∞
C
C C = (a, b)/a,b
{
Cn = (z1 , z2 ,...,zk ,...,zn )/zi
{
C
∈ R}
∈ C; i = 1, 2,...,k,...,n} C
z,w,...
C
⇒a=c∧b=d
(a, b) = (c, d) =
(a, b) + (c, d) = (a + c, b + d)
− bd, ad + bc) (a, b) (a, b)(c, −d) ac + bd −ad + bc = = , (c, d) (c, d)(c, −d) c2 + d2 c2 + d2
(a, b)(c, d) = (ac
£
(0, 0)
£
(1, 0)
/(c, d) = (0, 0)
C C
(a, b) + (0, 0) = (a, b) (a, b)(1, 0) = (a, b)
R ψ:
x
−→ (x, 0)
ψ
R
x, y ψ(x + y) = ψ(x) + ψ(y) ψ(x + y) = (x, 0) + (y, 0) = ψ(x) + ψ(y)
R
a = (a, 0)
−
(0, 1)(0, 1) = ( 1, 0) =
−1
”i” i = (0, 1) i2 =
−1
i=
√ −1
C
z1 , z2 ,... z = (a, b)
∈ C =⇒ z = (a, 0) + (0, b) = (a, 0) + (b, 0)(0, 1) =⇒z = a + bi C = a + bi/a, b
{
(x, 0)/x ∈ R
∈ R}
z = a + bi
z z¯ = a
− bi
z + w = z¯ + w ¯ (z.w) = z¯. w ¯
z w
=
z¯ W = 0 w ¯
z = a + bi
e(z) = a m(z) = b
a= b=
z+z 2 z
− z = −i
2i
− z
z
2
e(z + w) = e(z) + e(w) m(z + w) = m(z) + m(w) e(z) = e(z) m(z) = −m(z)
z
∈C
|z |
z = a + bi
|z| = ∈ C − {0} |z|
z z
a2 + b2
|z| ≥ 0 |z| = |z| |zw | = |z| |w| |z| z w ∈ C − {0 } = |w | w |z| = 0 ⇐⇒ z = 0 |z + w| ≤ |z| + |w| ||z| − |w| | ≤ |z| + |w| |e(z)| ≤ |z| |m(z)| ≤ |z| |z|2 = z.z |zn| = |z|n
z 2π
∈ C − {0} 1
z
z z
θ
∈ C − {0} ||
z = z (Cosθ + iSenθ) z z = z (Cosθ + iSenθ)
||
∀ ∈ N.
z = z (Cos(θ + 2πk) + iSen(θ + 2πk)) k
||
0
z b T gθ = a
z = a + ib
∈ C − {0} θ
z
z
∈ [−π , π > < −π, π] z ∈ C − {0}
Arg(z) + Arg(z) = 2π Arg(zw) = Arg(z) + Arg(w)
∈ C − {0} |Arg(z) − Arg(−z)| = π z ∈ C I =⇒ −z ∈ C II I =⇒ Arg(−z) − Arg(z) = π z ∈ C II =⇒ −z ∈ C IV =⇒ Arg(−z) − Arg(z) = π z ∈ C II I =⇒ −z ∈ C I =⇒ Arg(z) − Arg(−z) = π z ∈ C IV =⇒ −z ∈ C II =⇒ Arg(z) − Arg(−z) = π |Arg(z) − Arg(−z)| = π z
£
£
£
£
z
∈ C − {0}
C I C II C II I C IV
| | ∠θ
z= z
cisθ = eiθ = Cosθ + iSenθ = eiθ+2kπ (f ´ ormula de Euler)
n
∈Z
z
∈C
zn = z.z.z......
”n” factores
(z1 z2 )n = z1n z2n
|zn| = |z|n
z1 z2
n
z1n = n z2
(z n )m = z nm
θ Arg (z)
i4n = 1 i4n+1 = i i4n+2 = (z1 + z2 )n =
n
n k
n k=0
∈Z
−1
i4n+3 =
z1n−k z2k
−i
∀n ∈ Z+
(cosθ + iSenθ)n = Cos(nθ) + iSen(nθ)
n=1 n=h
n = h+1
(Cosθ + iSenθ)h+1 = (Cosθ + iSenθ)(Cosθ + iSenθ)h = (Cosθ + iSenθ)(Coshθ + iSenhθ) = (CoshθCosθ
− SenhθSenθ) + i(SenhθCosθ + SenθCoshθ)
n = h+1
(cisθ)n = cis(nθ) cisθ = cisθ
n
∈ Z+ z ∈ C
b
z
bn = z
b = z 1/n
n
∈ Z+
||
z = z (Cosθ + iSenθ) 1 z= z n
||
z
Cos
∈ C − {0} n
n
θ + 2kπ n
∈ Z+
z
+ iSen
n
θ + 2kπ n
,
k = 0, 1, 2,...,n
−1
z n
n
n
z
θ + 2kπ wk = z cis n n P (z ) = w z /n
1
||
−
k : 1, 2, 3,...,n
| | | | − ∧ n j=1 w j
e
z z
n j=0 cos
/n
1
θ + 2 jπ n θ + 2 jπ n sin j=0 n n j=0 cos
θ + 2kπ n
wk = cis n−1 k=1
2kπ cos n
,
=0
=0
=0
=0
k = 0, 1, 2,...,n n−1
=1
sin
k=1
− i)n = (z + i)n n ∈ Z+ z−i n z−i =1 =e
n j=1 w j
=0
θ + 2 jπ n θ + 2 jπ n j=0 sin n
/n
1
=0
n j=1 w j
m
−1
2kπ n
n 1
=0
(z
2kπi
z+i
k =0 z 1 = ze
⇒ −
2kπi
n
z+i
+ ie
n
k = 1, 2, 3,...,n
2kπi
n
z= (z 2 + 2iz
i(e 1
2kπi
+ 1)
n
−e
2kπi
, k = 1, 2, 3,...,n
n
−1
− 1)5 = (z + 2)10
[(z + i)2 ]5 = (z + 2)10 z=
2e 1
2kπi 10n
−e
−i
2kπi 10n
⇒
z+i =e z +2
k = 1, 2,..., 9
2kπi 10n
−1
=0
n n
P (z) =
ak z n−k
, ak
k=0
∈ R(´o C)
P (z)
α
P (z)
α
P (z) P (z )
a + ib (a + ib)2
a + ib = a
− ib
a
− ib = (a + bi)2
a = a2
− b2 ∧ −b = 2ab √ ⇒ a = −1/2 ∧ b = ± 3/2 √ 3 1 √ 3 √ 3 1 √ 3 1 3 1 1 2 P (z) = z −(− +i − 2 +i 2 − 2 −i 2 − −i 2 )z+ 4 + 4 2 2 2 P (z) = z 2 + z + 1 P (z) P (z)
−α = ei5π/4
α
P (z) = iz 4 + 2z 2 α
−
⇐⇒ −α
α = eiπ/4
−i
P (ei5π/4 ) = i(ei5π/4 )4 + 2(ei5π/4 )2
P (z)
− i = −i + 2i − i = 0
cumple!
P (z) α es ra´ız
⇐⇒ −α es ra´ız ⇐⇒ α es ra´ız ⇐⇒ −α es ra´ız P (z) =
n n−k k=0 ak z
, ak
∈ R, k = 0, 1,...,n
P (z )
P (z) = z 4 + 4 α,α, α
−
α = 1+i
−
P (z)
1 + i,
1
− i, −1 − i, −1 + i
P (z)
P (1/z) = z −n P (z)
n α es ra´ız de P (z) P (z ) = z 2 + bz + 1 b
α+ c+
c=d=
P (z) =
z3
6 + 3i P (z) = z 3
√ 2 2
− (5 −
P (z )
α = c + id
−
1 c id = c + id + 2 = α c + d2
c = c2 + d2
d
−
−
1 1 c id = = 2 α c + id c + d2
−b
− c2 +d d2 = 0
c2 + d2 = 1
P (z) 3 r=1 i i)z 2 + (10 2i)z 8
−
⇐⇒ α1 es ra´ız de P (z)
∈R
−b ∧ ⇒ b = −2c ∧ √ 2 −2
n=0
P (z) = z2
± √ 2z + 1
8
−
− (6 + 3i)z2 + (9 + 0i)z − (4 + 7i)
{ ∈ C/ |z − 1| ≤ |z + 1|}
R= z
|z − 1|2 ≤ |z + 1|2 =⇒ |z − 1||z − 1| ≤ |z + 1| |z + 1| =⇒ zz − z − z + 1 ≤ zz + z + z + 1 2(z + z) ≥ 0 ∴ e(z) ≥ 0
{ ∈ C/ |z + 1| < 2 |z − 1|
V = z
(x + 1)2 + y 2 < 4[(x
z = x + iy
3x2
− 10x + 3y2 + 3 > 0 =⇒ (x − 53 )2 + y2 > 169
( 53 ; 0)
2kπ n−1 ) k=1 Cos( n
=
−1
n−1 k=1
− 1)2 + y2]
4 3
Sen(
2kπ )=0 n
1 + cos72 + cos144 + cos216 + cos288 = 0 n=5
z1 , z2
∈C
z1
z2
d(z1 , z2 ) = z1
| − z2|
d(z1 , z2 ) > 0 d(z1 , z2 ) = 0
⇐⇒ z1 = z2
d(z1 , z2 ) = d(z2 , z1 )
∀ ∈C
d(z1 , z3 ) < d(z1 , z2 ) + d(z2 , z3 ) z2
(C, d)
z0
∈C
r > 0
z0 V r (z0 )
{ ∈ C/ |z − z0| < r }
V r (z0 ) = V (z0 , r) = z
(z0 ) V r (z0 ) = V (z0 , r) = V r (z0 )
z0
∈C
r>0
− {z0} = V (z0, r) − {z0}
V r (z0 )
{ ∈ C/ |z − z0| ≤ r}
V r (z0 ) = z V r (z0 )
A
⊂ V r (z0) ⊂C
z0 V (z0 , r)
∈C
A
∩ A = ∅
∀r > 0 z0
{
A = (x, y)/0 < x < 1; 0 < y < 1
}
A
1
A z0
⊂C
z0
z1 z2 = 1 + i/2
A V (z0 , r)
A
z0 A
⊂C
A
A
A
A = z
{ ∈ C/z es p.a. de A}
∈A
V (z0 , r)
z0
A
A A
⊂ C z0 ⊂ {C − A}
⊂C
∈A
A
A
∃ V (z0, r)
r > 0
∂A
A
A =< 0, 1 > . < 0, 1 >
A
⊂C
A
A
⊂C
A
A
⊂C
A
⊂C
A
(∂A )
A
A
A (C
A
− A)
∃ M > 0/|z| ≤ M ∀z ∈ A
(Aα )α∈I α
∈ I
⊂ ⊂ A
A (Aα )α∈I
Aα
z
∈A
Aα A
α∈I
{ }
I = 1, 2
A
(Aα )α∈I
A
⊂C
A
{
∪ A2
}
I = 1, 2, 3
A
V (z )z∈A V (z1 ), V (z2 ),...,V (zn )
z1 , z2 , z3 ,...,zn
A
Aα = A1
α∈I
z A
∈A
⊂C
⊂C
∅
A=
A
∂ (A)
{| − m| /z,m ∈ A}
∂ (A) = supremo z A a=5
b=4
∂ (A) = 10
z = x + iy iSen(y)
−
| |}
x+ z
z x
∈ R+
w=
± √ 12 {
||
x+ z +
w2 = z w2 = 12 [ x + z + iSen(y) x + z ]2 w2 = 12 [x + z + 2iSen(y) x+ z 2Sen 2 (y)( x + z ] w=
w=
|| − || || − | |− 1 2 (x + |x|), y = 0 1 2 (x + |z | + 2iy + x − |z |), y > 0 1 2 (x + |z | + 2iy + x − |z |), y < 0 1 2 (x + |x|), y = 0 0 z, y=
−
||
w2 = z
z = x + iy
z =0
1 z+ z
≤ ≤ z2 + 1
(x2
x2
− y2 + 2ixy + 1
− y2 + 1)2 + 4x2y2 ≤ (x2 + y 2 − 1)2 − 4y2 ≤ (x2 + y2 − 1 − 2y)(x2 + y2 − 1 + 2y) ≤ (x2 + y2
− 1 − 2y) = 0 =⇒ (x2 + y2 − 1 + 2y) = 0 =⇒
x2 + (y
≤
2
|| 2 |x + iy | 2 z
4(x2 + y2 ) 0 0
− 1)2 = 2
x2 + (y + 1)2 = 2
(3
−|z − i|−|z − 2i|)(3 −|z|) ≤ 0
3 (0, 1)
− |z − i| − |z − 2i| = 0 =⇒ |z − i| − |z − 2i| = 3
(0, 2)
a=
3 2 3
− |z| = 0 =⇒ |z| = 3
3
−
z +1 = c z 1
c=1
c > 0
|z + 1| = c |z − 1|
z=1
|x + 1 + iy|
z = x + iy
(x + 1)2 + y 2 = x2 + 2x + 1 + y2
− c2x2 + 2c2x − c2 − c2y2 (1 − c2 )x2 + (2 + 2c2 )x + (1 − c2 )y 2
1 + c2 x+ 1 c2
−
x+
1 + c2 1 c2
c=1
n k k=0 z
=
1
− zn+1 1−z
z = 0, 1
n k=0 Cos(kθ)
=
Sen θ2 + Sen(n + 12 )θ
n k=0 Sen(kθ)
=
Cos θ2
2Sen θ2
− Cos(n + 12 )θ 2Sen θ2
−
= 0 = c2
2
+y
2
| − 1 + iy| c2 [(x − 1)2 + y2 ]
= c x
=
2
+ y2 =
−1 c2 − 1 + 1 − c2
4c2 (1 c2 )2
−
2
1 + c2 1 c2
−
1
n=1 n k 0 1 k=0 z = z + z = 1 + z
− z1+1 = (1 − z)(1 + z) = 1 + z 1−z 1−z
z=0
n=h
h k k=0 z
=
1
n = h+1
h+1
− zh+1 1−z
h
z
k
= (
k=0 h+1
z k ) + z h+1
k=0
z
k
=
k=0 h+1
zk =
k=0 h+1
zk =
k=0
− zh+1 + zh+1 1−z 1 − z h+1 + z h+1 − z h+2 1−z 1 − z (h+1)+1 1−z 1
z = eiθ n
(eiθ )k =
k=0
n
(Coskθ + iSenkθ) =
k=0
− (eiθ )n+1 1 − eiθ 1 − (eiθ )n+1 e (e− − e ) i(e− − ei(n+1/2)θ ) 1
iθ
iθ
iθ
2
2
2
iθ 2
= n
(Coskθ + iSenkθ) =
2Sen θ2
Sen θ2 + Sen(n + 12 )θ + i Cos θ2 2Sen 2θ
k=0
Cos(nθ) = Sen(nθ) =
n k=0 (
−1)k/2Cos −k θ.Senk θ n (k−1)/2 Cos n−k θ.Sen k θ k=0 (−1)
k : par k : impar
− Cos(n + 12 )θ
(Cosθ + iSenθ)n = Cos(nθ) + iSen(nθ) (Cosθ + iSenθ)n = (i)2k = ( 1)k
−
n k=0 (
n
−
Cos(nθ) + iSen(nθ) =
( 1)
k/2
Cos
−1)(k−1)/2
n k
Cos n−k θ.(i)k Sen k θ
n
−k
k
θ.Sen θ + i.
k=0
k:P ar
−
( 1)(k−1)/2 Cos n−k θ.Senk
k=0
k:Impar
x = x(t), y = y(t)
x = x2
− y2
y = 2xy
x + iy = x2
− y2 + i2xy
z (t) = z 2 (t) 1 = t + k1 + ik2 z(t) 1 x + iy = (t + k1 ) + ik2 ((t + k1 ) + ik2 ) x + iy = (t + k1 )2 + ik22
−
−
A
z
∈
∈C
∈− ∈− ∃ ∈ ⇒
w A w A a A/w= V (w, r) =
−A = {−z/z ∈ A} w
−a
V (w, r)
|z − w| |−w − (−z)| |a − (−z)|
⊂ −A
< r < r < r
−z ∈ V (a, r) =⇒ −z < r =⇒ −(−z) ∈ −A
−A
A
−A
−A
A
{
∈ A} p ∈ C
A
A = z/z
0/V (a, r)
p + A = p + a/a y p+A = a
∈
⊂A V (y, r)
w
{
∈ V (y, r)
∈ A} ⇒ ∃ ∈ A/y = p + a
− ∈
∈
A, B
∈
{
}
< r < r
⊂ A =⇒ w − p ∈ A =⇒ ∃a1 ∈ A/w − p = a1 =⇒ w = p + a1
∃
− ∈
w a V (b, r2 ) = V (y, r) A + B A + B es abierto
⊂
A
a1
∈A
A+B
⊂ ⊂
|w − y| |w − a − b| |(w − a) − b| ⇒
⇒∃
< r
y A+B y a A, b B/y = a + b r1 , r2 > 0/V (a, r1 ) A, V (b, r2 ) r = min r1 , r2 V (y, r) A + B
∃ ∈
∈
p+A
⊂C
∈
a
p+A A= r >
⊂ p + A |w − y | |w − ( p + a)| |(w − p) − a|
w p V (a, r) w p + a
A
⊂ C/A
⊂B
A+B
⊂B
w
∈ V (y, r)
< r < r < r2
∃b1 ∈ B/w − a = b1 =⇒ w = a + b1 =⇒ w ∈ A + B
y
A+B
{
B = 1+i
− z/z ∈ A}
(A + B)o = A + B B
A
∈ B =⇒ ∃ a ∈ A/p = 1 + i − a V ( p, r) ⊂ B w ∈ V ( p, r) p
| p − w| |1 + i − a − w| |(1 + i − w) − a| (1 + i
− w) ∈ A V ( p, r) ⊂ B
a1 B es abierto
∃ r > 0/V (a, r) ⊂ A
A
< r < r < r
∈ A / 1 + i − w = a1 =⇒ w = 1 + i − a1 =⇒w ∈ B
{ ∈ C/ |z| − |z − i| ≥ 2 ∧
A= z
arg(z)
{ ∈ [0; π2 ]} B = {z − 2z/arg(z) ∈ [0;2π]} C = {z 2 /arg(z) ∈ [π; 2π]} 0} D = {(|z |)−1 z/z = A = {z ∈ C/e(z) + m(z) ∈ Q}
∈ [0;2π]}
A
A = 2z + z/arg(z)
{
A = (x, y)/x + y z A
∈C
∂A
∈ q, q ∈ Q} z
r>0
z
z ∂A = C
{ ||
B = z+ z / e(z)
∈ Q}
e(z +|z|) ≥ 0
∂B
∂B A, B
{ − |
∈C
A
∩ B =
| ∈ [1, 2]}
A
∩B
{ | | ∈ [1, 2]} = {z/ |z| ∈ [1, 2]} z − i = w ↔
A = z 1/ z + i z + i = w, A
A = w/ w
1
2
{ − i/ |ez | ≤ 2} = {(x, y)/ ex+1 ≤ 2} C = {z + 2z/ ezi ≤ 2} D = {z − 2z/ z 2 + 2zi − 1 ≤ 4} B= z
F = zi + 1/
{
z+1 z+i
A, B
≤ } 1
∈C
A+B
B = z/z
{
∈ B}
B
⇐⇒ B
∂ (A ∪ B) = ∂A ∪ ∂B ⊂C A = {z ⊂ C/e(z) > 0, m(z) > 0} B = {0, 1, −1}
A, B
A+B
E = z 2 / z < 1
{ | | } E P = {z ∈ C/e(z) = m(z)}
{ ∈ C/e(z) = −m(z)}
Q= z
P + Q
P + Q = C
−− −
a+b a+b a b z = a + ib = +i + 2 2 2 z = p + q
C
⊂ P + Q
P + Q
p=
⊂C
a+b a+b +i 2 2
i
∈ P
P + Q = C
a
b
2
q =
a
− b − ia − b ∈ Q 2
2
C
(P + Q)o = C
{ ∈ C/Arg(z) = πn , n ∈ Z+}
F = z
f : Ω = C Ω
⊂C
f (z ) = z 2
Ω g(z) = z /
1 2
g : g(z) = z /
1 2
z / = z cis 1 2
|| g1 (z) = |z | cis =⇒ g2 (z) = −g1 (z) AB
RS
θ + 2kπ 2 θ 2
k = 0, 1
||
g2 (z) = z cis
θ + 2π 2
g1 (z) h(z) = z
h
3 /
1 3
h1 (z) =
|z |
h2 (z) =
|z | /
h3 (z) =
|z | /
1 3
1 3
θ 3 θ + 2π cis 3 θ + 4π cis 3 cis
√ z2 − 1
T (z ) =
√ z2 − 1
/
1 3
w2 =
w2 = z 2
−1
w = (z + 1)(z z + 1, z
−1 ∈C
−
1 = r1 eiθ
1
z + 1 = r2 eiθ
2
z
w2 = r1 r2 ei(θ
1
+θ2 )
√ w = r1 r2 cis
r1 = z
| − 1| , θ1 : argumento de z − 1 r2 = |z + 1| , θ2 : argumento de z + 1
θ1 + θ2 + 2kπ 2
√ w1 = r1 r2 cis A
− 1)
⊂C
f : A f :
A
−→ R
,
π < θ1 < π, 0 < θ 2 < 2π
θ1 + θ2 2
−→ C A
−
⊂ C (Df = A)
,
w2 =
−w1
f
−→ R graf (g) ⊂ C2 graf (g) = {(x,y,z)/z = f (x, y)} g: A
A2
f :
A
⊂ C2
−→ C2
{
graf (f ) = (x, y), (z, w)/(z, w) = f (x, y)
} −→ graf (g) ⊂ C4
f (z) = z 2 z x2
∈ C =⇒ z = |z| eiθ =⇒ z2 = |z|2 ei2θ
⇒ f (z) = f (x, y) = (x + iy)2 =
z = x + iy =
− y2 + i2xy
f (z) = f (x + iy) = µ(x, y) + iν (x, y) µ, ν : R2
−→ C
µ(x, y) = x2
− y2
ν (x, y) = 2xy
x=y
y=x
µ(x, y) = 0 ν (x, y) = 2x2 ν
V r (z0 ) = V r (z0 )
− {z0}
V r (z0 )
z0 = z0
⇒
L
∩ A = ∅
−→ C
f : E
∈ E f : E −→ C z0 ∈ Df
ε>0 f (z)
|
E
⊂C
δ = δ (ε, z0 ) > 0
− L| < ε
f (z )
| − |
0 < z z0 < δ z
l´ım f (z) = L
z→z0
l´ım f (z) = L
z→z0
f : E
z0
∈ E 1 ∩ E 2
⇐⇒ ∀ε > 0 ∃δ = δ (ε, z0) > 0/z ∈ V δ (z0) ∩ Df =⇒ f (z) ∈ V δ (L)
−→ C E ⊂ C
f : E 1
−→ C
∃M > 0/ |f (z)| ≤ M ∀z ∈ E
g : E 1
−→ C
E 1 , E 2
lim f (z) ∧ lim g(z) ⊂ C ∃z→z z→z 0
0
lim (f + g)(z) = lim f (z) + lim g(z)
z→z0
z→z0
z→z0
lim (f.g)(z) = lim f (z).lim g(z)
z→z0
z→z0
z→z0
⇒ z→z lim
lim g(z) ==
z→z0
0
lim f (z) f z→z (z) = g lim g(z)
C E C f : E f (x + iy) = µ(x, y) + iν (x, y)
−→
z0
⊂
0
z→z0
z = x + iy µ, ν : E
f
f (z) =
−→ C2 E ⊂ C2
∈ E f : E −→ C ∃ z−→z l´ım
0
f (z)
⇐⇒ ∃ (x,y)→(x l´ım µ(x, y) ∧ ∃ l´ım ν (x, y) ,y ) (x,y)→(x ,y ) 0
0
0
0
z0 = x0 + iy0 l´ım f (z) =
l´ım
z−→z0
(x,y)→(x0 ,y0 )
µ(x, y) + i
l´ım
(x,y)→(x0 ,y0 )
ν (x, y)
f (z) = z z0 = 0 Df = C f (z) = f (x + iy) = (x + iy) = x
⇒ x0 = 0 ∧ y0 = 0
z0 = 0 + i0 =
l´ım
(x,y)→(0,0)
µ(x, y) = x
− iy µ(x, y) = 0
∧ ν (x, y) = −y
∧ (x,y)→(0,0) l´ım ν (x, y) = 0
=
l´ım f (z) = 0 + i0 = 0 ⇒ z→z 0
l´ım(x,y)→(0,0) µ(x, y)
⇒ l´ımz→z
l´ım(x,y)→(0,0) ν (x, y)
=
0
f (z)
z 3 + (z)3 lim =0 z→0 z + (z)2
δ 1 = 1/2
3
z3
+ (z) z + (z)2
| | ≤ | |
z 3 + (z)3 z + (z)2
3 | | ≤ |z| − |z|2
2 z
|z − 0| < 1/2 =⇒ |z| − |z|2 < 1/2 − 1/4 = 1/4 ⇒ |z| −1 |z|2 < 4
=
z3
3
+ (z) <8 z z + (z)2
z 3 + (z)3 <ε z + (z)2
| |3
| |3 < ε
ε > 0/8 z
| − 0| < 1/2 |z| <
0< z
⇒
| − 0| < δ =
0< z
z +1 =0 z→0 z 2 lim
z3
3
{
ε/8
δ = min 1/2, 3
+ (z) z + (z)2
−
0 <ε
} 3
ε/8
z +1 z2
−
z +1 0 = z2
|z| > 1 =⇒ |z| + 1 < 2 |z| z +1 2 |z | 1 − 0 = = z2 z 2 |z |2 z +1 −0 <ε z2 z +1 −0 <ε z2
M 1 = 1
M = m´ ax 1, 1/ε
{
}
⇒
|z − 0| > M = limz 2 =
z→0
∃
≤ | | ||
|| 1 si : |z | > 1 ∧ |z| < ε 1 si : |z | > 1 ∧ |z | < ε si : z > 1
z +1 z2
||
£
lim
£
£
z (z)2 + (z)2 z z→0 z3 + 1 lim
lim
z→∞
zSen(z) z2 + 1
lim (z 2
z→∞ £
π |z|
z3 + 1
z→0
£
− z)
z 8 + 10z 2 + 40z + 100 z→i z i 10
lim
| −|
z2 > N
| |2 > N √ N, si |z | > N
z2
> N, si z
z2
>
|z| > M =⇒ |z| > z 2 .Sen
0 <ε
∧ z ∈ Df =⇒
|z | ≤ | z |2
√ N
−
∞
N > 0, M > 0/ z > M
M =
z +1 z2
√
⇒ |z|2 > N =⇒
N =
z 2 > N
£
z +1 z→∞ z + 1
£
lim
lim
z→2
£
lim
z +2 z 2 + (1 i 3)z
z→2eiπ/4
− √ − i2
√ 3
√
z 4 + 16 z z i2 2
− −
∃ z→4+3i lim f (z)
A
f (z) =
limf (z )
Az z + 21
− 3i
|z| ≥ 5 |z| ≤ 5
z→0
f (z) =
∞
cos z 0
− cos z e(z) > 0 z e(z) ≤ 0
C
C
∞
x2 + y2 + z 2 = 1
σ : S
−∞
− {N } −→ C P
−→ z = LNP ∩ P xy
σ σ(P 1 ) = σ(P 2 ) =
⇒ P 1 = P 2
z
∈ C ∃P/σ(P ) = z ∃σ−1 ∞ N
S
P
N
f : Ω = C z0
∈Ω
C
Ω
f
f (z0 )
⇒ µ(x, y) = Re (f (z))
f (z) = µ(x, y) + iν (x, y) =
f
ν (x, y) = Im (f (z))
z0
f (z0 + h) h→0 h
f (z0 ) = l´ım
− f (z0) z→z − z0 f (z + z) − f (z) l´ım
f (z0 ) = l´ım
0
f (z) =
− f (z)
f (z) z
∆z
∆z→0
f (z0 ) f (z0 ) =
[µ(x0 + h1 , y0 + h2 ) + iν (x0 + h1 , y0 + h2 )] h1 + ih2 )→(0,0)
l´ım
(h1 ,h2
− [µ(x0, y0) + iν (x0, y0)]
z0 = x0 + iy0 = (x0 , y0 ) f (z0 ) =
µ(x0 + h1 , y0 + h2 ) h1 + ih2 )→(0,0)
l´ım
(h1 ,h2
− µ(x0, y0) + i ν (x0 + h1, y0 + h2) − ν (x0, y0) h1 + ih2
f (z)
−→ h2 = 0 −
µ(x0 + h1 , y0 ) µ(x0 , y0 ) ν (x0 + h1 , y0 ) +i h →0 h1 + h1 = µx (x0 , y0 ) + iν x (x0 , y0 )
f (z0 ) =
l´ım
− ν (x0, y0)
1
−→ h1 = 0 f (z0 ) = =
µ(x0 , y0 + h2 ) µ(x0 , y0 ) ν (x0 , y0 + h2 ) +i →0 ih2 ih2 iµy (x0 , y0 ) + iν y (x0 , y0 )
l´ım
h2
−
−
f (z0 ) µx (x0 , y0 ) = ν y (x0 , y0 ) µy (x0 , y0 ) =
−ν x(x0, y0)
− ν (x0, y0)
f f
f (z) =
f (z) =
=
=
z3 zz 0
z2 z 0
z=0 z=0
z=0 z=0
(x iy)3 x2 + y2 0
−
x3
(x, y) = (0, 0) (x, y) = (0, 0)
− i3x2y − 3xy2 + iy3 x2 + y 2
0
f (z) = µ(x, y) + iν (x, y)
µ(x, y) =
ν (x, y) =
(x, y) = (0, 0) (x, y) = (0, 0)
x3 3xy2 x2 + y2 0
−
y 3 3x2 y x2 + y2 0
−
(x, y) = (0, 0) (x, y) = (0, 0)
(x, y) = (0, 0) (x, y) = (0, 0)
h3 3h02 2 2 µ(h, 0) µ(0, 0) µx (0, 0) = l´ım = l´ım h + 0 h→0 h→0 h h
−
−
µy (0, 0) = l´ım
h→0
µ(0, h)
−0
− µ(0, 0) = l´ım −3(0)h2 − 0 h
h→0
h2
=1 =0
h3 3(02 )h 2 2 ν (0, h) ν (0, 0) ν y (0, 0) = l´ım = l´ım 0 + h h→0 h→0 h h
−0
=1
03 3h2 (0) 2 2 ν (h, 0) ν (0, 0) ν x (0, 0) = l´ım = l´ım h + 0 h→0 h→0 h h
−0
=0
−
−
−
−
µx (0, 0) = ν y (0, 0) µy (0, 0) =
−ν x(0, 0) z=0
f (0) f (0 + h) h→0 h f (h) 0 = l´ım h→0 h
f (0) =
l´ım
− f (0)
−
2
h 2 h h = l´ım = l´ım 2 h→0 h h→0 h (h1 ih2 )2 = l´ım (h ,h )→(0,0) (h1 + ih2 )2
−
1
h1 = 0
2
( ih2 )2 =1 f (0) = l´ım h →0 (ih2 )2
−
2
h2 = 0
(h1 )2 f (0) = l´ım =1 h →0 (h1 )2
1
h1 = h2
(h1 ih1 )2 (h1 )2 (1 i)2 f (0) = l´ım = = h →0 (h1 + ih1 )2 (h1 )2 (1 + i)2
−
−
1
−1
f (0)
f (z) = µ(x, y) + iν (x, y)
f : Ω = C Ω
µ(x, y)
C
z0
ν (x, y)
f (z0 )
∈Ω
£
£
µ ν µx µy ν x ν y
z0 = x0 + iy0
f (z0 ) = µx (x0 , y0 ) + iν x (x0 , y0 ) f (z0 ) = ν y (x0 , y0 )
− iµx(x0, y0)
(0, 0)
f (0)
f (z) = x2 + y2 + ix
µ(x, y) = x2 + y2 µ ν µx µy ν x ν y
ν (x, y) = x
(0, 0) µ ν µx (0, 0) = 0 µy (0, 0) = 0 ν x (0, 0) = 1 ν y (0, 0) = 0
µx (0, 0) = ν y (0, 0) µy (0, 0) = ν x (0, 0)
∴
f : Ω = C Ω f
f (0)
C z0 Ω z0 r > 0/ f (z) z
∈ ∃
C
∃
∀ ∈ V (z0, r)
f (z) = (z )3 f (0) f
z =0
⇒ f (z) = (x − iy)3 = (x3 − 3xy2)+i(y3 − 3x2y) = µ(x, y)+iν (x, y)
z = x+iy =
µx (0, 0) = 0 = ν y (0, 0) = 0 µy (0, 0) = 0 = µ ν µx µy ν x ν y z0 = 0
−ν x(0, 0) = 0
⇒ ∃ f (0) = 0 + i0 = 0
=
f (z0 )
µx (x0 , y0 ) = 3x20
− 3y02 µy (x0 , y0 ) = −6x0 y0
= ν y (x0 , y0 ) = 3y02 =
− 3x20
−ν x(x0, y0) = 6x0y0
x0 = y0 = 0
f (z0 ) z0
→ f
z =0
f : Ω = C Ω
C z0
∀z ∈ A
∀ ∈C
∈Ω
A
⊂C
f
Ω
←→ f
z
−
f (z ) = x2
g(z) = z 2
⇒ f (z) = g(z)
=
x2
z
y2 + i2xy
− y2 ≥ 0
f
|x| ≥ |y| f : Ω = C Ω
C
f : Ω = C Ω
f
C
A A
g(z) = f (z)
⊂Ω
f
f (z) = µ(x, y) + iν (x, y)
−
g(z) = µ(x, y) + i [ ν (x, y)] Ω f (z) :
z0 = x0 + iy0
µx (x0 , y0 ) = ν y (x0 , y0 ) µy (x0 , y0 ) =
g(z) :
−ν x(x0, y0)
−ν y (x0, y0) µy (x0 , y0 ) = − [−ν x (x0 , y0 )] µx (x0 , y0 ) =
f
Ω
f
g
µx (x0 , y0 ) = 0 µy (x0 , y0 ) = 0 ν x (x0 , y0 ) = 0 ν y (x0 , y0 ) = 0 =
⇒ ∀(x0, y0) ∈ Ω : µ(x, y) = p ∧ ν (x, y) = q
p, q
∈R
f (z) = p + iq g(z) = p
− iq f (z) = z 2
∧ g(z) = z2
−
f (z) = x2
y2 + i2 x y
| || |
∃ f (z) = y2 − x2 + i2xy ∧ f (z) = x2 − y2 + i2xy (RI )o ∪ (RII )o ∪ (RIV )o
−→f
f (z) = µ(x, y) + iν (x, y)
f
A (A
⊂ Ω, f : Ω = C)
f (z) = µx (x, y) + iν x (x, y) = µy (x, y) + iν y (x, y) A f (z) = µxx (x, y) + iν xx (x, y) = ν yx (x, y)
− iµyx (x, y) = ν xy (x, y) − iµxy (x, y)
A
f : Ω = C Ω
C
Ω f (z) = µx (z, 0) + iν x (z, 0)
f (z) = ν xy (z, 0)
− iµxy (z, 0)
−→ R
µ :
Rn
Ω µ
∈ C2Ω n
∆µ =
i=1
µ
∂ 2 µ = ∂x 2i
n
µxi xi
(x1 , x2 ,...,xn
i=1
n=2
∈ Ω)
µ: Ω
→R
µ = µ(x, y) Ω
⊂ R2
∂ 2 µ ∂ 2 µ ∆µ = ∆µ(x, y) = µxx (x, y) + µyy (x, y) = + 2 ∂x 2 ∂y ∆=
∂ 2 ∂ 2 + ∂x 2 ∂y 2
∆µ = µxx (x, y) + µyy (x, y) = 0 (x, y)
−→ R
µ:
Ω
−→ R
µ : Ω
∀
∆µ(x, y) = 0 (x, y)
µ
ν (x, y) = µ( p, q ), p =
∈Ω
Ω
∈Ω
⊂ R2 ⇐⇒ ν (x, y) = µ(−x, −y)
−x, q = −y −
ν x = µ p px + µq q x = µ p ( 1) + µq (0) = ν xx =
−µ p
− (µ pp px + µ pq q x) −
ν y = µ p py + µq q y = µ p (0) + µq ( 1) =
−µq
− (µqp py + µqq q xy ) ∀(x, y) ∈ Ω Ω ν yy =
ν xx + ν yy = µ pp + µqq = 0
µ:Ω
−→ R
Ω
R2 µ
Ω
⇐⇒ µ + k
∆ν (x, y) = 0.
k
µ(x, y) = ex cos y µxx + µyy = ex cos y ex cos y = 0 (µ + k)xx + (µ + k)yy = µxx + µyy = 0
−
µ = φ(x2 + y) µ = φ(t)/t = t(x, y) µx = µt tx
∧
µxx = µtt t2x + µt txx
µy = µt ty
∧
µyy = µtt t2y + µt tyy
∆µ = µtt (t2x + t2y ) + µt (txx + tyy ) = 0 µtt = µt
− txxt2 ++ tt2yy = h(t) x
y
µ = φ(t) ln (µ(t)) = µ=B
ˆ
ˆ ´ e
h(t)dt
h(t)dt + A
dt
|t=t(x,y) +C
µ = φ(x2 + y) t = x2 + y =
⇒ tx = 2x
txx = 2 ty = 1 tyy = 0
− txxt2 ++ tt2yy = − 4x22+ 1 x
t
y
µ = φ(x2 + y)
→
µ = φ(x2 + y2 ) t = x2 + y =
⇒ tx = 2x
txx = 2 ty = 2y tyy = 2
− txxt2 ++ tt2yy = − 4 (x22++2y2) = − 1t = h(t) x
y
´ ´ µ=B e
dt t
dt + C = B
´ 1 dt |
t=x2 +y 2
t
+C = B ln(x2 + y2 ) + C
µ : R2 R ν (x, y) = µ(ax + y, x + 2y)
ν : R2
→
→
R
∧
p = ax + y q = x + 2y ν x = µ p px + µq q x = aµ p + µq ν xx = a(µ pp px + µ pq q x ) + µqp px + µqq q x = a2 µ pp + 2aµ pq + µqq ν y = µ p py + µq q y = µ p + 2µq ν yy = µ pp py + µ pq q y + 2(µqp py + µqq q y ) = µ pp + 4µ pq + 4µqq
∧ ∧
∆ν = ν xx + ν yy = (a2 + 1)µ pp + (2a + 4)µ pq + 5µqq = 0
⇒ a2 + 1 = 5 ∧ 2a + 4 = 0 =⇒ a = −2
=
R2 µ = α(x)α(y)e−(x+y) α: R
→R
µx = (α (x) α(x)) α(y)e−(x+y) µxx = (α (x) 2α (x) + α(x)) α(y)e−(x+y) µyy = (α (y) 2α (y) + α(y)) α(x)e−(x+y)
−
∆µ =
α (x)
∧
−
− 2α(x) + α(x)
α(y) + α (y)
α (x)
∧ ν (x, y)
→C
C
Ω
− 2α(y) + α(y)
− 2α(x) + α(x) = 0
α(x) = (Ax + B) ex A, B
f : Ω
−
α(x) e−(x+y) = 0
µ(x, y) = (Ax + B) (Ay + B)
f (z) = µ(x, y) + iν (x, y)
Ω
µ(x, y)
Ω µx (x, y) = ν y (x, y)
∧
µy (x, y) =
∆µ = µxx + µyy = (µx )x + (µx )y = (ν x )x + ( ν x )y = ν yx
−
ν
µ(x, y) ∆ν = 0 ν
→
∨
⊂
− ν xy = 0
Ω Ω
→
µ : Ω ν : Ω R Ω R2 f (z) = µ(x, y) + iν (x, y) f (z) = ν (x, y) + iµ(x, y)
−ν x(x, y) ∀(x, y) ∈ Ω
→R
µ Ω
µ(x, y) = ex sin y
f (z ) = ex sin y + iν (x, y)
∨ f (z) = ν (x, y) + iex sin y
f (z) = ex sin y + iν (x, y) (ex sin y)x = ν y =
⇒ ν y = ex sin y =⇒ ν = ex cos y + φ(y)
(ex sin y)y =
−ν x =⇒ ν x = −ex cos y =⇒ ν = −ex sin y + ψ(x)
f (z) = ν (x, y) + iex sin y ν x = (ex sin y)y =
⇒ ν x = ex cos y =⇒ ν = ex cos y + φ(y)
ν y =
− (ex sin y)x =⇒ ν y = −ex sin y =⇒ ν = ex cos y + ψ(x)
φ(y) = Aψ(x) A f (z) = ex cos y + A + iex sin y = A + ez µ=
x(x 1) + y 2 (x 1)2 + y2
− −
µ(x, y) = 1 +
µ
x 1 = (x 1)2 + y2
−
− 1) − ((x2y(x − 1)2 + y2)2
− ´ 2y(x − 1) dx + ψ(y) = − y + ψ(y) µy = −ν x =⇒ v = (x − 1)2 + y2 ((x − 1)2 + y2 )2 x(x − 1) + y2 y f (z) = +i − + ψ(y) 2 2 (x − 1) + y (x − 1)2 + y 2 z(z − 1) f (z) = + ik (z − 1)2 z z=1∈ /Ω f (z ) = + ik z−1
k
ν (x, y) =
2xy ν : Ω (x2 + y2 )2
→R
Ω
⊂ R2 − {(0, 0)}
2x3 6xy2 = = µx (x, y) (x2 + y2 )3
ν f (z ) = µ(x, y) + iν (x, y) ν y (x, y) =
(x2 + y2 )2 2x
− 2xy
2(x2 + y2 )2y
(x2 + y 2 )4
−
Ω
2
µ(x, y) =
2
ˆ 2x(x − 6y ) (x2 + y 2 )3
dx + ψ(y) =
−
µ(x, y) = f (z) =
−
2x dx (x2 + y 2 )2
ˆ
A
B
A
B
A
1 7(02 ) + + ψ(0) + k z 2 + 02 2 (z 2 + 0)2
− z12 + k
∩B A∪B
∂ (A B
∂A = A
A
∅
B
B
A
B
A
∪ B) = ∂A ∪ ∂B A−B
∀ p, q ∈ C − {0} ∀ p, q ∈ C − {0} ( pA) ∪ (qB) ∀ p ∈ C − {0}
pA + qB
pA
A
B
A
B
A
B
A+B
A
B
AB
A
B
AB
A+B A+B
A+B
A
AB
A A
∩B
{
AB = ab/a
B
A
∂ (A + B) = A + B A
k : constante real
A
A
A
2x dx + ψ(y) (x2 + y 2 )3
⊂C
∩ B = ∅
∂B =
ˆ
1 7y2 + + ψ(y) x2 + y2 2 (x2 + y 2 )2
f (z) =
A, B
− 7y
2
B B
∀B ⊂ C
A
∩ B = ∅
∈ A ∧ b ∈ B}
