Komplex függvénytan

C
C
exp log
R 2

R 2
• (a1, b1) + (a2, b2) = (a1 + a2, b1 + b2)
• (a1, b1) · (a2, b2) = (a1a2 − b1b2, a1b2 + b1a2)
0 = (0, 0) 1 = (1, 0)
R 2
C R 2
±
R 2 C
(a2, b2) = (0, 0)
i = (0, 1)
(a1, b1)
(a2, b2) = a1a2 + b1b2
a2 2 + b2 ,
2
b1a2 − a1b2
a2 2 + b2
.
2
i 2 = (0, 1) · (0, 1) = (−1, 0) = −(1, 0) = −1.
{1, i} R 2 (a, b) ∈ R 2
(a, b) = a1 + bi
a + bi
a ∈ R (a, 0)
1 1 a ∈ R
a = a + 0i = a1 + 0i ∈ C
R ⊆ C.
x
C R
R(⊆ C)
a, b ∈ R ∼
ab ∼ (ab, 0) = (ab − 00, a0 − b0) = (a, 0) · (b, 0).
a + bi ∈ C
Re(a + bi) = a, Im(a + bi) = b,
a + bi = (Re(a + bi), Im(a + bi)) = Re(a + bi) + Im(a + bi)i
a + bi
i 2 = −1
a + bi
(a1 + b1i) · (a2 + b2i) = a1a2 + a1b2i + b1a2i + b1b2i 2 =

a1a2 + (a1b2 + b1a2)i − b1b2 = (a1a2 − b1b2) + (a1b2 + b1a2)i,
(a1, b1) (a2, b2) 1, i
(a1, b1) ·
(a2, b2) = (a1a2 −b1b2, a1b2 +b1a2) 1, i
R 2 D ⊆ C D ⊆ R 2
• a + bi = a − bi(= a + (−b)i)
• |a + bi| = √ a 2 + b 2
z, w ∈ C
w = 0
z
w
z ± w = z ± w, zw = z w,
= z
w .
zz = |z| 2 , |zw| = |z||w|,
w = 0
z
=
w
|z|
|w| .
a + bi
a + bi = r(cos ϕ + i sin ϕ),
r = √ a2 + b2 = |a + bi| tg ϕ = b
a
ϕ
ϕ ∈ [0, 2π) ϕ ∈ [−π, π)
(r1(cos ϕ1 +i sin ϕ1))·(r2(cos ϕ2 +i sin ϕ2)) = r1r2(cos(ϕ1 +ϕ2)+i sin(ϕ1 +ϕ2)),
r2 = 0
r1(cos ϕ1 + i sin ϕ1)
r2(cos ϕ2 + i sin ϕ2)
r1
= (cos(ϕ1 − ϕ2) + i sin(ϕ1 − ϕ2)).
r2
n ∈ Z
(r(cos ϕ + i sin ϕ)) n = r n (cos(nϕ) + i sin(nϕ)).
z = 0 n n. n ∈ N +
n
n r(cos ϕ + i sin ϕ) = √
r cos
ϕ + k2π
n
+ i sin
ϕ + k2π
,
n
k = 0, . . . , n − 1 n

C
R 2
R 2 R 2
·
U ⊆ C
∀ u ∈ U ∃ ɛ > 0 ∀ z ∈ C (|z − u| < ɛ ⇒ z ∈ U)
R 2
X ⊆ R 2 U, V ⊆ R 2
• X ⊆ U ∪ V
• X ∩ U = ∅ X ∩ V = ∅
• X ∩ U ∩ V = ∅
X X
a, b ∈ X f : [0, 1] → X
f(0) = a f(1) = b
X X
a, b ∈ X f : [0, 1] → X
f(0) = a f(1) = b f
X ⊆ R 2
X
∗
{(0, 0)} ∪ x, sin 1
: x ∈ (0, 1) ⊆ R
x
2
U ∩ V = ∅
D ⊆ R 2
D ⊆ R 2
• D

• D
• D
X ⊆ R 2 1
X γ : [0, 1] → X γ(0) = γ(1)
X p ∈ X H : [0, 1] 2 → X
t ∈ [0, 1] f(t, 1) = γ(t) f(t, 0) = p
∗ 1
1 p X
1 p ∈ X H
X ⊆ R 2 1 X
X p ∈ X H :
X × [0, 1] → X x ∈ X f(x, 1) = x
f(x, 0) = p
C
R 2
zn → z lim zn = z
∀ ɛ > 0 ∃ n0 ∈ N ∀ n ≥ n0 |zn − z| < ɛ
R 2
R 2
zn = an + bni z = a + bi
zn → z ⇐⇒ (an, bn) → (a, b).
(an, bn) → (a, b)
an → a bn → b
∞
zn lim zn = ∞
zn → ∞ |zn| → ∞
∀ K ∈ R ∃ n0 ∈ N ∀ n ≥ n0 |zn| ≥ K
R 2
zn K
n ∈ N |zn| < K
zn
∀ ɛ > 0 ∃ n0 ∈ N ∀ k, m ≥ n0 |zk − zm| < ɛ

R 2
zn → z wn → w z, w ∈ C ∪ {∞}
(zn ± wn) → z ± w znwn → zw
∞ ± ∞ ∞ · 0 0 · ∞
w = 0
zn
→ z
w ,
wn
∞
∞
∞ + ∞
n → ∞ −n → ∞ n − n = 0 → 0 n 2 → ∞ −n → ∞ n 2 − n → ∞
lim n2 +ni
i
= ∞ n 2 +ni
i
= |√n4 +n2 |
1
→ +∞
lim i n i n 1, i, −1, −i, 1, . . .
i n ɛ = 1
lim n+i
1+ni
= lim 1+ i
n
1
1 =
n +i i
lim 1+ni
n 2 +i =
lim √ 3+5ni
n+i
n =
lim in +n 2 −i
n 2 −n 2 i =
lim √ 3+5i
n+i
n =
= −i
lim 1
2 + √ 3
2 i n =
C
zn w ∈ C ∪ {∞} ∞
zn = w
lim(z0 + . . . zn) = w
∞
zn
n=0
∞
n=0
n=0
zn = z ∈ C ∪ {∞} ∞
wn = w ∈ C ∪ {∞}
∞
(zn ± wn) = z ± w ∞ ± ∞
n=0
n=0

∞
n=0
zn
ɛ > 0 n0 ∈ N k > m ≥ n0
k
zn
n=m+1
< ɛ
sn = z0 + · · · + zn n ∞
zn
n=0
z sn → z
ɛ > 0 n0 k, m ≥ n0
|sk − sm| < ɛ k > m sk − sm = k
zn
∞
n=0
n=m+1
zn ∞
|zn|
∞
zn
ɛ > 0 ∞
n=0
n=0
n0 k > m ≥ n0
k
zn
n=m+1
≤ k
n=m+1
n=0
|zn|
|zn| =
k
n=m+1
|z| < 1
∞
n=0
k
n=m+1
|zn| < ɛ
|zn|
z n = 1
1 − z ,
n z0 + z1 + · · · + zn = 1−zn+1
1−z
|z| n+1 → 0
< ɛ k > m ≥ n0
z n+1 → 0 |z n+1 | =
z
∞
n=1
1
1
=
(z + n)(z + n + 1) z .
D ⊆ C f : D → C
R 2
f : D → C z0 ∈ D ′ lim f = w ∈ C
z0
∀ ɛ > 0 ∃ δ > 0 ∀ z ∈ D (0 < |z − z0| < δ ⇒ |f(z) − w| < ɛ)
z0 ∈ D f z0
∀ ɛ > 0 ∃ δ > 0 ∀ z ∈ D (|z − z0| < δ ⇒ |f(z) − f(z0)| < ɛ)

R 2
R 2
f : D → C f : D → R 2 f
u, v u, v : D → R (x, y) ∈ D
f(x, y) = (u(x, y), v(x, y)) f = (u, v) f
f = u + iv
z0 ∈ D ′
∃ lim
z0
f ⇐⇒ (∃ lim
z0
lim
z0
f = lim
z0
u ∧ ∃ lim v)
z0
u + i lim v.
z0
z0 ∈ D f z0 u v
z0
D ⊆ C
f : D → C f z0 ∈ D
lim f = f(z0).
z0
lim f = ∞ lim f = w ∈ C lim f = ∞
z0
∞ ∞
D ⊆ C f : D → C
• z0 ∈ D ′ lim
z0
f = ∞ lim |f| = +∞
z0
∀ K ∈ R ∃ δ > 0 ∀ z ∈ D (0 < |z − z0| < δ ⇒ |f(z)| > K)
• ∞ ∈ D ′ D w ∈ C lim ∞ f = w
∀ ɛ > 0 ∃ K ∈ R ∀ z ∈ D (|z| > K ⇒ |f(z) − w| < ɛ)
• ∞ ∈ D ′ lim ∞ f = ∞
∀ K1 ∈ R ∃ K2 ∈ R ∀ z ∈ D (|z| > K2 ⇒ |f(z)| > K1)
∞ R 2

D ⊆ C f : D → C z0 ∈ D ′
lim
z0
f = w ∈ C ∪ {∞} zn ∈ D\{z0}
zn → z0 f(zn) → w
f, g : D → C z0 ∈ D ′ lim
z0
lim
z0
f ± g = w1 ± w2 lim fg = w1w2
z0
w2 = 0
lim 0
lim
−i
lim ∞
lim
0
|z|
z2 = ∞ |z|
z2 z+i
z2 +1 = lim
−i
iz+i
3−z 2 = lim ∞
lim 0
lim i
lim ∞
lim
z0
= |z|
|z 2 |
f
g
z+i
(z+i)(z−i) = lim
−i
i+ i
z i
3 =
z −z ∞
z
z lim 0
|z|
z = lim 0
z 2 +1
z6 +1 =
z 3 +z+1
z 2 −1 =
= 0
|z 2 |
z =
w1
= ,
w2
f = w1 lim g = w2
z0
|z|
= |z| 2 = 1
|z| → ∞ z → 0
1 1 i
z−i = −2i = 2 ( 1
n )
1 = 1 lim
n
0
f f(z) =
f(0)
( i
n )
i = −1
n
z Re(z)
|z| z = 0
D
fn
fn : D → C ∞
n=0
• D z ∈ D ∞
fn(z)
• D f
ɛ > 0 N m ≥ N z ∈ D
m−1
fn(z) − f(z) =
n=0
n=0
∞
fn(z) < ɛ.
n=m

∞
fn n |fn| ≤ an
∞
n=0
n=0
an

D ⊆ C f : D → C
f z0 ∈ D
lim
z0
f(z) − f(z0)
z − z0
f ′ (z0)
D D
O(D)
C O(C)
n ∈ N f(z) = z n
f ′ (z) = nz n−1
n = 0 n > 0
f ′ (z0) = lim
z0
z n − z n 0
z − z0
∈ C
= lim(z
z0
n−1 + z n−2 z0 + · · · + zz n−2
0
+ zn−1
f z0
0
) = nz n−1
0 .
f(z) = f(z0) + (f(z) − f(z0)) = f(z0) + f(z)−f(z0)
(z − z0)
z−z0
f(z0) + lim
z0
lim f = f(z0) + lim
z0
z0
f(z) − f(z0)
z − z0
f(z) − f(z0)
(z − z0) =
z − z0
lim(z
− z0) = f(z0) + f
z0
′ (z0)0 = f(z0).
f g z0 c ∈ C cf f ± g fg
z0
• (cf) ′ (z0) = cf ′ (z0)
• (f ± g) ′ (z0) = f ′ (z0) ± g ′ (z0)
• (fg) ′ (z0) = f ′ (z0)g(z0) + f(z0)g ′ (z0)

g(z0) = 0 f
g z0
f
′
g
(z0) = f ′ (z0)g(z0) − f(z0)g ′ (z0)
(g(z0)) 2 .
g z0 f g(z0) f ◦ g
z0
(f ◦ g) ′ (z0) = f ′ (g(z0))g ′ (z0).
f : D → C z0 ∈ D
f f f(z0)
f ′ (z0) = 0 f −1 f(z0) f −1
f(z0)
(f −1 ) ′ (f(z0)) = 1
f ′ (z0) .
f D f(D)
f −1 f(D)
z n
f(z) = anz n + · · · + a1z + a0
ai ∈ C
f(z) = anz n + · · · + a1z + a0
bmz m + · · · + b1z + b0
f(z) = z
lim
z0
z − z0 z − z0
= lim = lim
z − z0 z0 z − z0 0
z
z .
r(cos(−ϕ) + i sin(−ϕ))
lim
= lim cos(−2ϕ) + i sin(−2ϕ),
r→0 r(cos ϕ + i sin ϕ) r→0
f(z) = |z| 0 x
lim 0
|z| − |0|
z − 0
|x|
= lim
x→0 x ,

+1 −1 z0 = x0 + y0i = 0
y = y0
lim
z0
|z| − |z0|
z − z0
|(x + y0i)| − |x0 + y0i|
= lim
= lim
x→x0 (x + y0i) − (x0 + y0i) x→x0
x 2 + y 2 0 + x 2 0 + y2 0
lim
x→x0
(x 2 + y 2 0) − (x 2 0 + y 2 0)
(x − x0)( x2 + y2 0 + x2 0 + y2 = lim
0 ) x→x0
x0 + x0
x 2 0 + y 2 0 + x 2 0 + y2 0
=
x2 + y2 0 − x2 0 + y2 0
,
x − x0
x + x0
x 2 + y 2 0 + x 2 0 + y2 0
x0
x2 0 + y2 .
0
x = x0
lim
z0
|z| − |z0|
z − z0
x0
x0 + iy0 = 0 √
x2 0 +y2 0
x0 + iy0
|(x0 + yi)| − |x0 + y0i|
= lim
y→y0 (x0 + yi) − (x0 + y0i) =
=
y0
i √ x 2 0 +y2 0
y0
i x2 0 + y2 .
0
=
|z|
u, v : D → R f = u+iv : D → C
z0 = x0 + y0i
f ′ (z0) = lim
z0
f(z) − f(z0)
z − z0
lim
(x0,y0)
u(x, y) + iv(x, y) − u(x0, y0) − iv(x0, y0)
= lim
=
(x0,y0)
x + iy − x0 − iy0
u(x, y) − u(x0, y0) + i(v(x, y) − v(x0, y0))
.
x − x0 + i(y − y0)
y = y0
lim
x0
f ′ u(x, y0) − u(x0, y0)
(z0) = lim
x0 x − x0
u(x, y0) − u(x0, y0)
x − x0
+ i lim
x0
v(x, y0) − v(x0, y0)
x − x0
+ i v(x, y0) − v(x0, y0)
=
x − x0
x = x0
= u ′ x(x0, y0) + iv ′ x(x0, y0).
f ′ (z0) = −iu ′ y(x0, y0) + v ′ y(x0, y0) = v ′ y(x0, y0) + i(−u ′ y(x0, y0)).
f = u + iv : D → C f z0 ∈ D u
v z0
f ′ (z0) = u ′ x(z0) + iv ′ x(z0) = v ′ y(z0) + i(−u ′ y(z0)),
f ′ (z0) = u ′ x(z0) + i(−u ′ y(z0)) = v ′ y(z0) + iv ′ x(z0).

u v
f = u + iv : D → C f z0 u v
z0
f : D → C z0 ∈ D
rz0 : D → C A ∈ C f ′ (z0)
z ∈ D
f(z) = f(z0) + A(z − z0) + rz0(z) lim
z0
A ∈ C
lim
z0
rz 0 (z)
z−z0
f(z) − (f(z0) + A(z − z0))
|z − z0|
= 0 lim
z0
= 0.
rz0 (z)
|z−z0| = 0
f = u + iv z0 = x0 + y0i
lim
z0
u(z) + iv(z) − (u(z0) + iv(z0) + A(z − z0))
|z − z0|
u(x, y) + iv(x, y) − (u(x0, y0) + iv(x0, y0) + A((x − x0) + (y − y0)i))
lim
= 0.
(x0,y0)
|(x − x0) + (y − y0)i|
A = a + bi(= Re(f ′ (z0)) + Im(f ′ (z0))i)
A((x − x0) + (y − y0)i) = (a(x − x0) − b(y − y0)) + (a(y − y0) + b(x − x0))i =
(a, −b)(x − x0, y − y0) + (b, a)(x − x0, y − y0)i,
u(x, y) − (u(x0, y0) + (a, −b)(x − x0, y − y0))
lim
(x0,y0)
(x − x0) 2 + (y − y0) 2
= 0,
v(x, y) − (v(x0, y0) + (b, a)(x − x0, y − y0))
lim
(x0,y0)
(x − x0) 2 + (y − y0) 2
= 0,
u v z0
f ′ (z0) = u ′ x(z0) +
i(−u ′ y(z0)) = v ′ y(z0) + i(v ′ x(z0)) f ′ (z0) = a + bi
grad u(z0) = (a, −b) grad v(z0) = (b, a)
u, v : D → R u
v z0 ∈ D
z0
• u ′ x(z0) = v ′ y(z0)
• u ′ y(z0) = −v ′ x(z0)
=

f = u + iv z0 u
v z0 z0
f(z) = z
u(x, y) = x v(x, y) = −y u ′ x = 1 = −1 = v ′ y f
u v
f
f
f = u + iv : D → C z0 ∈ D
f z0 u v z0
z0
f = u + iv : D → C z0 ∈ D z0
u v z0
z0 f z0
f(z) = z 2 z + z 2
f(x + yi) = (x − yi)(x − yi)(x + iy) + (x 2 − y 2 + 2xyi) =
(x−yi)(x 2 +y 2 )+(x 2 −y 2 +2xyi) = (x 3 +xy 2 )−i(x 2 y +y 3 )+(x 2 −y 2 +2xyi) =
(x 3 + x 2 + xy 2 − y 2 ) + i(−y 3 − x 2 y + 2xy).
u v u ′ x = 3x 2 + 2x + y 2 u ′ y =
2xy − 2y v ′ x = −2xy + 2y v ′ y = −3y 2 − x 2 + 2x
u ′ x = v ′ y 3x 2 + 2x + y 2 = −3y 2 − x 2 + 2x x 2 = −y 2
x = y = 0
u ′ y = −v ′ x 2xy − 2y = 2xy − 2y
u v 0 f
0 f
f(z) = |z|
f(z) = z|z|
f(z) = z + i|z|
D ⊆ R 2 u : D → R D
grad u ≡ (0, 0) u D

f ∈ O(D) f ′ ≡ 0 f D
f ′ = u ′ x + iv ′ x = v ′ y + i(−u ′ y) u ′ x ≡ u ′ y ≡ v ′ x ≡ v ′ y ≡ 0
u v D
u v D f D
f = u + iv : D → C f ∈ O(D) u ≡ 0 f
v ≡ 0 f f D
u ≡ 0 u ′ x ≡ u ′ y ≡ 0 f ′ = u ′ x + i(−u ′ y) ≡ 0
f, f ∈ O(D) f f D
f + f = 2u f − f = 2iv
u v D f
f ∈ O(D) |f| D f
D
|f| ≡ 0 f ≡ 0 |f| ≡ c = 0 c 2 ≡ |f| 2 =
ff f = c2
f
D

an n ∈ N
z0 ∈ C
∞
an(z − z0) n
n=0
z0
1
R =
lim sup n |an| .
z0 ∈ C ɛ > 0 R2 S(z0, ɛ) = {z ∈ C : |z − z0| < ɛ}
z0 ɛ
B(z0, ɛ) = {z ∈ C : |z − z0| ≤ ɛ}
z0 ɛ
∞
an(z − z0) n
n=0
R > 0
• ∞
an(z − z0) n S(z0, R)
n=0
S(z0, r) 0 < r < R
B(z0, R)
K ⊆ C S(z0, R) ⊆ K ⊆
B(z0, R)
• f(z) = ∞
an(z − z0) n S(z0, R)
n=0
z ∈ S(z0, R)
f ′ (z) =
∞
ann(z − z0) n−1 .
n=1
f
S(z0, R)
• n ∈ N an = f (n) (z0)
n!

R = 0
∞
∞
i
n=0
n (z − i) n n=0
∞
n=0
n 3 (z+1−i) n
n!
z n
(1−i)(1−2i)···(1−ni)
exp log
exp(z) =
∞
n=0
z n
n! .
1
R =
lim sup n
1 = lim n√ n! = +∞,
n!
exp exp ∈ O(C)
exp
∀ x ∈ R ⊆ C exp(x) = e x
exp ′ = exp
∀ z1, z2 ∈ C exp(z1 + z2) = exp(z1) exp(z2)
∀ x, y ∈ R exp(x + yi) = e x (cos(y) + i sin(y))
exp 2πi
∀ z, w ∈ C (exp(z) = exp(w) ⇐⇒ ∃ k ∈ Z z − w = k2πi)
∀ z ∈ C exp(z) = 0

exp ′ (z) =
∞
n=1
n zn−1
n! =
∞
n=1
z n−1
(n − 1)! =
∞
n=0
z n
n!
= exp(z).
a ∈ C f(z) = exp(z) exp(a − z)
f ′ (z) = exp ′ (z) exp(a − z) + exp(z) exp ′ (a − z)(−1) =
exp(z) exp(a − z) − exp(z) exp(a − z) = 0,
f ≡ c ∈ C c = f(0) = exp(0) exp(a) =
exp(a) a, z ∈ C exp(z) exp(a − z) = exp(a)
a = z1 + z2 z = z1
exp(z1) exp((z1 + z2) − z1) = exp(z1 + z2),
exp(z1) exp(z2) = exp(z1 + z2)
exp(x+yi) = exp(x) exp(yi) = e x
n=0
∞
n=0
e x ∞
∞
n y2n
(−1) + i (−1)
(2n)!
n=0
(yi) n
n! = ex ∞
n y2n+1
n=0
i2ny2n (2n)! +
∞
n=0
= e
(2n + 1)!
x (cos(y) + i sin(y)).
z = x1 + iy1 w = x2 + iy2
i2n+1y2n+1
=
(2n + 1)!
exp(z) = exp(w) ⇐⇒ e x1 (cos(y1) + i sin(y1)) = e x2 (cos(y2) + i sin(y2)).
exp(z) = exp(w) e x1 = e x2 k ∈ Z
y1 = y2+k2π x1 = x2 k ∈ Z y1 = y2+k2π
∃ k ∈ Z z − w = k2πi
e x = 0 cos(y) + i sin(y) = 0 cos sin
0
exp(z) e z
e iπ + 1 = 0
E = {z ∈ C : −π ≤ Im(z) < π}
exp ↾ E : E → C
E C\{0} w ∈ C\{0}
z ∈ E exp(z) = w

E z = w z − w = k2πi k ∈ Z
exp(z) = 0 z
w ∈ C\{0} w
arg(w) ∈ [−π, π) w = |w|(cos(arg(w)) + i sin(arg(w)))
log(w) = ln(|w|) + i arg(w) ∈ E w
exp(log(w)) = e ln(|w|) (cos(arg(w)) + i sin(arg(w))) =
|w|(cos(arg(w)) + i sin(arg(w))) = w.
log : C\{0} → C
R − R − 0 0
log C\R − 0
R −
w ∈ C\R − 0 ɛ > 0 δ > 0
∀ s ∈ S(w, δ)\{0} log(s) ∈ S(log(w), ɛ)
δ1 > 0
(ln(|w|) − δ1, ln(|w|) + δ1) × (arg(w) − δ1, arg(w) + δ1) ⊆ S(log(w), ɛ).
δ1 log(w)
log(w)
ln |w| δ ′ 1 > 0
∀ r > 0 (||w| − r| < δ ′ 1 ⇒ | ln(|w|) − ln(r)| < δ1)
S = {s ∈ C\{0} : ||w| − |s|| < δ ′ 1 | arg(w) − arg(s)| < δ1}
w s ∈ S log(s) ∈ S(log(w), ɛ)
log(S) ⊆ S(log(w), ɛ)
δ > 0 S(w, δ) ⊆ S δ log
w
r ∈ R− an = |r|(cos(π− 1
n
1
1
n ) + i sin(π + n )) an → r bn → r
)+i sin(π− 1
n )) bn = |r|(cos(π+
log(an) = ln(|r|) + i π − 1
→ ln(|r|) + iπ,
n
log(bn) = ln(|r|) + i − π + 1
→ ln(|r|) + i(−π),
n
log R −

log C\R − 0
C\R − 0 w ∈ C\R− 0
log ′ (w) =
1 1
=
elog(w) w .
w ∈ C\{0} z ∈ C
w z = e z log(w) .
(−1) i = e i log(−1) = e i(ln(1)+iπ) = e i2 π = e −π
ii = ei log(i) π
i(ln(1)+i = e 2 ) π
π
i2 = e 2 − = e 2
sin(z) =
cos(z) =
∞
(−1)
n=0
∞
(−1)
n z2n+1
(2n + 1)! ,
n z2n
(2n)! .
n=0
z sin cos
+∞
sin cos
sin ′ = cos cos ′ = − sin
sin ′ ∞
(z) = (−1) n z
(2n + 1)
2n
(2n + 1)! =
∞
(−1)
n=0
n=0
n z2n
= cos(z).
(2n)!
sin(z) = 1
2i (eiz − e −iz ) cos(z) = 1
2 (eiz + e −iz )
1
2i (eiz − e −iz ) = 1
∞
(iz)
2i
n=0
n
1
2i
∞
n=0
(i n (1 − (−1) n )) zn
=
n!
n! −
= 1
2i
∞
(−1)
n=0
∞ (−iz) n
n=0
∞
n=0
n z2n+1
n!
= 1
2i
∞
n=0
2n+1 z2n+1
2i
(2n + 1)! =
= sin(z).
(2n + 1)!
(iz) n − (−iz) n
∞
n=0
n!
2n z2n+1
i
(2n + 1)!
=

e iz = cos(z) + i sin(z)
sin(z ± w) = sin(z) cos(w) ± sin(w) cos(z),
cos(z ± w) = cos(z) cos(w) ∓ sin(z) sin(w),
sin 2 (z) + cos 2 (z) = 1.
sin cos 2π
e z 2πi e iz 2π
sin
sin(z) = 0 ⇐⇒ e iz = e −iz ⇐⇒ ∃ k ∈ Z iz − (−iz) = k2πi ⇐⇒
∃ k ∈ Z z = kπ
sh(z) = 1
2 (ez − e −z ),
ch(z) = 1
2 (ez + e −z ).
sh(−z) = − sh(z) ch(−z) = ch(z) ch 2 (z) − sh 2 (z) = 1
z ∈ C
• sh(iz) = i sin(z) sin(iz) = i sh(z)
• ch(iz) = cos(z) cos(iz) = ch(z)
sin(z) + cos(z) = 0

a <
b ∈ R γ : [a, b] → C γ = x + iy x, y : [a, b] → R
t ∈ [a, b] γ(t) = x(t) + iy(t)
γ = (x, y) γ
∀ t ∈ [a, b] ∀ ɛ > 0 ∃ δ > 0 ∀ s ∈ [a, b] (|t − s| < δ ⇒ |γ(t) − γ(s)| < ɛ)
γ
γ
∀ ɛ > 0 ∃ δ > 0 ∀ s, t ∈ [a, b] (|t − s| < δ ⇒ |γ(t) − γ(s)| < ɛ)
γ = x + iy : [a, b] → C
• γ(a) = γ(b)
•
• t0 ∈ [a, b] t0
x y t0
γ = x + iy t0
˙γ(t0) = x ′ (t0) + iy ′ (t0)
γ : [a, b] → C
γ − : [a, b] → C, γ − (t) = γ(a + b − t)
γ − −γ
γ(t) = r(t)e iϕ(t) γ = (r, ϕ)
γ : [a, b] → R 2
R 2 \ ran(γ)
[a, b] a = t0 < · · · < tn = b
δ
∀ k = 1, . . . , n |tk − tk−1| < δ

t0 < · · · < tn
ξk ∈ [tk−1, tk] k = 1, . . . , n
a < b ∈ R g : [a, b] → R g [a, b] b
g = I
a
ɛ > 0 δ > 0 δ a = t0 < t1 <
· · · < tn = b ξk k = 1 . . . , n
I −
n
g(ξk)(tk − tk−1) < ɛ.
k=1
a < b ∈ R g : [a, b] → C g [a, b]
b
g = I ∈ C ɛ > 0 δ > 0 δ
a
a = t0 < t1 < · · · < tn = b ξk k = 1 . . . , n
n
I − g(ξk)(tk − tk−1) < ɛ.
k=1
a < b ∈ R g : [a, b] → C g = u + iv g
[a, b] u v [a, b]
b b b
g = u + i v.
a
a
∗
γ : [a, b] → C a = t0 < . . . tn = b
γ δ k = 1, . . . , n γ [tk−1, tk]
δ
a < b ∈ R γ : [a, b] → D f : D → C
f γ
f = f(z)dz = I ∈ C ɛ > 0
γ γ
δ > 0 γ δ a = t0 < t1 < · · · < tn = b
ξk k = 1 . . . , n
I −
n
f(γ(ξk))(γ(tk) − γ(tk−1)) < ɛ.
k=1
f D
ran(γ)
γ [a, b]
γ : [a, b] → [a, b] ⊆ C γ(t) = t
γ : [a, b] → C
a

γ
a = t0 < · · · < tn = b k = 1, . . . , n γ
[tk−1, tk]
γ γ
b b
|γ| = | ˙γ| =
a
a
(x ′ ) 2 + (y ′ ) 2 ,
γ = x+iy
[a, b] γ −→ D f −→ C f D
γ
b
f = f(γ(t)) ˙γ(t)dt.
a
γ
f
r > 0 γ : [0, 2π] → C γ(t) = r(cos(t) + i sin(t))
r f : C\{0} → C
f(z) = 1
z
2π
1
f =
r(− sin(t) + i cos(t))dt =
r(cos(t) + i sin(t))
2π
γ
0
2π
2π
2π
0
− sin(t) + i cos(t)
dt =
cos(t) + i sin(t) 0
idt =
0
0dt + i
0
1dt = 2πi.
w ∈ C r > 0 γ : [0, 2π] → C γ(t) =
w + r cos(t) + ir sin(t) w r
dz = 2πi
1
γ z−w
γ [0, 1 + i] f(z) = Re(z)
γ |z| = 2 f(z) = z+2
z
[a, b] γ −→ D f −→ C f D ϕ : [c, d] → [a, b]
ϕ(c) = a ϕ(d) = b
γ
[a, b] −−−−→ D
⏐
⏐
ϕ⏐
γ◦ϕ⏐
[c, d] [c, d]
γ
f =
γ◦ϕ
f
−−−−→ C
f.

γ
b
d
= f(γ(t)) ˙γ(t)dt = f(γ(ϕ(s))) ˙γ(ϕ(s))ϕ ′ (s)ds =
a
d
c
c
f((γ ◦ ϕ)(s)) (γ ◦˙ ϕ)(s)ds = f.
γ◦ϕ
t = ϕ(s)
(γ ◦˙ ϕ)(s) = ˙γ(ϕ(s))ϕ ′ (s)
[a, b] γ −→ D f −→ C f D
γ −
f = −
ϕ : [a, b] →
[a, b] ϕ(s) = a + b − s
γ
a
b
b
a
= f(γ(t)) ˙γ(t)dt = f(γ(ϕ(s))) ˙γ(ϕ(s))ϕ ′ (s)ds =
a
f(γ − (s)) ˙
γ − (s)ds = −
b
b
a
γ
f.
f(γ − (s)) ˙ γ−
(s)ds = −
γ− f.
t = ϕ(s)
γ− (s) = ˙
˙ (γ ◦ ϕ)(s) = ˙γ(ϕ(s))ϕ ′ (s)
a
b = − b
a
γ1 : [a, b] → C γ2 : [b, c] → C γ1(b) = γ2(b)
γ ⌢ 1 γ2 γ = γ ⌢ 1 γ2 : [a, c] → C
γ(t) =
γ1(t), t ∈ [a, b]
γ2(t), t ∈ (b, c]
γ1 : [a, b] → D γ2 : [b, c] → D γ1(b) = γ2(b) f : D → C
f = f + f.
γ ⌢ 1 γ2
γ1
γ2

γ : [a, b] → D D
C(D)
: C(D) → C, f ↦→ f
γ
f1, f2 ∈ C(D) c1, c2 ∈ C
(c1f1 + c2f2) = c1 f1 + c2 f2.
γ
[a, b] γ −→ D f −→ C f D |f| ≤ M
ran(γ)
γ
γ
f
≤ M|γ|.
a = t0 < t1 < · · · < tn = b ξk
k = 1 . . . , n
n
f(γ(ξk))(γ(tk) − γ(tk−1)) ≤
k=1
M
γ
γ
n
|f(γ(ξk))||γ(tk) − γ(tk−1)| ≤
k=1
n
|γ(tk) − γ(tk−1)| → M|γ|,
k=1
ɛ > 0 δ > 0 γ δ
a = t0 < · · · < tn = b
γ
f −
n
f(γ(ξk))(γ(tk) − γ(tk−1)) < ɛ.
k=1
γ δ ′ > 0 tk
δ ′ γ δ δ ′
δ ′ tk
|γ| −
n
|γ(tk) − γ(tk−1)| < ɛ.
k=1
γ
f−ɛ
<
n
f(γ(ξk))(γ(tk)−γ(tk−1)) ≤ M
k=1
n
|γ(tk)−γ(tk−1)| < M(|γ|+ɛ),
γ f
< M|γ| + (M + 1)ɛ ɛ
γ f
≤
M|γ|
k=1

D
∞
fn D γ
f = ∞
n=0
γ n=0
m ≥ N z ∈ D ∞
∞
fn =
γ n=0
γ
n=0
∞ ∞
fn =
n=0
γ
fn.
fn ɛ > 0 N
n=m
fn(z)
< ɛ m ≥ N
f0 + f1 + · · · + fm−1 +
γ
γ
γ
ɛ > 0 N m ≥ N
∞
fn −
γ n=0
∞
n=0
γ
m−1
n=0
fn = lim
m→∞
fn
γ
m−1
=
n=0
fn
γ
∞
fn
γ
n=m
∞
fn
γ n=m
=
∞
fn.
n=m
≤ ɛ|γ|
γ n=0
≤ ɛ|γ|,
∞
fn.
γ : [a, b] → D F ∈ O(D) F ′
D
γ
F ′ = F (γ(b)) − F (γ(a)).
µ : [a, b] → C µ(t) = F (γ(t))
˙µ−t F
µ γ
˙µ(t) = F ′ (γ(t)) ˙γ(t).
µ = x + iy
γ
F ′ b
= F ′ b b
(γ(t)) ˙γ(t)dt = ˙µ(t)dt = x ′ b
+ i
a
a
a
a
y ′ = [x] b a + i[y] b a =
x(b) − x(a) + i(y(b) − y(a)) = µ(b) − µ(a) = F (γ(b)) − F (γ(a)).

f : D → C
F ∈ O(D) F ′ = f D
∗
f ≡ c ∈ C γ : [a, b] → C γ = x + iy
γ
b
f =
a
b
f(γ(t)) ˙γ(t)dt = c ˙γ(t)dt = c
a
b
x ′ b
+ i y ′
=
c([x] b a + i[y] b a) = c(x(b) + iy(b) − (x(a) + iy(a))) = c(γ(b) − γ(a)).
F (z) = cz f = F ′
γ
f = F (γ(b)) − F (γ(a)) = cγ(b) − cγ(a).
f(z) = z2 +1 γ [1, i] γ
γ : [0, 1] → C γ(t) = (1−t)1+ti = (1−t)+it
f F (z) = z3
3
γ
f =
γ
+ z
F ′ = F (γ(1)) − F (γ(0)) = F (i) − F (1) =
i 3
3
2
+ i − (1 + 1) = −4 +
3 3 3 i.
γ : [a, b] → C
e z sin(z) cos(z) sh(z) ch(z)
f(z) = n
ak(z − wk) dk ak, wk ∈ C dk ∈ Z\{−1}
k=0
f(z) = 1
(z−i) 2 − i
z 5 + (1 − i)(2 − i + z) 3
f(z) = 1
z f γ : [a, b] → C\R− 0
log(z)
f : D → C
γ : [a, b] → D
0
1
z C\R− 0
a
a

f : D → C D
f
z0 ∈ D
z ∈ D
z
F (z) = f,
z0 z γ
F ∈ O(D) F ′ = f
z S ⊆ D z ξ ∈ S
[z, ξ] D
[z,ξ]
F (ξ) =
γ ⌢ [z,ξ]
f =
γ
f +
z0
[z,ξ]
F (ξ) − F (z) =
f
[z,ξ]
[z,ξ]
f(z)ds +
f =
[z,ξ]
[z,ξ]
f(s)ds =
[z,ξ]
f = F (z) +
[z,ξ]
f.
[z,ξ]
f,
f(z) + (f(s) − f(z))ds =
(f(s) − f(z))ds = f(z)(z − ξ) +
[z,ξ]
f(s) − f(z)ds,
rz(ξ) =
[z,ξ]
f(s) − f(z)ds.
ɛ > 0 z δ > 0 s ∈ D
|s − z| < δ |f(s) − f(z)| < ɛ |ξ − z| < δ s ∈ [z, ξ]
|s − z| < δ
|rz(ξ)| =
[z,ξ]
f(s) − f(z)ds
≤ ɛ|ξ − z|.
ɛ > 0 δ > 0 0 < |ξ −z| < δ |rz(ξ)|
|ξ−z|
lim z
rz(ξ)
= 0.
ξ − z
≤ ɛ
F z
F ′ (z) = f(z) ξ ∈ S
F (ξ) = F (z) + f(z)(z − ξ) + rz(ξ) lim z
rz(ξ)
ξ−z
= 0

f : D → C
f
f D
f D
⇒
⇒ γ µ D
γ : [a, b] →
D µ : [b, c] → D γ(a) = µ(b) γ(b) = µ(c) γ ⌢ µ −
0 =
γ ⌢ µ −
1
z
f =
γ
f +
µ −
f =
γ
f −
C\{0}
f 0
2πi = 0
D F (z)
[z0, z]
F (ξ) = f
γ ⌢ [z,ξ]
z0, z, ξ D
f D
D [a, b] γ −→ D f −→ C γ
f ∈ O(D)
γ
f = 0.
D
T0 l0 γ0
I = |
γ
µ
f.
f| 4
T k 0 k = 1, 2, 3, 4 γ

γk 0
γ
f =
4
k=1
k |
γk f| ≥
0
I
4 T1 = T k 0 γ1 = γk 0
T1 l1 = l0
2
T1 Tn
ln = l0
2 n γn
γn
γ k 0
f
≥ I
.
4n z0 ∈ ∞
Tn f
n=0
z0 rz0 : D → C
f(z) = f(z0) + f ′ rz (z) 0 (z0)(z − z0) + rz0 (z) lim = 0
z−z0 z0
γn
f = f(z0)
γn
dz + f ′
(z0)
γn
f.
(z − z0)dz +
γn
rz0(z)dz.
1 z − z0
0
rz (z) 0 lim z−z0 z0
= 0 ɛ > 0 δ > 0 0 < |z − z0| < δ
rz 0 (z)
< ɛ |rz0(z)| < ɛ|z−z0| rz0(z0) = 0 |z−z0| < δ
z−z0
|rz0 (z)| ≤ ɛ|z − z0| ɛ > 0 δ > 0
n ∈ N Tn ⊆ S(z0, δ) z ∈ Tn |z − z0| < δ
|rz0 (z)| ≤ ɛ|z − z0| |z − z0| < ln Tn
ran(γn) |rz0 | ≤ ɛln
I
≤
4n
γn
f
=
γn
rz0(z)dz
≤ ɛl 2 n = ɛ l2 0
,
4n I ≤ ɛl 2 0 ɛ > 0 I = 0
D f ∈ O(D) f
D z1, . . . , zn ∈ D f ∈ O(D\{z1, . . . , zn})
k = 1, . . . , n lim f(z)(z − zk) = 0 D\{z1, . . . , zn}
zk
γ
γ
f = 0.

n = 1 z0
D\{z0} D
z0
0
T γ
z0 T
z0 d z0
T γ µ
γ
γ1, γ2, . . . , γk
f 0
µ
γ
f −
µ
f = 0.
lim f(z)(z − z0) = 0 ɛ > 0 δ > 0
z0
0 < |z − z0| < δ |f(z)(z − z0)| < ɛ |f(z)| < ɛ
|z−z0|
d
d µ
f = γ µ f d < δ
S(z0, δ) z d
2 ≤ |z −
z0| ≤ √ 2d
2 < δ |f(z)| < ɛ d =
2
2ɛ
d
f
≤
µ
2ɛ
4d = 8ɛ.
d
ɛ > 0
µ f = 0
γ
f = 0
γ : [a, b] → C γ s ∈ C\ ran(γ) γ
s
n(γ, s) = 1
1
2πi z − s dz.
γ n(γ, s)
s n k
n(γ, s) = n−k s γ : [0, 2π] → C s
dz = 2πi n(γ, s) = 1
γ
1
z−s
n(γ − , s) = −1 γ
γ

γ
1 0
γ : [a, b] → C γ
n(γ, ·) : C\ ran(γ) → Z
C\ ran(γ)
0
D [a, b] γ −→ D f −→ C γ
f ∈ O(D) s ∈ D\ ran(γ)
n(γ, s)f(s) = 1
f(z)
2πi z − s dz.
g : D\{s} → C
f
lim s g(z)(z − s) = lim s
g(z) =
γ
f(z) − f(s)
.
z − s
f(z) − f(s)
(z − s) = lim f(z) − f(s) = 0
z − s
s
g D
s γ
γ
g =
γ
γ
f(z) − f(s)
dz = 0.
z − s
f(s) f(z)
dz =
z − s γ z − s dz.
f(s) 1
2πi
f(z) = cos(z)
z γ
γ
f =
γ
cos(z)
dz = 2πin(γ, 0) cos(0) =
z − 0
γ
1
dz = 2πi.
z
1

D 1 D
γ s /∈ D n(γ, s) = 0
γ1, . . . , γn
D s /∈ D n
n(γk, s) = 0
f ∈ O(D)
n
k=1
γk
f = 0.
D 1 f ∈ O(D) γ D
γ
f = 0
D 1 f ∈ O(D) f
γ1 γ2
γ1, γ2 γ2
γ1 D
f ∈ O(D)
f = f.
γ1
γ1 γ2
γ1 γ2
γ1, γ2 µ ν µ ν γ1
γ10, γ11 γ2 γ20, γ21
γ ⌢ 10ν ⌢ γ −⌢
20 µ− γ ⌢ 11µ ⌢ γ −⌢
21 ν− D 1
f 0
γ ⌢ 10 ν⌢ γ −⌢
20 µ−
γ ⌢ 11 µ⌢ γ −⌢
21 ν−
f =
f =
γ10
γ11
f +
ν
f +
µ
γ2
f +
f +
k=1
γ −
20
γ −
21
f +
f +
µ −
ν −
f = 0,
f = 0.
γ1, γ − 2
f + f = 0
f −
γ2
f = 0
γ1
γ −
2
γ1

γ, γ1, . . . , γn
γ1, . . . , γn γ D
γ γk k = 1, . . . , n
f ∈ O(D)
γ
f =
n
k=1
γ −
γ1, . . . , γn
n
f + f = 0,
γ
n
f =
k=1
γk f
γ −
k=1
γk
γk
f.
f(z) = sh(z) + 1
(z−i) 3 + i γ : [1, 2] → C γ(t) = ln(−t 2 + 3t − 1) + π
t i
f C\{i}
F (z) = ch(z) −
γ
1
+ iz,
2(z − i) 2
1
1
f = ch(γ(2)) −
+ iγ(2) − ch(γ(1)) −
+ iγ(1) =
2(γ(2) − i) 2 2(γ(1) − i) 2
π
ch
2 i
1
−
2( π + iπ
2 i − i)2 2 i
1
− ch(πi) − + iπi =
2(πi − i) 2
π
2 π
1
cos + − − cos(π) − + π = . . .
2 (π − 2) 2 2 2(π − 1) 2
f(z) = 1
z − ez γ : [0, 2π] → C γ(t) = 1 + 2 cos(t) + (3 + sin(t))i
f C\R − 0 1 γ
f = 0
1
0
γ

f(z) = sh(z2 )+e sin(z)
cos(z) γ : [0, π] → C γ(t) = 1 + cos(2t) + (2 + sin(2t))i
1 + 2i 1 cos
f
γ
f = 0
γ : [0, 2π] → C γ(t) = 2+2 cos(t)+i2 sin(t) 2
f(z) = z
z2−1 2 g(z) = z
z+1
γ
f =
γ
z
z+1
z − 1 =
γ
g(z)
1
= 2πin(γ, 1)g(1) = 2πi · 1 · = πi.
z − 1 1 + 1
f(z) = i sin(z2 )+sh(z 2 )
z2−2z+2 γ : [0, 2π] → C γ(t) = 2 cos(t) + i2 sin(t)
0 2 z2 − 2z + 2 =
(z − (1 + i))(z − (1 − i)) 1 + i 1 − i
γ1 γ2 γ 1+i 1−i
γ
f = f + γ γ1 γ2 f
g1(z) = i sin(z2 )+sh(z 2 )
z−(1−i) g2(z) = i sin(z2 )+sh(z 2 )
z−(1+i) gi
γi
γ
f =
γ1
f +
γ2
f =
γ1
i sin(z 2 )+sh(z 2 )
z−(1−i)
z − (1 + i) +
γ2
i sin(z 2 )+sh(z 2 )
z−(1+i)
z − (1 − i) =
g1(z)
γ1 z − (1 + i) +
g2(z)
γ2 z − (1 − i) = 2πin(γ1, 1+i)g1(1+i)+2πin(γ2, 1−i)g2(1−i).
i sin(z) = sh(iz) sh(−z) = − sh(z)
2 2 sh(i(1 + i) ) + sh((1 + i) )
2πi
(1 + i) − (1 − i)
sh(−2) + sh(2i)
2πi
+
2i
sh(2) + sh(−2i)
+ sh(i(1 − i)2 ) + sh((1 − i) 2 )
=
(1 − i) − (1 + i)
=
−2i
π(sh(−2) + sh(2i) − sh(2) − sh(−2i)) = 2π(sh(2i) − sh(2)).
γ
f
γ [1 + i, 2 − i, 2 + i] f(z) =
3z 2 + 2z
γ |z − 2 + i| = 1 f(z) = sh(z)
z+i
γ i 3
sin(z)
2 f(z) = z2 +1
γ − 1
2
+ i
2
1 i − 2 − 2
f(z) = e2z
z 3 −1
2

γ |z + 1| = 4
3
f(z) =
1
z(z2−1)
γ 0 3
f(z) = 1
z 4 −1
D ⊆ C u : D → R
u v : D → R
u + iv ∈ O(D) v u
u + iv ∈ O(D)
u + iv ∈ O(D) ⇐⇒ −v + iu ∈ O(D).
u v f = u + iv ∈ O(D)
f D f ′ = u ′ x+iv ′ x =
v ′ y + i(−u ′ y) u v
u v D
f ′ = u ′ x + iv ′ x
u ′′
xx = (u ′ x) ′ x = (v ′ x) ′ y = v ′′
xy f ′ = v ′ y + i(−u ′ y)
v ′′
yx = (v ′ y) ′ x = (−u ′ y) ′ y = −u ′′
yy
u ′′
xx + u ′′
yy ≡ 0.
v ′′
xx + v ′′
yy ≡ 0
u
∆u = u ′′
xx + u ′′
yy.
D ⊆ R 2 u : D → R
∆u ≡ 0
f = u + vi : D → C f ∈ O(D) u v
u : D = R 2 \{(0, 0)} → R u(x, y) =
ln(x 2 + y 2 ) u
∆u ≡ 0 u
v : C\{0} → R u + iv ∈ O(C\{0})
g : R → R g(t) = v(cos(t), sin(t)) g
g(0) =
g(2π) ξ ∈ (0, 2π) g ′ (ξ) = 0
t ∈ R
g ′ (t) = (v ′ x(cos(t), sin(t)), v ′ y(cos(t), sin(t)))(− sin(t), cos(t)),

(−u ′ y(cos(t), sin(t)), u ′ x(cos(t), sin(t)))(− sin(t), cos(t)) =
2 sin 2 (t)
sin 2 (t) + cos 2 (t) +
2 cos2 (t)
sin 2 (t) + cos2 = 2,
(t)
D ⊆ C u : D → R
D
u
g = u ′ x+i(−u ′ y) D
u ′′
xx = −u ′′
yy u ′′
xy = u ′′
yx = −(−u ′′
yx)
u ′ x −u ′ y D g D
D
g f0 = U + iV g
g = f ′ 0 = U ′ x + i(−U ′ y) = u ′ x + i(−u ′ y) U ′ x = u ′ x
−U ′ y = −u ′ y U u
c U = u + c
f = f0 − c
u
1
D ⊆ C 1 u : D → R
D
u(x, y) = x 3 − 3xy 2 + 2y D = C
u ∆u ≡ 0
u v : C → R u + iv ∈ O(C)
v ′ y(x, y) = u ′ x(x, y) = 3x 2 − 3y 2
v(x, y) = 3x 2 − 3y 2 dy = 3x 2 y − y 3 + c(x).
−v ′ x(x, y) = u ′ y(x, y) = −6xy + 2
−(6xy − 0 + c ′ (x)) = −6xy + 2.
c ′ (x) = −2 c(x) = −2x + c v(x, y) = 3x 2 y − y 3 − 2x + c

u(x, y) = e x cos(y) D = C
u ∆u ≡ 0 u
v : C → R u + iv ∈ O(C)
v ′ y(x, y) = u ′ x(x, y) = e x cos(y)
v(x, y) =
e x cos(y)dy = e x sin(y) + c(x).
−v ′ x(x, y) = u ′ y(x, y) = −e x sin(y)
−(e x sin(y) + c ′ (x)) = −e x sin(y).
c ′ (x) = 0 c(x) = c v(x, y) = e x sin(y) + c
u(x, y) = x 2 − y 2 + xy u(x, y) = x
x 2 +y 2
v(x, y) = 2y(x + 1) v(x, y) = arctg y
x

z0 ∈ C f S(z0, R)
f(z) =
∞
an(z − z0) n
n=0
γ z0 r < R
n
an = 1
f(ξ)
dξ.
2πi γ (ξ − z0) n+1
z ∈ S(z0, r) n(γ, z) = 1
z
f(z) = 1
2πi
|ξ − z0| = r z ∈ S(z0, r) z−z0
1
ξ − z =
1
ξ − z0 − (z − z0) =
z ∈ S(z0, r) z−z0
n=0
ξ−z0
γ
ξ−z0
f(ξ)
ξ − z dξ.
< 1
1
(ξ − z0) 1 − z−z0
ξ−z0
= |z−z0|
r
= q < r
n=0
=
∞
n=0
∞
n
(z − z0)
(ξ − z0) n+1
= 1
∞
q
r
n 1
=
r(1 − q) ,
(z − z0) n
.
(ξ − z0) n+1
ξ
ξ
∞
n=0
(z − z0) n
−
(ξ − z0) n+1
∞
n=N+1
N (z − z0) n
(ξ − z0) n+1
=
n=0
n
(z − z0)
(ξ − z0) n+1
= 1
r
∞
n=N+1
∞
n=N+1
(z − z0) n
(ξ − z0) n+1
≤
q n = qN+1
r(1 − q) .
ξ z ∈ S(z0, r)
f(z) = 1
∞
f(ξ) 1
dξ = f(ξ)
2πi γ ξ − z 2πi γ
n=0
(z − z0) n
dξ =
(ξ − z0) n+1

∞
n=0
1
2πi γ
f(ξ)
dξ (z − z0)
(ξ − z0) n+1 n .
r
S(z0, r)
r
f ∈ O(D) f
D
D z0 f
f
f s
γ s
n
f (n) (s) = n!
2πi
γ
f(z)
dz.
(z − s) n+1
D 1 f ∈ O(D)
γ D s ∈ D\ ran(γ)
n(γ, s)f (n) (s) = n!
2πi
γ
f(z)
dz.
(z − s) n+1
f(z) = eπz
(z2 +1) 2 γ |z − i| = 3
2
z2 + 1 = (z − i) 2 (z + i) 2 i f(z) = g(z)
(z−i) 2
g(z) = eπz
(z+i) 2 g ′ (z) = e πz
(z+i) 2
′ e
= πz π(z+i) 2 −e πz 2(z+i)
(z+i) 4 = eπz π(z+i)−2
(z+i) 3
γ
f =
γ
g(z) 2πi
dz =
(z − i) 2 2! g′ πi −2 + 2πi
(i) = πie = −
−8i
π π2
+
4 4 i.
γ
f
γ 0 2 f(z) =
sh(z)
z 2 −2iz−1
γ |z − 2| = 3 f(z) = ez
z 4 −z 3
γ |z + 1| = 3
cos(z)
(z2−z)(z2−2iz−1)
2
f(z) =

z0 ∈ C an ∈ C n ∈ Z
∞
n=−∞
an(z − z0) n
z0
∞
an(z−z0)
n=−∞
n z
−1
n=−∞
an(z − z0) n =
∞
n=1
a−n
1
(z − z0) n
∞
an(z − z0)
n=0
n z
z
∞
an(z − z0) n
n=−∞
R1 = lim sup n |a−n| , R2 =
1
lim sup n |an| ,
{z ∈ C : R1 < |z − z0| < R2}
R1 < r1 ≤ r2 < R2
{z ∈ C : r1 ≤ |z − z0| ≤ r2}
H(w) = ∞
∞
n=−∞
n=1
H
a−nwn h(z) = ∞
an(z − z0) n
n=0
an(z − z0) n
1
= H + h(z).
z − z0
1
rH =
lim sup n√ a−n
= 1
.
R1
H S 0, 1
R1
1
z−z0 ∈ S0, 1
R1
1 < z−z0
1
R1 |z − z0| > R1 H
1
z−z0
{z ∈ C : |z − z0| > R1} S(z0, R1)
h
1
rh =
lim sup n√ = R2.
an

h S(z0, R2)
{z ∈ C : |z − z0| > R1} ∩ S(z0, R2) = {z ∈ C : R1 < |z − z0| < R2}
f z0 ∈ C {z ∈ C : R1 < |z − z0| < R2}
f(z) =
∞
n=−∞
an(z − z0) n ,
γ z0 R1 < r < R2
n
an = 1
f(ξ)
dξ.
2πi (ξ − z0) n+1
γ
f S(z0, R)
0
n < 0 γ z0 r < R
f(z)
= f(z)(z − z0)
γ (z − z0) n+1
γ
−n−1 = 0,
f(z)(z − z0) −n−1 S(z0, R) −n − 1 ≥ 0
f, g ∈ O(D) a ∈ D n > 0
k = 0, 1, . . . , n−1 f (k) (a) = g (k) (a) = 0 g (n) (a) = 0
lim a
f
g = f (n) (a)
g (n) (a) .
f g a
g
lim a
f(z) = f (n) (a)
(z − a)
n!
n + f (n+1) (a)
(n + 1)! (z − a)n+1 + . . .
f(z)
g(z) = lim a
f (n) (a)
n! (z − a) n + f (n+1) (a)
(n+1)! (z − a)n+1 + . . .
g (n) (a)
n! (z − a) n + g(n+1) (a)
(n+1)! (z − a)n+1 + . . . =

lim a
f (n) (a)
n!
g (n) (a)
n!
+ f (n+1) (a)
(n+1)! (z − a) + . . .
+ g(n+1) .
(a)
(n+1)! (z − a) + . . .
f g a
a f (n) (a)
n!
g(n) (a)
n! = 0
1
z 2 −z
f(z) =
f
D b 0 = {z ∈ C : 0 < |z − 0| < 1},
D k 0 = {z ∈ C : 1 < |z − 0| < ∞}
0
D b 1 = {z ∈ C : 0 < |z − 1| < 1},
D k 1 = {z ∈ C : 1 < |z − 1| < ∞}
1 f
C\{0, 1}
D b 0 f(z) = 1
z
1
z−1
1
z
D k 0 |z| > 1 1
D b 1 f(z) = 1
z
0 |z| < 1
1 −1
=
z − 1 1 − z
= −
∞
z n ,
n=0
f(z) = − 1
∞
z
z
n=0
n = −1
z +
∞
−z
n=0
n .
< 1
f(z) = 1 1
z z
z
1 1
=
z − 1 z
1
1 − 1
z
∞
n=0
1
z−1 1
z−1
1
z =
1
1 − (1 − z) =
1
z−1
z
=
1
n =
z
∞
n=0
= 1
z
∞
n=2
1
1 − 1
z
1
n ,
z
1
n =
z
−2
n=−∞
z n .
1 |1 − z| < 1
∞
(1 − z) n ∞
=
n=0
n=0
(−1) n (z − 1) n ,

f(z) = 1
z − 1
∞
n=0
D k 1 |z − 1| > 1 1
1 1
=
z z − 1
1
1 − −1
z−1
(−1) n (z − 1) n = 1
=
f(z) = 1 1
z − 1 z − 1
1 z
z−1
∞
n=0
z−1
z − 1 +
< 1
= 1
z − 1
1
1 + 1
z−1
−1
n =
z − 1
0
n=−∞
0
n=−∞
(−1) n (z − 1) n =
f(z) = (z+1)2
z
∞
n=0
= 1
z − 1
(−1) n+1 (z − 1) n .
1
1 − −1
z−1
(−1) n (z − 1) n ,
−2
n=−∞
0
f(z) = z2 + 2z + 1
z
= 1
+ 2 + z.
z
f(z) = e 1
z−i i
f(z) =
∞
n=0
1
n z−i
n! =
∞
n=0
1
=
n!(z − i) n
−1
n=−∞
(−1) n (z − 1) n .
1
(−n)! (z − i)n + 1.
z
(z+1)(z+2)
e 2z
(z−1) 3
1−cos(z)
z
z
(z2 +1)
z
(z+1) 3
a ∈ C R > 0 f S(a, R) |f| ≤ M
n ∈ N
|f (n) (a)| ≤ n!M
.
Rn

γ a r < R
f (n) (a) = n!
2πi
γ
f(z)
dz.
(z − a) n+1
|f (n) (a)| ≤ n!
2π
M n!M
2rπ =
rn+1 rn r < R
|f| ≤ M f ′ ≡ 0
f
a R > 0 |f ′ (a)| ≤ M
R f ′ (a) = 0 a
p(z) = anz n +· · ·+a1z +a0 n > 0 an = 0
z0 ∈ C p(z0) = 0
p
lim ∞ p = ∞ p f(z) = 1
p(z)
lim ∞ f = 0 R > 0 z |z| > R
|f(z)| < 1 f f B(0, R)
f
f ∈ O(D)
f ≡ 0
z0 ∈ D n ∈ N f (n) (z0) = 0
{z ∈ D : f(z) = 0} D
⇒ z0 ∈ D f R > 0
S(z0, R) ⊆ D f f(z0) = 0
n > 0 f(z0) = f ′ (z0) = · · · = f (n−1) (z0) = 0 f (n) (z0) = 0
f S(z0, R)
f(z) =
g(z) =
∞
ak(z − z0) k .
k=n
∞
ak(z − z0) k−n .
k=n
g S(z0, R) f(z) = (z − z0) n g(z) g(z0) = an = 0 g
0 < r < R z ∈ S(z0, r) g(z) = 0

z0 f a ∈ S(z0, r) a = z0
f(a) = 0 g(a) = f(a)
(a−z0) n = 0
⇒ A = {z ∈ D : ∀ n ≥ 0 f (n) (z) = 0}
A D\A D A = ∅
D\A = ∅ D = A
A z0 ∈ A R > 0 S(z0, R) ⊆ D
f
f(z) =
∞
n=0
f (n) (z0)
(z − z0)
n!
n = 0,
f 0 S(z0, R) ⊆ A
D\A D\A
w ∈ D\A ɛ > 0 S(w, ɛ) D\A R > 0
S(w, R) ⊆ D 0 < ɛ < R S(w, ɛ) ∩ A = ∅
ak ∈ A lim ak = w f (n) n
n 0 = lim f (n) (ak) = f (n) (w) w ∈ A
sin(2z) = 2 sin(z) cos(z) f(z) = sin(2z) −
2 sin(z) cos(z) f 0
0
f ≡ 0
f ∈ O(D) z0 /∈ D
R > 0 S(z0, R)\{z0} ⊆ D
f ∈ O(D) z0 /∈ D f
f(z) = ∞
n=−∞
an(z − z0) n f z0 z0
• f n > 0 a−n = 0
• n f n > 0 a−n = 0
m > n a−m = 0
• f n > 0 a−n = 0
f(z) = 1
sin 1 0 0
z
f
f(z) = ez −1
z 0
e z − 1
z
1
=
z
∞
n=1
z n
n! =
∞
n=0
z n
(n + 1)! .

f(z) = ez −1
z 2 0
e z − 1
z 2
1
=
z2 ∞
n=1
z n
n!
= 1
z +
∞
n=0
z n
(n + 2)! .
f(z) = e 1
z 0 e 1
z = −1
n=−∞
1
(−n)! zn + 1
f ∈ O(D) z0 /∈ D f f(z) =
∞
n=−∞
an(z − z0) n f z0
z0 f ⇐⇒ lim f(z)(z − z0) = 0
z0
z0 n f ⇐⇒
lim(z
− z0)
z0
n+1f(z) = 0 lim(z
− z0)
z0
nf(z) = 0
z0 f ⇐⇒ lim f = ∞
z0
z0 f ⇐⇒ ∄ lim f
z0
ch(z)
z 4
1
(sin(z)−1) 2
1
z(z−1) 3
z
e 1 z −1
z0 f
w ∈ C ∪ {∞} zn → z0 (zn = z0)
f(zn) → w
w ∈ C w ∈
C ɛ, δ > 0
= ∞
z 0 < |z − z0| < δ |f(z) − w| ≥ ɛ lim
z0
f(z)−w
z−z0

f(z)−w
z−z0 z0
f(z)−w
n lim z−z0 z0
(z − z0) n+1 = 0
|f(z)(z − z0) n | ≤ |(f(z) − w)(z − z0) n | + |w(z − z0) n |
lim f(z)(z − z0)
z0
n = 0 f n − 1
z0
z0 f
δ > 0 f(S(z0, δ))
C
z0 f f(z) = ∞
n=−∞
f z0 f z0
Resz0 f = a−1.
an(z − z0) n
D z1, . . . , zm ∈ D
f ∈ O(D\{z1, . . . , zm}) γ1, . . . , γn D\{z1, . . . , zm}
s /∈ D n
n(γk, s) = 0
n
k=1
γk
f = 2πi
l=1
k=1
k=1
m n
n(γk, zl) Reszl f.
γ D
γ z1, . . . , zm γ
f ∈ O(D\{z1, . . . , zm})
γ
f = 2πi
m
Reszl f.
l = 1, . . . , m γl zl γ
γ
γ1, . . . , γm
m
f(z)
f = f l f = γ γl γl γl (z−zl)
l=1
0 dz =
2πi Reszl f
l=1

f(z) = 1
z 2 −z
γ
1 γ 2 + i 2
f = 2πi(Res0 f + Res1 f).
f S(0, 1)\{0} S(1, 1)\{1} f
0 1
f = 2πi(−1+1) =
γ
0
z0 f Resz0 f = 0
z0 n f
Resz0 f =
lim
z0
d n−1
dzn−1 [(z − z0) nf(z)] .
(n − 1)!
f
z0 n z0 f(z) = ∞
ak(z − z0)
k=−n
k
a−n = 0
(z − z0) n f(z) =
∞
k=−n
ak(z − z0) k+n ,
n − 1
∞
ak(n + k)(n + k − 1) · · · (k + 2)(z − z0) k+1 .
d n−1
dz n−1 [(z − z0) n f(z)] =
k=−1
dn−1
dz n−1 [(z − z0) n f(z)] z0
z0 0
a−1(n − 1)!
f(z) = ze 1
z γ 1 0
f 0
e 1
z = −1
Res0 f = 1
2
n=−∞
1
(−n)! zn + 1 f(z) = −1
γ
f = 2πi 1
= πi.
2
n=−∞
1
(−n+1)! zn + 1 + z

f(z) =
1
z(1−e z )
γ 1 0
f 0
0 f
lim 0 z 2 f(z) = lim 0
z 1
= −
1 − ez exp ′ = −1,
(0)
lim 0 z 3 f(z) = 0
Res0 f = lim 0
d
dz [z2 f(z)] = lim 0
−e z + e z + ze z
2(1 − e z )(−e z ) = lim 0
(1 − e z ) − z(−e z )
(1 − e z ) 2
z
lim
0
2(ez − 1) = lim 1 1
=
0 2ez 2 ,
3. 5.
1 f = 2πi 2 = πi
γ
γ
f
γ i 2 f(z) = ez
z 2 −1
γ 0
f(z) =
1
(z−1) 2 (z2 +1)
γ |z| = 5 f(z) = z+1
sin(iz)
γ 0 f(z) = sin 1
z
∞
−∞
x 2
1 + x 4 dx = π √ 2 .
f(z) = z2
1+z 4 −1
zk = e
π kπ i( 4 + 2 ) = cos
π
4
kπ
π
+ + i sin
2 4
=
kπ
+ ,
2
k = 0, 1, 2, 3
f
lim(z
− z0)
z0
2 f(z) = lim(z
− z0)
= 0,
z0 (z − z1)(z − z2)(z − z3)
lim(z
− z0)f(z) = lim
z0
z0
z 2
z 2
(z − z1)(z − z2)(z − z3) =

z2 0
1 − i
=
(z0 − z1)(z0 − z2)(z0 − z3) 4 √ = 0,
2
z0
f z0 z1
Resz0
1 − i
f = lim(z
− z0)f(z) =
z0
4 √ 2
−1 − i
Resz1 f = lim(z
− z1)f(z) =
z1
4 √ 2 .
R > 1 γR R
γR z0 z1
f = 2πi Resz0 f + Resz1 f = π √ .
2
γR
µR [−R, R] νR
f = f
γR
µR
µR
R
f =
f +
νR
−R
x 2
1+x 4 > 0
∞
−∞
x2 R
dx = lim
1 + x4 R→∞ −R
lim νR R→∞
|1+z 4 | ≥ R4 −1 z 2
νR
0 R → ∞
∞
x2 dx
1 + x4 x2
π√2
dx = lim −
1 + x4 R→∞
f = 0
1+z 4
≤ R 2
R 4 −1
f
≤ R2
R4 − 1 Rπ = R3π R4 − 1 ,
0
sin(x) π
dx =
x 2 .
νR
f ,
f(z) = eiz
z 0 0 < r < R
γr,R r R 0
γ f
1
f = 0
γr,R
µr νR
γr,R
R
f =
r
eix
dx +
x νR
eiz −r
dz +
z −R
eix
e
dx +
x µr
iz
z dz.

sin(x) = 1
2i (eix − e −ix )
R
r
R
sin(x) 1
dx =
x 2i r
e ix − e −ix
x = −t − R
lim
R→∞
∞
e iz
z
lim
R→∞
0
lim
r→0,R→∞
νR
lim
r→0,R→∞
νR
e iz
z
x
r
dx = 1
R
2i r
e −ix
x dx = − −R
−r
e
µr
iz
z
dz = 0 lim
r→0
R
sin(x)
dx = lim
x r→0,R→∞ r
1
2i
f −
γr,R νR
1
2i
−
νR
eiz dz −
z
eiz dz −
z
µr
eix −r
1 e
dx +
x 2i −R
ix
x dx.
e −i(−t)
−t (−1)dt = −r e
−R
it
t dt
dz = −πi
sin(x)
dx =
x
µr
eiz z dz
eiz z dz
=
= π
2 .
e iz
z dz = 0 z = x + iy R
= e −y
R ɛ > 0 νR
ν1 R = νR ↾ [0, ɛ] ν2 R = νR ↾ [ɛ, π − ɛ] ν3 R = νR ↾ [π − ɛ, π]
ν1 R ν3 R e iz
1
z ≤ R ν2 R
e iz −R sin(ɛ)
e
z ≤ R
R0 R > R0 e−R sin(ɛ) < ɛ
π
R > R0
µr
lim
r→0
νR
eiz z dz
≤
ν 1 R
eiz z dz
+
ν 2 R
eiz z dz
+
ν 3 R
eiz z dz
≤
sin(ɛ)
1 e−R
ɛR + (π − 2ɛ)R +
R R
1
R ɛR < 2ɛ + πe−R sin(ɛ) < 3ɛ.
e
µr
iz
z dz = −πi eiz−1 z 0
S(0, 1) e iz
−1
z ≤ M r < 1
e iz −1
z dz ≤ Mπr
0 = lim
r→0
µr
µr
eiz
− 1
dz = lim
z
r→0
µr
e iz
z dz → −πi r → 0
eiz
dz − lim
z r→0
µr
1
dz = lim
z r→0
µr
∞
0
ln(x)
dx = 0,
1 + x2 ∞
0
ln(x)
(1 + x2 dx = −π
) 2 4 ,
∞
0
eiz dz − (−πi),
z
cos(x)
(1 + x2 π
dx =
) 2 2e .

N Z R
A B A × B
A × B = {(a, b) : a ∈ A, b ∈ B}.
A, B A B f f :
A → B A f −→ B f dom(f) =
A f
ran(f) = {b ∈ B : ∃ a ∈ A f(a) = b}
f : A → B b ∈ B
a ∈ A f(a) = b f ≡ b
f : A → B X ⊆ A f(X) X
f
f(X) = {b ∈ B : ∃ x ∈ X f(x) = b}
f : A → B A ′ ⊆ A f A ′
f ↾ A ′ : A ′ → B
a ∈ A ′ (f ↾ A ′ )(a) = f(a)
f : A → B
• A f B
∀ a1, a2 ∈ A (a1 = a2 ⇒ f(a1) = f(a2))
• B A
ran(f) = B
∀ b ∈ B ∃ a ∈ A f(a) = b
•
A B
f : A → B
f −1 : ran(f) → A
b ∈ ran(f) f −1 (b) a ∈ A f(a) = b
f −1 (f(a)) = a
g : A → B f : B → C f g
f ◦ g : A → C (f ◦ g)(a) = f(g(a))

T ⊕ ⊙
n, e ∈ T 〈T, ⊕, ⊙, n, e〉
∀ a, b ∈ T a ⊕ b = b ⊕ a
∀ a, b, c ∈ T a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c
a ⊕ n = a n
∀ a ∈ T ∃ b ∈ T a ⊕ b = n
∀ a, b ∈ T a ⊙ b = b ⊙ a
∀ a, b, c ∈ T a ⊙ (b ⊙ c) = (a ⊙ b) ⊙ c
a ⊙ e = a e
∀ a ∈ T \{n} ∃ b ∈ T a ⊙ b = e
∀ a, b, c ∈ T a ⊙ (b ⊕ c) = (a ⊙ b) ⊕ (a ⊙ c)
0 1
〈R, +, ·, 0, 1〉.
Q
{n, e}
n ⊕ n = e ⊕ e = n n ⊕ e = e ⊕ n = e n ⊙ n = n ⊙ e = e ⊙ n = n
e ⊙ e = e
Q( √ 2) {a + b √ 2 : a, b ∈ Q}
C
〈C, +, ·, (0, 0), (1, 0)〉.
(a1, b1) · (a2, b2) = (a1a2 − b1b2, a1b2 + b1a2) (a2, b2) · (a1, b1) =
(a2a1 − b2b1, a2b1 + b2a1) (a1, b1) · (a2, b2) = (a2, b2) · (a1, b1)
(a1, b1) · ((a2, b2) · (a3, b3)) = (a1, b1) · (a2a3 − b2b3, a2b3 + b2a3) =
(a1(a2a3 − b2b3) − b1(a2b3 + b2a3), a1(a2b3 + b2a3) + b1(a2a3 − b2b3)) =
(a1a2a3 − a1b2b3 − b1a2b3 − b1b2a3, a1a2b3 + a1b2a3 + b1a2a3 − b1b2b3).

((a1, b1) · (a2, b2)) · (a3, b3) = (a1a2 − b1b2, a1b2 + b1a2) · (a3, b3) =
((a1a2 − b1b2)a3 − (a1b2 + b1a2)b3, (a1a2 − b1b2)b3 + (a1b2 + b1a2)a3) =
(a1a2a3 − b1b2a3 − a1b2b3 − b1a2b3, a1a2b3 − b1b2b3 + a1b2a3 + b1a2a3).
(a, b) · (1, 0) = (a1 − b0, a0 + b1) = (a, b)
(a, b) = (0, 0)
a
(a, b) ·
a2 −b
,
+ b2 a2 + b2 2 a
=
a2 −b2
−
+ b2 a2 −ab
,
+ b2 a2 ab
+
+ b2 a2 + b2
= (1, 0).
(a1, b1) · ((a2, b2) + (a3, b3)) = (a1, b1) · (a2 + a3, b2 + b3) =
(a1a2 + a1a3 − b1b2 − b1b3, a1b2 + a1b3 + b1a2 + b1a3) =
(a1a2 − b1b2, a1b2 + b1a2) + (a1a3 − b1b3, a1b3 + b1a3) =
((a1, b1) · (a2, b2)) + ((a1, b1) · (a3, b3)).
{(a, 0) : a ∈ R} ⊆ C (0, 0) (1, 0)
a, a ′ ∈ R
(a, 0) + (a ′ , 0) = (a + a ′ , 0) (a, 0) · (a ′ , 0) = (aa ′ , 0)
Q Q( √ 2)
C
ϕ : R → C ϕ(a) = (a, 0)
a, a ′ ∈ R ϕ(a + a ′ ) = ϕ(a) + ϕ(a ′ ) ϕ(aa ′ ) = ϕ(a) · ϕ(a ′ )
ϕ
R ⊆ C (a, 0)
a
R 2
a = (x, y) ∈ R 2
a = x 2 + y 2 .
. : R 2 → R λ ∈ R a, b ∈ R 2
a ≥ 0 a = 0 ⇐⇒ a = 0 = (0, 0)
λa = |λ|a
a + b ≤ a + b ()

a = (x1, y1) b = (x2, y2) ∈ R 2
d(a, b) = a − b = (x1 − x2) + (y1 − y2) 2 .
a ∈ R 2 0
d : R 2 × R 2 → R a, b, c ∈ R 2
d(a, b) ≥ 0 d(a, b) = 0 ⇐⇒ a = b
d(a, b) = d(b, a)
d(a, c) ≤ d(a, b) + d(b, c) ()
a ∈ R 2 ɛ > 0 a ɛ
S(a, ɛ) = {p ∈ R 2 : d(a, p) < ɛ},
a ɛ
B(a, ɛ) = {p ∈ R 2 : d(a, p) ≤ ɛ}.
U ⊆ R 2 a ∈ U ɛ > 0
S(a, ɛ) ⊆ U
∅ R 2
a ∈ R 2 ɛ > 0 S(a, ɛ)
S = {a}
∞
n=1
a, 1
n
H ⊆ R 2 a ∈ R 2
ɛ > 0 S(a, ɛ) ∩ H H
H ′
F ⊆ R 2
F ′ ⊆ F
F ⊆ R 2 R 2 \F

B(z0, ɛ)
S(z0, ɛ)
∆u
arg(w)
ch(z)
cos(z)
dom(f)
exp(z)
γ −
γ ⌢ 1 γ2
Im(a + bi)
∞
f γ
log(w)
C
O(D)
ran(f)
Re(a + bi)
Resz0 f
sh(z)
sin(z)
a + bi
ez
f ↾ A ′
n(γ, s)
wz

B(z0, ɛ)
S(z0, ɛ)
∆u
arg(w)
ch(z)
cos(z)
dom(f)
exp(z)
γ −
γ ⌢ 1 γ2
Im(a + bi)
∞
f γ
log(w)
C
O(D)
ran(f)
Re(a + bi)
Resz0 f
sh(z)
sin(z)
a + bi
ez
f ↾ A ′
n(γ, s)
wz