
2n (n+ 1)1
(z + 1)n+2.
3. 1sin z =
n= anzn pi Laurent f(z) = 1sin z
= {z C : pi < |z| < 2pi} z0 = 0. pi an, n < 0.
. pi Laurent an pi pi
an =1
2pii
C+(0, r)
1
zn+1 sin zdz ,
pi C+ (0, r) 0, r, pi < r < 2pi
.
(i) n = 1: pipi pi, 0 pi f(z) = 1sin z C+ (0, r) pi pi. pi
-
82 1.
pipi
a1 =1
2pii
C+(0, r)
1
sin zdz = Res
(1
sin z, pi
)+ Res
(1
sin z, 0
)+ Res
(1
sin z, pi
)=
1
(sin z)
z=pi
+1
(sin z)
z=0
+1
(sin z)
z=pi
=1
cos (pi) +1
cos 0+
1
cospi= 1 .
(ii) n 2: pipi
g(z) =1
zn+1 sin z=z(n+1)
sin z, (n+ 1) 1
pi pi pi g C+ (0, r) pi pi. pi 0
pi g(z) = z(n+1)sin z , 0 pi . , pi
pipi
an =1
2pii
C+(0, r)
1
zn+1 sin zdz = Res
(z(n+1)
sin z, pi
)+ Res
(z(n+1)
sin z, pi
)
=z(n+1)
(sin z)
z=pi
+z(n+1)
(sin z)
z=pi
=(pi)(n+1)
cos (pi) +pi(n+1)
cospi
= pi(n+1)[(1)n 1]
=
0 n ,
2pi(n+1) n pi .
4. pi
I =1
2pii
C+(0, 2)
[z3 cos
(z1)
+sinpiz
(z 1)2]dz ,
pi C+ (0, 2) 0, 2 .
. 0 f(z) = z3 cos(z1)
C (0, 2). pi
z3 cos(z1)
= z3(
1 z2
2!+z4
4! z
6
6!+
)= z3 z
2!+z1
4! z
3
6!+ ,
-
1.5. 200607 83
Res(z3 cos
(z1), 0)
= 1/4!. 1 g(z) =
sinpiz/ (z 1)2 C (0, 2). pi 1 pi pi pi, 1 pi pi
g. pi, pi pi
Res
(sinpiz
(z 1)2 , 1)
= 2(sinpiz)((z 1)2
)z=1
= 2pi cospiz
2
z=1
= pi .
, pi pipi
I = Res(z3 cos
(z1), 0)
+ Res
(sinpiz
(z 1)2 , 1)
=1
4! pi = 1
24 pi .
. , z0 pi pi g,
pipi g z0 pi pi
Res (g, z0) = limzz0
(z z0) g (z) .
pi,
Res
(sinpiz
(z 1)2 , 1)
= limz1
(z 1) sinpiz(z 1)2 = limz1
sinpiz
z 1(LHpital)
= limz1
pi cospiz = pi .
5. a > 0, pi
pi/20
a
a2 + sin2 d =
1
2
2pi0
a
2a2 + 1 cos d =pi
2
1 + a2.
.
pi/20
a
a2 + sin2 d =
pi/20
2a
2a2 + 1 cos 2 d
= a
pi0
1
2a2 + 1 cos d ( = 2)
=1
2
2pi0
a
2a2 + 1 cos d .
( f () = a/(2a2 + 1 cos) pi pi 2pi)
-
84 1.
z = ei, cos = ei + ei
2=
1
2
(z +
1
z
) dz = ieid = izd d = dz
iz. -
pi,
pi/20
a
a2 + sin2 d =
1
2
2pi0
a
2a2 + 1 cos d
=1
2
|z|=1
a
2a2 + 1 12(z + 1z
) dziz
= ai
|z|=1
1
z2 2 (1 + 2a2) z + 1 dz .
pi z1,2 = 1+2a22a
1 + a2 pi z22 (1 + 2a2) z+1 = 0, 1+2a22a1 + a2 pi pi g(z) = 1
z22(1+2a2)z+1 .
1+2a22a1 + a2 |z| = 1, 1+2a2+2a1 + a2 |z| = 1. , pi pipi
pi/20
a
a2 + sin2 d = a
i
|z|=1
1
z2 2 (1 + 2a2) + 1 dz
= ai2piiRes
(1
z2 2 (1 + 2a2) z + 1 , 1 + 2a2 2a
1 + a2
)= 2api 1
(z2 2 (1 + 2a2) z + 1)z=1+2a22a1+a2
= 2api 12(
1 + 2a2 2a1 + a2) 2 (1 + 2a2)
=pi
2
1 + a2.
6. pi pi
I =
cosx
(x2 + 1)2dx .
.
J =
sinx
(x2 + 1)2dx , I + iJ =
eix
(x2 + 1)2dx .
I J . i
-
1.5. 200607 85
pi 2 f (z) = eiz/(z2 + 1
)2. i pipi Res
(eiz
(z2 + 1)2, i
)= lim
zi
((z i)2 e
iz
(z2 + 1)2
)= lim
zi
(eiz
(z + i)2
)= lim
zi
(ieiz
(z + i)2 2e
iz
(z + i)3
)=ie1
(2i)2 2e
1
(2i)3= i
2e.
f pi pi pi pi
pi pipi R, z () = Rei, 0 pi [R,R]. R z = i f R.
pi pipi RR
eix
(x2 + 1)2dx+
R
eiz
(z2 + 1)2dz = 2piiRes
(eiz
(z2 + 1)2, i
)=pi
e. (1.6)
, pi Jordan
limR
R
eiz
(z2 + 1)2dz = 0 .
pi, pi (1.6) pipi
eix
(x2 + 1)2dx = lim
R
RR
eix
(x2 + 1)2dx =
pi
e.
,
I =
cosx
(x2 + 1)2dx =
pi
e J =
sinx
(x2 + 1)2dx = 0 .
-
86 1.
7. a C, |a| 6= 1. pi C+(0, 1)
z6
az7 1 dz ,
pi C+ (0, 1) 0, 1 .
.
(i) |a| < 1, f (z) = z6/ (az7 1) pi - |z| = 1. pi, pi Cauchy
C+(0, 1)
z6
az7 1 dz = 0 .
(ii) |a| > 1, C+(0, 1)
z6
az7 1 dz
=2pii
7a
{1
2pii
C+(0, 1)
(az7 1)az7 1 dz
}=
2pii
7a { f (z) = az7 1 |z| = 1}
( )
=2pii
7a 7 = 2pii
a.
,C+(0, 1)
z6
az7 1 dz =
0 |a| < 1 ,2piia |a| > 1 .
8. pi
P (z) = 3z9 + 8z6 + z5 + 2z3 + 1
= {z C : 1 < |z| < 2}.. |z| = 2 P (z) 3z9 = 8z6 + z5 + 2z3 + 1
8 |z|6 + |z|5 + 2 |z|3 + 1= 561 < 1536 =
3z9 .
-
1.5. 200607 87
pi pi Q1 (z) = 3z9 9 |z| < 2, pi Rouch pi P 9 |z| < 2. |z| = 1
P (z) 8z6 = 3z9 + z5 + 2z3 + 1 3 |z|9 + |z|5 + 2 |z|3 + 1= 7 < 8 =
8z6 .pi pi Q2 (z) = 8z6 6 |z| < 1, pi Rouch pi P 6 |z| < 1. , pi P 9 6 = 3 = {z C : 1 < |z| < 2}.
9. > 1. pi
zez = 1
D (0, 1) = {z C : |z| < 1} R.. f (z) = zez 1. |z| = 1, |e0 = 1, pi Bolzano pi (0, 1) g () = 1 e = 1.
10. f D(0, 1) = {z C : |z| 1}.pi |f(z)| = 1, |z| = 1 f .
i. pi f D (0, 1) = {z C : |z| < 1}.ii. pi f pi D (0, 1).
w0 C, |w0| < 1, pi z0 C, |z0| < 1, f (z0) = w0.
-
88 1.
pi. pi Rouch.
pi. i. pi f |f (z)| = 1, |z| = 1, pi D (0, 1) f () D (0, 1). g (z) := f (z) f ().
|g (z) f (z)| = |f ()| < 1 = |f (z)| , |z| = 1 .
pi g D (0, 1), pi Rouch
f D (0, 1).
ii. |w0| < 1. h (z) = f (z) w0,
|h (z) f (z)| = |w0| < 1 = |f (z)| , |z| = 1 .
pi pi () f D (0, 1), pi
Rouch h D (0, 1). pi z0 C,|z0| < 1, f (z0) = w0.. pi pi, () pipi
f D (0, 1). , f pi D (0, 1)
.
-
2
2.1 201112
&
10 , 2012
1. () a R pi u(x, y) = ax3y + 4xy3 + x R2. v u
pi f = u + iv, f(0) = i. f z = x+ iy.
(1,5 .)
()
={z C : |=z| pi
2
} z-pipi. , pi
. x = x0 R, pi/2 y pi/2, w = ez. w = ez; (1 .)
.
89
-
90 2.
() uxx + uyy = 6axy + 24xy. u pipi
uxx+uyy = 0 6axy+24xy = 0 (x, y) R2. pi a = 4 pi u(x, y) = 4x3y+ 4xy3 +x. , pi piC pi v u. f = u+ iv .
pi CauchyRiemann ux = vy
vy = 12x2y + 4y3 + 1 .
pi,
v(x, y) = 6x2y2 + y4 + y + c(x) .
pi CauchyRiemann vx = uy pipi
12xy2 + c(x) = 4x3 12xy2 c(x) = 4x3 . c(x) = x4 + c .
v(x, y) = 6x2y2 + y4 + y + x4 + c pi
f(z) = u(z, 0) + iv(z, 0) = z + i(z4 + c) ,
c R. f(0) = i, pi c = 1. ,
f(z) = iz4 + z i .
() pi w = ez pi
w 6= 0. y = pi/2, w = exipi/2 = iex. pi y = pi/2pi y = pi/2 pi . pi
.
x = x0 R, pi/2 y pi/2, w = ex0eiy, pi/2 y pi/2, O R = ex0 pipi w-pipi. pi,
x0 R, pi pipi
w-pipi.
, w = ez
= {w C : |
-
2.1. 201112 91
2. f : C C ,
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.
() ( Liouville) pi pi k 0 pi A 0 B > 0,
|f(z)| A+B|z|k |z| > R0 > 0 . (2.1)
R > R0, pi Cauchy pi
|an| A+BRk
Rn.
pi f pi pi k. pi
f z C; (1,3 .)
() lim|z|
f(z)
z= 0, pi f . (1,2 .)
pi. () C(0, R) = {z C : |z| = R} R > R0. M =max|z|=R |f(z)|, pi (2.1)
M A+BRk .
-
92 2.
, 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.
f z C, (2.1) k = 0 pi f ( Liouville).
() pi lim|z|
f(z)
z= 0, pi R0 > 0
f(z)z < 1 |z| > R0.,
|f(z)| < |z| |z| > R0 .
pi () pi f pi pi 1, f(z) = az + b
a, b C. pi pi
lim|z|
(az + b
z
)= lim|z|
(a+
b
z
)= 0
pi a = 0. f(z) = b z C, f .
3. () pi , pi
C.
pi f : 1 < |z| < 3 . |f(z)| 1 |z| = 1 |f(z)| 9 |z| = 3, pi |f(2i)| 4. (1,2 .)
() f : D(0, 1) C D(0, 1) = {z C : |z| < 1} C pi z(t) = reit, 0 t 4pi, pi 0 < r < 1. 0 pi f , pi
1
2pii
C
f(z)
z3dz
pi : (i) pi Cauchy pi (ii)
pipi. (1,3 .)
.
-
2.1. 201112 93
() : f ,
G C G G, |f | pi G G f G.
g(z) := f(z)z2
. g ,
: 1 < |z| < 3 pi |z| = 1 |z| = 3.pi pi |g(z)| 1 |z| = 1 |g(z)| 1 |z| = 3.pi |g(z)| 1 z . ,
|f(z)| |z|2 z
pi |f(2i)| |2i|2 = 4.() pi C pi 0 I(C, 0) = 2.
(i) pi pi Cauchy pi
1
2pii
C
f(z)
z3dz =
1
2
{2!
2pii
C
f(z)
z3dz
}=
1
2I(C, 0) f (0) = f (0) .
(ii) pi 0 pi f , f(0) = 0, f (0) 6= 0 3 z3, 0 pi 2 w = f(z)/z3. , pi
pipi
1
2pii
C
f(z)
z3dz = I(C, 0) Res
(f(z)
z3, 0
)= 2 lim
z0
[z2 f(z)
z3
]= 2 lim
z0
[f(z)
z
]= 2 lim
z0zf (z) f(z)
z2
= 2 limz0
zf (z)2z
( LHpital)
= f (0) .
4. ()
f (z) =1
z (z2 + 5)2.
-
94 2.
pi( ) Laurent f z0 = 0
pi pi 2 2i. (1 .)
() pi Laurent f(z) = 1cospiz
=
{z C : 1
2< |z| < 3
2
} z0 = 0 .
pi pipi pi -
z1 z2 pi( ) Laurent f(z) = 1cospiz -
.
(1,5 .)
.
() 1 : 0 < |z|
5. 22i 2. 1/ (1 + w) =
n=0(1)nwn,
|w| < 1,
1(1 + w)2
=
n=1
(1)nnwn1 1(1 + w)2
=
n=1
(1)n1nwn1 , |w| < 1 .
pi,
f(z) =1
z51
(1 + 5/z2)2
=1
z5
n=1
(1)n1n(
5
z2
)n1(|5/z2| < 1 |z| > 5)
=
n=1
(1)n1n5n1 1z2n+3
=n=0
(1)n(n+ 1)5n 1z2n+5
.
pi pipi pi Laurent f
2 : |z| >
5.
() f(z) = 1/ cospiz pi
pi 1
cospiz=
n=
anzn z ,
-
2.1. 201112 95
pi pi pi pi
. an pi pi
an =1
2pii
C+(0, r)
1
zn+1 cospizdz ,
pi C+(0, r) 0, r, 1/2 < r < 3/2
.
(i) n = 1: a1 =
1
2pii
C+(0, r)
1
cospizdz .
1/2 1/2 f(z) = 1/ cospiz C+(0, r) pi pi. pi pipi
a1 =1
2pii
C+(0, r)
1
cospizdz
= Res
(1
cospiz, 1
2
)+ Res
(1
cospiz,
1
2
)=
1
(cospiz)
z=1/2
+1
(cospiz)
z=1/2
=1
pi 1pi
= 0 .
(ii) n = 2:
a2 =1
2pii
C+(0, r)
1
z1 cospizdz =
1
2pii
C+(0, r)
z
cospizdz .
1/2 1/2 g(z) = z/ cospiz C+(0, r) pi pi. pi pipi
a2 =1
2pii
C+(0, r)
z
cospizdz
= Res
(z
cospiz, 1
2
)+ Res
(z
cospiz,
1
2
)=
z
(cospiz)
z=1/2
+z
(cospiz)
z=1/2
= 12pi 1
2pi= 1
pi.
-
96 2.
5. () z0 C R > 0, pi f
0 < |z z0| < R
z0.
(i) z0 pi N N f , pi z0 pi N + 1 f . (0,8 .)
(ii) z0 pi (pi) f ; -
z0 pi
f .
(0,7 .)
() pi f : C \N C . f . pi . (1 .)
.
() (i) z0 pi N f pi g
D(z0, R) = {z C : |z z0| < R}, g(z0) 6= 0
f(z) =g(z)
(z z0)N z D(z0, R), z 6= z0 .
,
f (z) =(z z0)g(z)Ng(z)
(z z0)N+1 z D(z0, R), z 6= z0 .
pi
f (z) =h(z)
(z z0)N+1 z D(z0, R), z 6= z0 ,
pi h(z) := (z z0)g(z) Ng(z) D(z0, R) h(z0) = Ng(z0) 6= 0. , z0 pi N + 1 f .(ii) z0 pi (pi) f pi
Laurent f(z) =
n= an(z z0)n, 0 < |z z0| < R, an = 0 n 1.
f(z) =
n=0
an(z z0)n , 0 < |z z0| < R .
-
2.1. 201112 97
. f(z0) = a0, pi pi
|z z0| < R. z0 pi f
limzz0
f(z) pi pipi
pi M > 0, > 0, |f(z)| < M 0 < |z z0| < lim
zz0(z z0)f(z) = 0 .
() pi pi pi M > 0, |f(z)| M z C \ N. n pipi . f
pi 0 < |z n| < 1 n pi n pi f .
f(z) =
k=0
ak(z n)k , 0 < |z n| < 1 .
pi limzn f(z) = a0, |a0| M . f(n) = a0, pi pi n. pi n N, fpi C( ) |f(z)| M z C., pi Liouville f .
pi 4
-
98 2.
&
pipi
30 , 2012
1. f : A C, f(z) = f(x + iy) = u(x, y) + iv(x, y), pi A pi C z0 = x0 + iy0 A.
() pi f pi z0. f
pi x, y, pi pi fx(x0, y0) =
ux(x0, y0) + ivx(x0, y0), fy(x0, y0) = uy(x0, y0) + ivy(x0, y0) pi
f (z0) = fx(x0, y0) = ify(x0, y0) .
(1 .)
() f : A R, f pi pi , pi f pi z0 f (z0) = 0.
(0,5 .)
.
() pi pi pi
f (z0) = limh0
f(z0 + h) f(z0)h
pi.
(i) h R z0 + h A, z0 + h = x0 + iy0 + h = (x0 + h) + iy0 (x0 + h, y0) A pi
f (z0) = limh0hR
f(z0 + h) f(z0)h
= limh0hR
f(x0 + h, y0) f(x0, y0)h
= fx(x0, y0) .
(ii) h = ik, k R, z0 + h A, z0 + h = x0 + iy0 + ik = x0 + i(y0 + k)
-
2.1. 201112 99
(x0, y0 + k) A pi
f (z0) = limh0
h=ik,kR
f(z0 + h) f(z0)h
= limk0kR
f(x0, y0 + k) f(x0, y0)ik
= i limk0kR
f(x0, y0 + k) f(x0, y0)k
= ify(x0, y0) .
pi fx(x0, y0), fy(x0, y0) pi f (z0) = fx(x0, y0) =
ify(x0, y0).
() pi f pi z0. pi f pi pi-
, pi () pi f (z0) = fx(x0, y0) R if (z0) = fy(x0, y0) R. pi f (z0) = 0.
2. f : A C, f(z) = f(x + iy) = u(x, y) + iv(x, y), pi A pi C. f
A.
: A R , pi g := x iy A. (1 .)
. f A f ( pi
x, y) pi A pi
() Cauchy-Riemann
fx = ify
ux = vy
uy = vx
.pi A, pi 2 A
pi Laplace: xx + yy = 0. pi xy = yx.
U := x V := y, U , V pi A
Ux Vy = xx + yy = 0 Uy + Vx = xy yx = 0 .
-
100 2.
pi U , V pi ()Cauchy-Riemann pi
g = x iy = U + iV A.
3. () f : C C, f(z) = u(x, y)+ iv(x, y), ( C). 0 (x, y) R2, pi Liouville pipi pi pi f . (1 .)
() pi , pi
C.
pi f D(0, 1) = {z C : |z| < 1}, |z| = 1
|f(z)| = e|z| |z| 1;
(1 .)
.
() 1 pi. g(z) := ef(z)
|g(z)| = |ef(z)| = |eu(x,y)iv(x,y)| = eu(x,y) < e0 = 1(u(x, y) > 0 (x, y) R2)
z C. pi, pi Liouville g C. |g(z)| = eu(x,y) pi 0 .
-
2.1. 201112 101
() : f , -
G C, G G f(z) 6= 0 z G, |f | pi G G f G.
pi pi f ,
|z| = 1 |f(z)| = e|z| |z| 1. f . pi |z| = 1 |f(z)| = e1 = e |f(0)| = e0 = 1, |f | pi |z| = 1. pi pi f pipi .
pi |f(z)| = e|z| |z| < 1. pi. , pi f .
4. ()
f(z) =1
(z 1)(z + 2)2 .
pi( ) Laurent f z0 = 1
pi pi 2 3i. (1 .)
() pi Laurent f
= {z C : R1 < |z| < R2}, 0 R1 < R2 , z0 = 0 .
n= cnzn
n= dnz
n pi( ) Laurent f(z) =1
sinpiz 1 : 0 < |z| < 1 2 : 1 < |z| < 2, . -pi Laurent pipi, pi
dn cn = 2pi
pi n .
(1,5 .)
.
() 1 : 0 < |z 1| < 3 2 : |z 1| > 3. 2 3i 2. 1/(1 + w) =
-
102 2.
n=0(1)nwn, |w| < 1,
1(1 + w)2
=n=1
(1)nnwn1 1(1 + w)2
=n=1
(1)n1nwn1 , |w| < 1 .
pi,
f(z) =1
(z 1) [(z 1) + 3]2
=1
(z 1)3 [1 + 3/(z 1)]2
=1
(z 1)3n=1
(1)n1n(
3
z 1)n1
(|3/(z 1)| < 1 |z 1| > 3)
=
n=1
(1)n1n3n1 1(z 1)n+2
=
n=3
(1)n1(n 2)3n3 1(z 1)n .
pi pipi pi Laurent f
2 : |z 1| > 3.
() f pi pi
f(z) =
n=anz
n z ,
pi pi pi pi
. an pi pi
an =1
2pii
|z|=
f(z)
zn+1dz ,
pi |z| = 0, , R1 < < R2 .
(i) 1 : 0 < |z| < 1.
1
sinpiz=
n=
cnzn z 1 ,
pi
cn =1
2pii
|z|=1
1
zn+1 sinpizdz 0 < 1 < 1 .
-
2.1. 201112 103
0 g(z) = 1zn+1 sinpiz
|z| = 1. pi pipi
cn = Res
(1
zn+1 sinpiz, 0
).
(ii) 2 : 1 < |z| < 2.
1
sinpiz=
n=
dnzn z 2 ,
pi
dn =1
2pii
|z|=2
1
zn+1 sinpizdz 1 < 2 < 2 .
0, 1 g(z) = 1/zn+1 sinpiz |z| = 2. 1 pi pi g. pi pipi
dn = Res
(1
zn+1 sinpiz, 1
)+ Res
(1
zn+1 sinpiz, 0
)+ Res
(1
zn+1 sinpiz, 1
).
-
104 2.
pi
dn cn = Res(zn1
sinpiz, 1
)+ Res
(zn1
sinpiz, 1
)=
zn1
(sinpiz)
z=1
+zn1
(sinpiz)
z=1
=(1)n1pi cos(pi) +
1
pi cospi
= 1pi
((1)n1 + 1)
= 2pi. ( n pi)
5. () z0 C R > 0, pi f
: 0 < |z z0| < R
z0. f(z) =
n= an(z z0)n pi() Laurent f z0 , pi z0
f ; z0 f . (0,5 .)
() g(z) = z cos(1/z), z 6= 0. pi limn g(zn) limn g(n), pi zn = i/n n = 1/n, n N. pi limz0 g(z); g 0; pi Res (g, 0).
(1 .)
.
() z0 f , pi( ) Laurent
f(z) =
n=
an(z z0)n
f : 0 < |z z0| < R an 6= 0 pi pi
-
2.1. 201112 105
n < 0.
z0 f
limzz0
f(z) pi ( limzz0
|f(z)| 6=)
f pi z0 limzz0
|f(z)| 6= .
() limn zn = limn n = 0. pi cos z = 12(ein + ein),
g(zn) =i
ncos(ni
)= i
en + en
2n g(n) =
cosn
n.
pi,
limn g(zn) = i limn
en + en
2n
= i limn
en en2
( LHpital)
=
limn g(n) = 0. limz0 g(z) pi . pi 0 g.
pi
g(z) = z cos1
z= z
(1 1
2!z2+
1
4!z4
)= z 1
2!z+
1
4!z3 ,
Res (g, 0) = 12 .
6. () f pi
|z| = 1, pi |z|=1
z f(z) dz = pii f (0) .
(1,2 .)
() pi pi 2pi0
15 + cos t
dt .
-
106 2.
(1,3 .)
. |z| = 1 z = eit, 0 t 2pi, dz = ieitdt =izdt.
()
|z|=1
z f(z) dz = 2pi0
eit f(eit) ieit dt
=
2pi0
f(eit) ie2it dt
= 2pi0
f(eit)ie2it dt
= ( 2pi
0f(eit)ie2it dt
)=
( 2pi0
f(eit)
e3itieit dt
)
= (|z|=1
f(z)
z3dz
)
=2pii
2!
(2!
2pii
|z|=1
f(z)
z3dz
)= pii f (0) . ( pi Cauchy pi)
() pi cos t = 12(eit + eit
)= 12
(z + z1
) dt = (1/iz)dz,
2pi0
15 + cos t
dt =
|z|=1
15 + 12 (z + z
1)1
izdz
=2
i
|z|=1
1
z2 + 2
5 z + 1dz
=2
i
|z|=1
1
(z +
5 2)(z +5 + 2) dz .
5 2 pi pi f(z) = 1/(z+5 2)(z+5 + 2). pi 5 + 2 |z| = 1, pi
-
2.1. 201112 107
pipi 2pi0
15 + cos t
dt =2
i2pii Res(f,
5 + 2)
= 4pi limz5+2
(z +
5 2) 1(z +
5 2)(z +5 + 2)
= 4pi 14
= pi .
-
108 2.
2.2 200910
5 , 2010
1. () f : G C, f (z) = u (x, y) + iv (x, y), pi G pi C.
-
2.2. 200910 109
(yu+ xv)xx + (yu+ xv)yy = 0
(2vx + yuxx + xvxx) + (2uy + yuyy + xvyy) = 0 2vx + 2uy + y (uxx + uyy) + x (vxx + vyy) = 0 uy = vx . (uxx + uyy = 0, vxx + vyy = 0)
pi u, v pi
G pi Cauchy-Riemann: {ux = vy, uy = vx}, pi f = u+ iv G.
() pi |f (z)| < |f (z)|, z C, f C. , f (z0) = 0 pi z0 C, |f (z0)| < 0 (pi). h (z) := f (z) /f (z), z C, h ( C) |h (z)| < 1 z C. , pi Liouville h C.
h (z) =f (z)f (z)
= c z C, pi |c| < 1.
pi f C, pi
g : C C eg(z) = f (z) , z C.
g (z) eg(z) = f (z) g (z) = f (z)f (z)
.
pi, g (z) = c z C pi g (z) = cz + d. ,
f (z) = ecz+d = edecz = Aecz , pi A = ed 6= 0 .
,
f (z) = Aecz , A, c C, A 6= 0 |c| < 1.
-
110 2.
2.
sin(a(z + z1
))=
n=
cnzn , a C ,
pi Laurent f (z) = sin(a(z + z1
)) -
= {z C : 0 < |z|
-
2.2. 200910 111
3. pi C+(0, 1)
=zz (z 1/2) dz ,
pi C+ (0, 1) .
(1,5 .)
. z = x+ iy C+ (0, 1),
=z = y = 12i
(z z) = 12i
(z z1) .
( z C+ (0, 1), |z| = 1 |z|2 = 1 zz = 1 z = z1)
2 pi: z = ei, pi pi,
=z = sin = 12i
(ei ei
)=
1
2i
(z z1) .
C+(0, 1)
=zz (z 1/2) dz =
1
2i
C+(0, 1)
z z1z (z 1/2) dz
=1
2i
C+(0, 1)
z2 1z2 (z 1/2) dz ,
piC+(0, 1)
=zz (z 1/2) dz =
1
2i2pii
[Res
(z2 1
z2 (z 1/2) , 0)
+ Res
(z2 1
z2 (z 1/2) , 1/2)]
( pipi)
= pi
[limz0
(z2
z2 1z2 (z 1/2)
)+ limz1/2
(z 1/2) z2 1
z2 (z 1/2)
]
= pi
[limz0
z2 z + 1(z 1/2)2 + limz1/2
(1 z2)]
= pi[22 + 1 22] = pi .
-
112 2.
pipi
29 pi, 2010
1. () f : C C
f (z) =
z2 sin (1/z) z 6= 0
0 z = 0 .
pi
limz0z=xR
f (z) f (0)z 0 limz0
z=iy,yR
f (z) f (0)z 0 .
f pi 0; (1 .)
() u : R2 R,
(x, y) := xu (x, y)
R2. v u
pi f (z) = u (x, y) + iv (x, y). (1 .)
.
()
limz0z=xR
f (z) f (0)z 0 = limx0
x2 sin (1/x)
x= lim
x0x sin (1/x) = 0
limz0
z=iy,yR
f (z) f (0)z 0 = limy0
(iy)2 sin (1/iy)
iy
= limy0
y2 sin (i/y)
iy
= limy0
y sinh (1/y) . (sin (i/y) = i sinh (1/y))
-
2.2. 200910 113
,
limy0+
y sinh (1/y) = limy0+
sinh (1/y)
1/y
= limt+
sinh t
t
= limt+ cosh t (LHpital)
= limt+
et + et
2= +
limy0 y sinh (1/y) = limy0+ y sinh (1/y) = +. limy0 y sinh (1/y) =+. , pi f (0) pi.
() pi u R2, u pi
2 R2 pi Laplace
uxx + uyy = 0 .
(x, y) = xu (x, y) R2 pi
(xu)xx + (xu)yy = 0 (2ux + xuxx) + xuyy = 0 2ux + x (uxx + uyy) = 0 ux = 0 (uxx + uyy = 0)
pi u (x, y) = c1 (y) Laplace uxx + uyy = 0 pi
c1 (y) = 0. ,
u (x, y) = c1 (y) = ay + b , a, b R .
v u, pi CauchyRiemann vy = uxpi
vy = 0 , pi v (x, y) = c2 (x) .
pi CauchyRiemann uy = vx pipi
a = c2 (x) c2 (x) = a .pi , c2 (x) = ax+ c .
, v (x, y) = ax+ c
f (z) = u (z, 0) + iv (z, 0) = b+ i (az + c) = iaz + b+ ic ,
-
114 2.
c R.
2.
f (z) =20
(z4 4) (z4 + 16) .
pi pi Laurent f z0 = 0
pi pi 3i/2
f (z) =
n=
anz4n an =
4(n+1) n 1
(1/16)n+1 n 0 .
; (1,5 .)
.
f (z) =20
(z4 4) (z4 + 16) =1
z4 4 1
z4 + 16.
,
1
1 w =n=0
wn 11 + w
=n=0
(1)nwn , |w| < 1 . ( )
={z C : 2 < |z| < 2}, 3i/2 pi Laurent f
f (z) =1
z4 4 1
z4 + 16
=1
z41
1 4/z4 1
16
1
1 + z4/16
=1
z4
n=0
(4
z4
)n 1
16
n=0
(1)n(z4
16
)n=n=0
4n
z4(n+1)+
n=0
(1)n+1 116n+1
z4n
=
1n=
4(n+1)z4n +n=0
( 1
16
)n+1z4n .
-
2.2. 200910 115
3. f () D (0, R) =
{z C : |z| R}, pi 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
(Reit
)dt .
(1,5 .)
. pi
Reit
Reit z +z
Reit z =R2 |z|2
(Reit z) (Reit z) =R2 |z|2|Reit z|2 ,
pi
f (z) =1
2pi
2pi0
[Reit
Reit z +z
Reit z]f(Reit
)dt .
|z| < R, z 6= 0,
1
2pi
2pi0
[Reit
Reit z +z
Reit z]f(Reit
)dt
=1
2pi
2pi0
[Reit
Reit z +zReit
R2 zReit]f(Reit
)dt
=1
2pi
2pi0
[1
Reit z 1
Reit R2/z]Reitf
(Reit
)dt
=1
2pii
C+(0, R)
[1
w z 1
w R2/z]f (w) dw
( w = Reit, dw = iReitdt)
=1
2pii
C+(0, R)
f (w)
w z dw 1
2pii
C+(0, R)
f (w)
w R2/z dw ,
pi C+ (0, R) 0 R > 0 . pi
pi Cauchy
f (z) =1
2pii
C+(0, R)
f (w)
w z dw .
pi |z| < R, R2/z > R pi pi Cauchy 1
2pii
C+(0, R)
f(w)
w R2/z f(w) dw = 0 .
-
116 2.
,
f(z) =1
2pi
2pi0
[Reit
Reit z +z
Reit z]f(Reit
)dt .
z = 0,
1
2pi
2pi0
[Reit
Reit z +z
Reit z]f(Reit
)dt =
1
2pi
2pi0
f(Reit
)dt
=1
2pii
C+(0, R)
f (w)
wdw
( w = Reit, dt = dw/iw)
= f (0) .
-
2.3. 20089 117
2.3 20089
17 , 2009
1. C tan z = i.
f (z) =1
tan z i pi
|z|=2
1
tan z i dz .
(1 .)
. {tan z = i
}{ (
eiz eiz) /2i(eiz + eiz) /2
= i
}{eiz eizeiz + eiz
= 1}{eiz = 0
}.
pi C C \ {0}. pi tan z = i C.
w = tan z z = kpi + pi/2, k Z. pi, zk = kpi + pi/2, k Z, f(z) = 1/ (tan z i). pi 1tan z i
= 1|tan z i| 1|tan z| 1 zkpi+pi/2 0 , zk = kpi + pi/2, k Z, pi () f . pi pi2 |z| = 2. pi,
|z|=21
tan z i dz = 0 .
2. f (z) =
n= anzn pi Laurent ()
f = {z C : 0 < |z| < 1} z0 = 0. pi
|f(z)| ln 2|z| , z .
-
118 2.
pi
|an| 1rn
ln2
r, 0 < r < 1 .
f 0;
pi limz0 f(z); limz0 |f(z)| ln 2; (2,5 .)
. pi Laurent an pi pi
an =1
2pii
C+(0, r)
f (z)
zn+1dz ,
pi C+ (0, r) 0, r, 0 < r < 1
. pi
|an| = 12pi
C+(0, r)
f (z)
zn+1dz
1
2pi
C+(0, r)
|f (z)||z|n+1 |dz|
12pirn+1
ln2
rC+(0, r)
|dz| (|f (z)| ln 2|z| = ln 2r )
=1
2pirn+1ln
2
r 2pir = 1
rnln
2
r.
n < 0,
limr0+
1
rnln
2
r= lim
r0+ln (2/r)
rn(LHpital)
= limr0+
1
nrn= 1
nlimr0+
rn = 0 .
pi an = 0 n = 1,2,3, . . . . , z0 = 0 pi () f pi f pi ()
D (0, 1) = {z C : |z| < 1}. limz0 f (z) = a0 pi
limz0|f (z)| = |a0| lim
r1ln
2
r= ln 2 .
3. pi , pi 2pi0
cosn
5 + 4 cos d , n = 1, 2, 3, . . . .
-
2.3. 20089 119
(1,5 .)
. z = ei, 0 2pi. pi dz = ieid =izd cos = 12
(z + z1
),
2pi0
eni
5 + 4 cos d =
C+(0, 1)
zn
iz [5 + 2 (z + z1)]dz
= iC+(0, 1)
zn
2z2 + 5z + 2dz
= iC+(0, 1)
zn
(2z + 1) (z + 2)dz .
1/2 2 pi pi f (z) = zn/ (2z + 1) (z + 2). pi 1/2 C+ (0, 1), pi pipi
2pi0
eni
5 + 4 cos d = 2pii (i) Res
(zn
(2z + 1) (z + 2), 1
2
)= 2pi lim
z1/2
(z +
1
2
)zn
(2z + 1) (z + 2)
= 2pi(1/2)n
2 (1/2 + 2) = pi(1)n
3 2n1 .
, 2pi0
cosn
5 + 4 cos d = 0. pi
limR
R
z4
z8 + 1dz = 0 .
, f(z) = z4z8+1
pi pi pi
pi pi 0 pi R, R
Reipi/4 pi 0, pi 0
x4
x8 + 1dx =
pi
8 sin (5pi/8).
(2 .)
. z R z4z8 + 1 = |z|4|z8 + 1| |z|4|z|8 1 = R4R8 1 .
R piR/4 piR
z4
z8 + 1dz
piR54 (R8 1) = pi4 1/R31 1/R8 R 0 .,
limR
R
z4
z8 + 1dz = 0 .
f pi pi (
) pi pi pi [0, R], R
[Reipi/4, 0
].
pi [0, Reipi/4
] z (x) = xeipi/4, 0
x R. z8 + 1 = 0 z8 = 1 z8 = eipi
zk = e(2kipi+ipi)/8 , k = 0, 1, 2, . . . , 7 .
-
2.3. 20089 123
zk, k = 0, 1, 2, . . . , 7, pi pi f z0 = eipi/8
pi/4 R > 1.
Res(f, eipi/8
)=
z4
8z7
z=eipi/8
=1
8
e5ipi/8
eipi= 1
8e5ipi/8 .
pi pipi
R0
x4
x8 + 1dx+
R
z4
z8 + 1dz
R0
(xeipi/4
)4(xeipi/4
)8+ 1
eipi/4 dx
= 2piiRes(f, eipi/8
)= pii
4e5ipi/8 .
pi, R0
x4
x8 + 1dx+
R
z4
z8 + 1dz + eipi/4
R0
x4
x8 + 1dx = pii
4e5ipi/8
(1 + eipi/4
) R0
x4
x8 + 1dx+
R
z4
z8 + 1dz = pii
4e5ipi/8 . (2.2)
pi (2.2) R pi(1 + eipi/4
) 0
x4
x8 + 1dx = pii
4e5ipi/8 .
pie5ipi/8
1 + eipi/4=
1
e5ipi/8 + e3ipi/8=
1
e5ipi/8 e5ipi/8 = 1
2i sin (5pi/8),
0
x4
x8 + 1dx =
pi
8 sin (5pi/8).
-
124 2.
2.4 20078
11 , 2008
1. () f : G C, f = u + iv, pi G, pi u =
-
2.4. 20078 125
pi c C. pi pi f C \ {0}, f C \ {0}. w = Log z C \ {0}. pi,pi w = Log z (, 0].
2. T 0, 1 1 + i. f (z) = ez2 , |f | T .
. pi f(z) = ez2 C, pi
|f(z)| =ez2 pi
pi T pi 0, 1 1 + i.
1 pi.
(i) [1, 0], z = x, 1 x 0, |f (z)| = ex2 .pi,
maxz[1, 0]
|f (z)| = f (1) = e minz[1, 0]
|f (z)| = f (0) = 1 .
(ii) [0, 1 + i], z = x+ ix, 0 x 1,
|f (z)| =e(x+ix)2 = e2ix2 = 1 .
(ii) [1, 1 + i], z = 1+(2+i)t = 2t1+it, 0 t 1,
|f (z)| =e(2t1+it)2 = e3t24t+1 e2it(2t1) = e3t24t+1 .
-
126 2.
g (t) = 3t2 4t+ 1, 0 t 1,
maxt[0, 1]
g (t) = g (0) = 1 mint[0, 1]
g (t) = g
(2
3
)= 1
3.
pi,
maxz[1, 1+i]
|f (z)| = f (1) = e minz[1, 1+i]
|f (z)| = f(
1 + 2i
3
)= e1/3 .
,
maxzT|f (z)| = f (1) = e min
zT|f (z)| = f
(1 + 2i
3
)= e1/3 .
2 pi.
|f (z)| =e(x+iy)2 = ex2y2 e2ixy = ex2y2 ,
pi 0, 1 1 + i. pipi
(i) y = 0, 0 x 1, (ii) y = x, 0 x 1 (iii) 2y = x+ 1, 1 x 1 .
3.
f (z) =z2 z i
(z 1 3i) (z2 2z + 1 + 2i) =1
z 1 3i +1
z2 2z + 1 + 2i .
pi Laurent f z0 = 1
pi pi 1/2.
.
f (z) =1
z 1 3i +1
(z 1)2 + (1 + i)2 .
, 1/ (1 w) = n=0wn 1/ (1 + w) = n=0 (1)nwn, |w| < 1 ().
pi |1 + i| = 2, = {z C : 2 < |z 1| < 3}, 1/2 piLaurent f
f (z) = 13i 1
1 z13i+
1
(z 1)2 1
1 +(1+iz1)2
= 13i
n=0
(z 1
3i
)n+
1
(z 1)2n=0
(1)n(
1 + i
z 1)2n
= n=0
1
(3i)n+1(z 1)n +
n=0
(1)n (1 + i)2n 1(z 1)2(n+1)
.
-
2.4. 20078 127
={z C : 2 < |z 1| < 3}, 1/2 ,
pi pipi pi Laurent f .
. f
f (z) =1
z 1 3i +1
(z 1 i (1 + i)) (z 1 + i (1 + i))=
1
z 1 3i +1
(z 1 + (1 i)) (z 1 (1 i))=
1
z 1 3i +1
2 (1 i) 1
z 1 (1 i) 1
2 (1 i) 1
z 1 + (1 i) .
4. R, z () = Rei, 0 pi, pipi 0 R > 0. pi
limR
R
1
(z2 + 1)3dz = 0 .
, pi pipi pi
0
1
(x2 + 1)3dx .
. z R 1(z2 + 1)3 = 1|z2 + 1|3 1(|z|2 1)3 =
1
(R2 1)3 .
R Rpi piR
1
(z2 + 1)3dz
Rpi(R2 1)3 R 0 .,
limR
R
1
(z2 + 1)3dz = 0 .
i pi 3 f (z) = 1/ (z2 + 1)3. f pi pi pi pi pi pipi R,
z () = Rei, 0 pi [R,R]. R i f
R.
-
128 2.
pi pipi RR
1
(x2 + 1)3dx+
R
1
(z2 + 1)3dz = 2piiRes (f, i) . (2.3)
Res (f, i) = Res
(1
(z2 + 1)3, i
)=
1
2!limzi
[(z i)3 1
(z2 + 1)3
]=
1
2!limzi
[1
(z + i)3
]=
1
2!limzi
12
(z + i)5= 3
16i .
pi, pi (2.3) pipi
1
(x2 + 1)3dx = lim
R
RR
1
(x2 + 1)3dx = 2pii
( 3
16i
)=
3pi
8.
, 0
1
(x2 + 1)3dx =
1
2
1
(x2 + 1)3dx =
3pi
16.
-
2.4. 20078 129
pipi
3 , 2008
1. () pi f : G C C, f(z) = u(x, y) + iv(x, y), pi ( x, y) G.
f
z:=
1
2
(f
x if
y
) f
z:=
1
2
(f
x+ i
f
y
),
pi f G fz = 0. f
, pi f = fz .
() pi
f (z) = zz +z
z, z 6= 0 ,
pi pi pi.
.
() f G, pi pi pi
CauchyRiemann: ux = vy uy = vx. pif
x= if
y f
z= 0 .
pi f = f/x,
f =f
x=
1
2
(f
x if
y
)=f
z.
, pi f/z = 0 f/x = if/y, pi- CauchyRiemann G. pi f
pi ( x, y), pi pi f
G.
() C \ {0} f pi ( x, y). f pi pipi pi
CauchyRiemann.
f
z= 0 z z
z2= 0 z2 = 1 z2 = 1 z = 1 .
-
130 2.
pi, f pi 1. pi f (z) =z + 1/z, f (1) = 2.
2. f : D (0, 1) C , pi D (0, 1) = {z C : |z| < 1} . pi
1
2pii
C+(0, r)
f (z) f (z)z2
dz ,
pi C+ (0, r) 0, r, 0 < r < 1 .
d = supz,wD(0,1)
|f (z) f (w)|
pi f , pi
f (0) d2.
. z D (0, 1), z D (0, 1) pi g (z) := f (z) f (z) D (0, 1). pi pi Cauchy
pi
g (0) =1
2pii
C+(0, r)
g (z)
z2dz 2f (0) = 1
2pii
C+(0, r)
f (z) f (z)z2
dz .
pi, r (0, 1)
f (0) = 14pi
C+(0, r)
f (z) f (z)z2
dz
1
4pi
C+(0, r)
|f (z) f (z)||z|2 |dz|
14pi
d
r2
C+(0, r)
|dz|
=1
4pi
d
r22pir =
d
2r.
, f (0) limr1
d
2r=d
2.
-
2.4. 20078 131
3. ze2piz1 =
n= anz
n pi Laurent f(z) = ze2piz1
= {z C : 1 < |z| &