Esercizi svolti di analisi reale e complessa - Dipartimento di ...

Rn
Rn
Rn Rm
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Rn Rm
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Rn

Rn
Rn
Rn Rm
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Rn Rm
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Rn
Rn
Rn Rm
. . .
Rn
Rn
Rn
Rn
Rn
Rn Rm
. . .
Rn
Rn
Rn

R n
R n
. : C([0, 1], R) → R f ↦→ f ≡
1
|f(x)| dx
0
C([0, 1], R)
.∞
x + y∞ < x∞ + y∞
A = {x ∈ R 2 : 1 < x < 2}
E ⊂ Rn
C(E, Rm )
(E, R m ⎧
⎨
) ≡
⎩ f ∈ C(E, Rm ⎫
|f(x) − f(y)|
⎬
) : fLip ≡ sup
+ sup |f(x)| < ∞
x,y∈E |x − y| x∈E
⎭
x=y
.
((E, Rm ), · Lip)

ℓ 1 ℓ ∞
x = {xn}n ∈ R N
x1 ≡
n∈N
R N
|xn| e x∞ ≡ sup |xn| .
n∈N
ℓ 1 ≡ {x ∈ R N : x1 < ∞}
ℓ ∞ ≡ {x ∈ R N : x∞ < ∞} .
1 (ℓ 1 , · 1) (ℓ ∞ , · ∞)
2 ℓ 1 ⊂ ℓ ∞ (ℓ ∞ , · ∞)
(ℓ 1 , · ∞)
3 ∗∗ ℓ 1 · ∞
(ℓ 1 , · 1)
Ω := {x ∈ ℓ 1 : x1 ≤ 1}
D := {x ∈ ℓ 1 : |xk| ≤ 1 ∀ k, xk = 0 ∀ k > 10} .
R n R m
. . .
f(x) = xi
x
ɛ > 0 δ |f(x) − f(x0)| < ɛ
|x − x0| < δ
f = |x| α , α > 0 x ∈ R 4 , x0 = (0, 1, 1, 2)
x0 = (0, .., 0)
f = sin 1
x1x2 , x ∈ R
3
3 , x0 = (−1, 0, −1)

f = log[cos ( n
i=1 xi)] , x ∈ R n , x0 = (0, .., 0)
f = +∞
k=0 e−k|x|2
, x ∈ R n , x0 = (1, .., 1)
f = tanh |x|1 , x ∈ R n , x0 = (1, .., 1)
f = ( |x| 3
2 , tanh |x|1 ) , x ∈ R 4 , x0 = (0, 1, 1, 2)
S 2 ≡ { x ∈ R 3 : |x| = 1 } , x ≡ (2, 0, 0) , x0 = (1, 0, 0) ,
f ≡ x1x2(sin |x − x|) −1 .
δ |f(x) − f(x0)| < ɛ x ∈ S 2 |x − x0| < δ
f : R 4 −→ R 2
x ↦−→
1
(f1(x), f2(x)) ≡ , sin(x1x4) .
1 + |x|
x0 = (0, 0, 0, 0)
f : E ⊂ R n −→ R m
i = 1, . . . , m
fi : E ⊂ R n −→ R
L > 0
|f(x) − f(y)| ≤ L|x − y| ∀ x, y ∈ Ω
x ∈ Rn f | · |
(i)
1
f(x) =
2 − |x| ,
sin
n
xi
,
Ω = B1(0) ;
(ii) f(x) =
1
2 − |x| 1
2
i=1
, Ω = B1(x0), x0 = (2, . . . , 2)
oppure x0 = (0, . . . , 0) (per il primo dominio
dobbiamo supporre che n = 3, 4, 5, 6 );
(iii) f(x) = e |x|2
x , Ω = Br(0), r > 0 .

P ≡ {(x, y) ∈ R 2 y = x 2 (x, y) = (0, 0)}
f(x, y) =
0 (x, y) ∈ P
1 (x, y) ∈ P
f ∂f
∂ξ
ξ = 0
∂|x|α
∂xi
, ∀ x ∈ R n \ {0} , e ∀ α ∈ R
(0) = 0
f : R 2 → R , α, β > 0 :
α
|x1x2|
f(x) = |x| β x = 0
0 x = 0
f
f
f
f C 1 ({0})
f : A ⊂ R n → R m A
f ∈ C 1 ({x0}, R n ) x0 ∈ A f
x0
∂ f
∂ x
f : R 2 −→ R 2
(x, y) ↦−→ (sin(xy), e xy2
) .
∂ f
(x, y) (x, y)
∂ y
f (0, 0)

g : R −→ R
t ↦−→ tgh t + 1 + t 2
F (t) ≡ f(g(t), 1 − g 2 (t)) F ′ (0)
f ∈ C 1 (R 3 , R) h ∈ C 1 (R 2 , R)
∂
f(x, h(x, z), z) e
∂ x
∂
f(x, h(x, z), z) .
∂ z
y = f(x) C 2 x = 0
x 2 + sinh y + e xy = 1
f(0) = 0
x ∈ R2 y ∈ R3 f : R5 →
R6 f ∈ C1 ∂f
∂x ∂f
∂y
g : R → R3 C1 ∂
∂t f(x, g(t) )
f(x, y, t) = (sin(tx1), |x|, (y1+y2 2x3) 1+t )
x ∈ R3 ,
y ∈ R2 t ∈ R .
∂f ∂f
∂x , ∂y ∂f
∂t x = 0 t = −1
g(t) = (tanh t, ln[ln t] ), ∂
∂t [f(x, g(t), t)] x = 0
t > 1
f(x) = e |x|2
D1f(0) (ξ) D3f(0) (1, 2, .., n) 3
(x1 + x 4 n ) x ∈ R n

∂ 5 f
(1, 1, .., 1) n > 5
∂x1 ∂x2 ...∂x5
∂ (1,0,..,10)
x f(x0) x0 = 0 x0 = (−1, 1, −1, .., (−1) n )
∂f
∂x (0)
f(x, y) =
e (x+y) x = y
1 + (x + y) + (x+y)2
2
x = y
f R 2
ɛ > 0 δ > 0 |f(x, y)−f(0, 0)| < ɛ |(x, y)| <
δ
f (0, 0)
k f ∈ C k ({(0, 0)})
k f (0, 0)
x = 0
x1
1−x2x3x4
f(x) = x3e x1+x2
x = 0
δ |f(x)| < 1
4 |x| < δ
sin |x|
f(x) = |x| x = 0
1 x = 0
f ∈ C1 (Rn )
f(x, y) =
f
x 2 + y 2 x = y
4y 2 x = y

f(x, y) = (x + 3y)e −xy
f(x, y) = x 2 y
D ≡ {x 2 + y 2 ≤ 1}
∂
∂xi g(|x|) x ∈ Rn \ {0} g
C 1 ((0, +∞))
log (s + t)
(s0, t0) = (1, 0)
log (x − y 2 )
(x0, y0) = (1, 0)
f(x, y) ≡
ye
0 x = 1
y
−( x−1 )2 x = 1
f ∈ C∞ (R2 \ {(1, 0)})
f (1, 0)
f
δ > 0 |f(x, y) − f(x0, y0)| < 1
100 |(x, y) − (x0, y0)| < δ
(x0, y0) = (1, 1) f (1, 0) (x0, y0) = (1, 0)
∂f
∂x
f(x) = n
i=1 x2 i
f(x) = (x1 + x 2 2 , cos (x1x2)) x ∈ R n n ≥ 2
f(x, y, z) = y
z−x 2
∂f
∂x
N x0 = 0
|x| sin |x| (x ∈ R n )

z = z(x, y)
(1, 1)
z 3 − 2xy + y = 0 z(1, 1) = 1.
(1, 1)
z = z(x, y)
z = z(x, y)
(1, 1) z3 − 2xy + z = 0
z(1, 1) = 1
z3 − 2xy − 3z = 0 z(1, 1) = 1
1
1
sin
s(t) ≡ t t = 0
cos c(t) ≡ t t = 0
0 t = 0 0 t = 0
f(x, y) = x 2 s(x)+y 2 c(y) f
f(x, y) = 1+x−y √
1+x2 +y2 {x2 + y2 ≤ 4}
D ≡
x 2 + y 2 = R 2
f(x, y) = x 2 − xy 2
K x2 + y2 ≤ 1 [− 1 1
2 , 2 ] ×
[−2, 2]
f R 2
f
f K

f(x) =
4
i=1
x i i
D ≡ {x ∈ R 4 : xi ≥ 0,
x ∈ R 4 ,
4
xi = 1 }.
i=1
(∗∗)
f(x, y) ≡
f
1
x2 +y2
A ≡ {(x, y) : xy + 1
sin (xy) > 1} .
2
R 2
⎧
⎪⎨
⎪⎩
˙x = − πy2
cos
(1 − 2y) 2
˙y = 2y 2
x 1
2
1 = y 2 = 1 .
π
2
y
(1 − 2y)
(α, β)
lim dist (x(t), y(t)), y =
t↓α 1
= 0
2
lim |(x(t), y(t))| = +∞ .
t↑β
P ∈ [−1, 1]× 1
2
tk ↓ α
(x(tk), y(tk)) k→+∞
−→ P .

K ⊂ R2 \ y = 1
2 0 < δ < β−a
t0 ∈ (α, α + δ) t1 ∈ (β − δ, β)
(x(t0), y(t0)) ∈ K
(x(t1), y(t1)) ∈ K .
R 2
⎧
⎪⎨
1 + x2
β 2
˙x = β
⎪⎩
π
4
˙y = β y
x(0) = 0, y(0) = 1
β ∈ R \ {0} x(t, β)
β Iβ
L > 0
|x(t0, β) − x(t0, β ′ )| ≤ L |β − β ′ |
β, β ′ ∈ K ⊂ R \ {0} t0 ∈ C ⊂ ∩β∈KIβ
un ∈ C([a, b]) un (a, b) un
[a, b]
un ∈ C([a, b]) un (a, b) un(a) un
(a, b)
fn 0 x ∈ (0, 1
1 (0, 1
2n
limn→∞ fn(x) = 0 x
fn [0, 1]
lim 1
0 fn = 1
0 lim fn = 0
1 1 ) ( 2n , n )
n
) x = 1
2n

fn ∈ C([0, 1])
f ∈ C([0, 1])
lim 1
0 fn = 1
f 0
k ∈ N
fn ∈ Ck ([0, 1]) f ∈ Ck ([0, 1])
lim 1
0 fn = 1
f 0
C∞ ([0, 1])
fn(x) = n x + 1
n − √
x
x > 0 f(x) fn(x) n
fn f
x
α
∞
n=0
∞
∞
n=2
n=1
∞
∞
n=1
n=1
e −αn x n
x αn
n x
(x sin n) n
1 + n 2 x
(xn) n
x + n!
un(x) un(x) ≡ (
n
j=1
j x ) −1

(i)
∞
n=0
n 3
2 n
(ii)
∞
n=1
n 3 x n
un = 1
n2 1
0 e−nxt4dt
u(x) ≡
n≥1 un(x) v(x) ≡
x u(x) v(x)
n≥1 u′ n(x)
x m
(1−x) n =
+∞ n − 1 − m + k
k=m n − 1
e x = +∞
k=0 xk
k!
∀x
log (1 + x) = +∞
xk
k=1 (−1)k+1 k
log ( 1+x
1−x ) = 2 +∞
(1 + x) α = +∞
k=0
k=0 x2k+1
2k+1
α
k
sin x = +∞ x2k+1
k=0 (−1)k (2k+1)!
cos x = +∞ x2k
k=0 (−1)k (2k)!
arcsin x = +∞
k=0
(2k−1)!!
(2k)!!
arccos x = π
2 − +∞
k=0
x k n, m ∈ N , n ≥ 1 |x| < 1
|x| < 1
|x| < 1
xk α ∈ R \ Z , |x| < 1
∀x
∀x
1
2k+1x2k+1 |x| < 1
(2k−1)!!
(2k)!!
arctan x = +∞ x2k+1
k=0 (−1)k 2k+1
sinh x = +∞
k=0 x2k+1
(2k+1)!
cosh x = +∞
k=0 x2k
(2k)!
∀x
∀x
1
2k+1x2k+1 |x| < 1 |x| < 1
α
0
≡ 1 ,
α
1
≡ α
α
k
α(α−1)...(α−k+1)
k ≥ 2
k!
0
x ∈ (−1, 1) |x| < θ < 1 ⇒ ɛ > 0 θ(1 + ɛ) < 1
1
k0 ∀ k ≥ k0 α k
≤ 1 + ɛ k0
k
π
arccos x = − arcsin x
2 sinh x = −i sin ix
cosh x = cos ix
≡

x = +∞
x = +∞
k=0 x2k+1
2k+1
k=0 (−1)k (2k−1)!! 1
(2k)!! 2k+1x2k+1 |x| < 1
|x| < 1
z π
iπ= −1
ϕε(x) ∈ C ∞ , ϕε(x) ≥ 0 (ϕε) = [0, ε]
gε ∈ C ∞ gε(x) = 0 x ≤ 0 gε(x) = 1
x ≥ ε gε x = 0 x = ε
M t t → M(t)
M(t) ⇐⇒ Mi,j(t) ∀ i, j
M(t) ⇐⇒ ∀ ɛ > 0, ∃ δ M(t) − M(t0) < ɛ , ∀ |t − t0| < δ
f : y ∈ R 2 ↦−→ f(y) = (f1(y), f2(y)) ∈
R 2
f1 = y1 + y 2 1 cos y2 f2 = y2 + y 2 1
f y0 = (0, 0)
r
f(x, y) = |x| 2 + y 2 − 2x1 + 4x2 − 6y − 11
x ∈ R 2 y ∈ R
g C ∞ x0 = (1, −2)
f(x, g(x)) = 0
g(x0) > 0 g x0
x = −i arcsin ix
1 1+x
x = −i arctan ix x = log ( 2 1−x )

v ∈ R n 1
f(x) ≡ x + v sin |x| 2 .
f x = 0
r > 0 f −1 Br(0)
(0, 0)
e x2 +y 2
− x 2 − 2y 2 + 2 sin y = 1
y = f(x)
f
f(x)
lim
x→0 x2 .
v ∈ R n 1
f(x) ≡ x + v sin |x| 2 .
f(x) − y = 0
x = g(y) y = 0 r > 0
g Br(0)

R n
R n
f ≡ 0 ([0, 1]\Q)∪{0} f(x) = 1
n x = m
n 0 ≤ m ≤ n
m n
f Q ∩ (0, 1]
f ∈ R([0, 1])
A ⊂ Rn
⇐⇒ ∀ε > 0, ∃E1, E2 E1 ⊂ A ⊂ E2 nE2 −
nE1 < ε
nA = inf{nE2 : A ⊂ E2, E2 } = sup{nE1 :
E1 ⊂ A, E1 }
A
A, o
A e ∂A
Q n ∩ E E
X ⊂ Rn o
X = ∅

X ⊂ R n o
X = ∅ X
Qx , Q y ⊂ R
Qx × Q y ⊂ R 2
Q ⊂ R 2 ∀x, y ∈ R, Qx ≡ {y : (x, y) ∈ Q}
Q y ≡ {x : (x, y) ∈ Q} R
X ⊂ R {xn}n
X
(X) = 0
i)
ii)
iii)
iv)
x
D
2
y2 dx dy D ≡ {(x, y) ∈ R2 : 1 ≤ x ≤ 2, 1
x ≤ y ≤ x}
D x2 y2 dx dy D ≡ {(x, y) ∈ R2 : x2 + y2 ≤ 1}
D y3ex dx dy D ≡ {(x, y) ∈ R2 : y ≥ 0, x ≤ 1, x ≥ y2 }
D xy dx dy D ≡ {(x, y) ∈ R2 : x + y ≥ 1, x2 + y2 ≤ 1}
1 1
x − y 1
dx
dy =
(x + y) 3 2
0
0
1
0
1
dy
0
x − y
dx = −1
(x + y) 3 2 .
D =
D
x2 dx dy
y2
(x, y) ∈ R2 : 1 ≤ x ≤ 2 , 1
≤ y ≤ x
x
y
D
3 e x dx dy
D = (x, y) ∈ R 2 : y ≥ 0, x ≤ 1 , x ≥ y 2 .
.

xy dx dy
D
D = (x, y) ∈ R2 : x + y ≥ 1 , x2 + y2 ≤ 1 .
a > 1 R2
y = ax y = x
a y = a2x2 a
x 2 dx dy dz
D
D ≡ {(x, y, z) ∈ R3 : x2
a2 + y2
b2 + z2
c2 ≤ 1} a, b, c > 0
R 3 . 1
{(x, y, z) ∈ R 3 : |x| + |y| + |z| ≤ 1 })
R 4 . 1
∗
n
I2 =
I3 =
Dn ≡ {(x1, . . . , xn) ∈ R n : 0 ≤ x1 ≤ x2 ≤ . . . ≤ xn ≤ 1};
D2
xy dx dy
xyz dx dy dz
D3
∗ In =
Dn (x1 . . . xn) dx1 . . . dxn
x 2 + y 2 ≤ 1 z = x 2 + y 2 − 2
x + y + z = 4

E = {(x, y) ∈ R 2 : x ≥ 0, (x 2 + y 2 ) 2 ≤ (x 2 − y 2 )} .
E
E
k > 0
Ek ≡ {(kx, ky) : (x, y) ∈ E} .
G = {(x, y, z) ∈ R 3 : x ≥ 0, (x 2 + y 2 ) 2 ≤ (1 − z 2 )(x 2 − y 2 ), |z| ≤ 1} .
x
T
2 (y − x 3 )e y+x3
dxdy
T ≡ {(x, y) ∈ R 2 : x 3 ≤ y ≤ 3, x ≥ 1}.
u = y − x 3 v = y + x 3
D
< x, z > z ∆
D z
x dx dz
D (∆) = 2π 2(D).
dx dz D
D
r
(a, 0)
r < a
h r
h r
h R
r

Ω z = x 2 + y 2
x 2 + y 2 + z 2 = 1
Ω
(xe 1+z2
ln (1 + z 2 ) − y sin z + 1) dx dy dz
⎧
⎨
⎩
x 2 + y 2 − 2y = 0
4z = x 2 + y 2
z = 0
x
D
|yz| dx dy dz.
D xe−xy dx dy D = {(x, y) ∈ R2 :
x < y, x, y > 0}
f D
{Dk}k D = ∪kDk f
Dk k supk |f| < ∞ f = limk→∞
Dk
D
Dk f
α ∈ R p > 0 z α
Fp ≡ {(x, y, z) ∈ R 3 : 0 < z < 1 , x 2 + y 2 ≤ z 2p } ,
Fp
z α dx dy dz .
R n
Γ R 3 {y = x 2 }
{z = x3 } {x = 1} {x = 2} Γ
log |z|
f ds f ≡ √
1+4y+9xz
Γ

R 3
Γ ≡ {(u(t), v(t)) : t ∈ (a, b)}
(0, ∞) × R α (0, 2π]
S ≡ {(x, y, z) ∈ R 3 : x = u(t) cos θ, y = u(t) sin θ, z = v(t) t ∈ (a, b) θ ∈ (0, α)}
R 3
(a, b)
γ R 2
ρ = a (1 + cos θ) θ ∈ [0, 2π) .
a > 0
T3 r R r < R
R3
z
r z
R > r
R 2
f(x, y) = (1 + xy, x) ed A = {(x, y) ∈ R 2 : (x − 2) 2 + y 2 < 1, y > 0} .
Ω R 3 ∂Ω
Ω 1
∂Ω F (x) = x (x ∈ R 3 )
ω(x, y, z) = x 3 dx + y 2 dy + z dz
ω(x, y) = x
x 2 +y 2 dx + 2y
x 2 +y 2 dy
ω(x, y, z) = 1
1+y 2 dx − 2xy
(1+y 2 ) 2 dy
ω(x, y, z) = Ax+By
x 2 +y 2 dx + Cx+Dy
x 2 +y 2 dy A, B, C, D ∈ R

1
ω(x, y) = x 2 dx + xy 2 dy
ϕ [0, 1] × [0, 1]
T ≡ {(x, y, z) ∈ R3 : |x| + |y| + |z| ≤ 1 }.
T
T |z|γ dx dy dz γ ∈ R
α β γ |x| + |y| + |z| dx dy dz α, β, γ ∈ R
T F (x, y, z) = (x + y +
z, x + y + z, x + y + z)
F T
• ω =
x dy − y dx
x2 + y2 ,
+ z2 +∂S
ω
• S {x 2 + y 2 ≤ 1, 0 ≤ z ≤ 1}
C ≡ {(x, y, z) ∈ R 3 : z ≤ 4, x 2 + y 2 ≤ z, x 2 + y 2 ≤ 1}
C
F (x, y, z) = (x, 0, 0) F C
F
C
C
E = {(x, y, z) ∈ R 3 : x 2 + y 2 + z 2 ≤ 1, z ≥ x 2 + y 2 } .
∂E

F
F (x, y, z) = (x, y, z) ;
F
∂E
E
R 3
E =
(x, y, z) ∈ R 3 : x 2 + y 2 + z 2 ≤ 1 , 0 ≤ z ≤ x 2 + y 2
∂E
F (x, y, z) = (x, y, z)
∂E
E
1
ω(x, y) = (y3 − x 2 y) dx + (x 3 − y 2 x) dy
(x 2 + y 2 ) 2
ω ω
α > 0 γα = +∂Bα(0)
α
γα
(∗) γ R2 \{0}
ω = 0 .
γ
ω .
.
.
ω

2π
f(x) = 1 − 2|x|
π
[−π, π]
(i)
n≥0
x ∈ [−π, π]
1
(2n+1) 2 ; (ii)
n≥1 1
n2 ; (iii)
n≥0
1
(2n+1) 4 ; (iv)
n≥1 1
n4 .
⎧ ⎨
⎧⎪ ⎨
⎪⎩
⎩
∂u
∂t − ∂2u ∂x2 = 0 0 < x < π, t > 0
u(0, t) = u(π, t) = 0 t ≥ 0
u(x, 0) = x 0 ≤ x ≤ π ;
∆u ≡ ∂2 u
∂x 2 + ∂2 u
∂y 2 = 0 0 < x < π, 0 < y < π
u(x, 0) = x 2 0 ≤ x ≤ π
u(x, π) = x 2 0 ≤ x ≤ π
u(0, y) = 0 0 ≤ y ≤ π
u(π, y) = π 2 0 ≤ y ≤ π.

Im {(1 + i) n + (1 − i) n }
Re {(1 + i) n + (1 − i) n }
i i
(−1) 2i
4 √ i
D
| sin z| D = C
| sin z| D = {z ∈ C : |Im z| < R }
z−i
D = {z ∈ C : Im z > 0}
z+i
|e z−i
z+i | D = {z ∈ C : Im z > 0}
f(z) Ω ⇐⇒ f(z) Ω
Re f = f
Ω ≡ {z : z ∈ Ω}

Im f = f
P (x, y) = ax 3 + bx 2 y + cxy 2 + dy 3 .
2z+3
z+1 z − 1
(1 − z) −m m > 0 z
∞
n=0 zn
n!
∞
n=0 n!zn
∞
n=0 n!zn!
∞
n=0 zn!
∞
n=0 nn z n2
∞
n=0
∞
n=0
∞
n=0
(z+i) n
(1+i) n+1
z n
n(n+1)
z n
n √ n+1
(1)
∞
n=0 anz n R
∞
n=0 a2 nz n
∞
anz 2n
n=0
∞ n=0 a2nz 2n

x dz σ 1 + i
σ
x dz
|z|=R
x = z+z
2
|z|=2
dz
z2−1 = 1
2
e
|z|=1
z
zn dz n ∈ Z
|z|=2
|z|=ρ
dz
z2 +1
z + R2
z
dz
|z−a| 2 |a| = ρ
sin z
|z|=1 zn dz n ∈ Z
|z|=2 zn (1 − z) m dz n, m ∈ Z
{|z| = R}
f Ω |f(z)| ≤ M |z| ≤ R
BR(0) ⊂ Ω) 0 < ρ < R
sup |f
|z|≤ρ
(n) (z)| .
|f (n) (z)| ≥ n!n n n
• f {|z| < 1}
• f g |z| < R R > 1 f ≡ g
{|z| < 1}

f : C → C f
z ∈ C f(z+ω1) = f(z+ω2) =
f(z) ω1, ω2 ∈ C R
f |f(z)| < 1
|z| < 1 f 0 m f(z) = λzm
|λ| = 1
|f(z)| < |z| m
∀ |z| < 1 .
f : Ω → C |f(z) − 1| < 1
z ∈ Ω
γ
f ′
f
= 0 γ ⊂ Ω .
z → z
Imz > 0
z0 Imz0 > 0
|z| = 2 |z + 1| = 1 −2 → 0 0 → i
|z| = 1 |z − 1 1
4 | = 4
1 | = 2 x > 0
|z| = 1 |z − 1
2
R(z) = z+1
z−1
R x = c c ∈ R

f Imz ≥ 0
∃ lim f(z) < ∞ .
z→∞
z0 Imz0 > 0
f(z0) = Imz0
π
+∞
−∞
f(t)
dt.
|t − z0| 2
f h z0
1
Resz0f =
(h − 1)! Dh−1 z [(z − z0) h f(z)] |z=z0 .
f Ω g
z0 ∈ Ω Resz0 fg
(a)
(b)
(c)
1
z2 +5z+6
1
(Z2−1) 2
1
sin z
(d) cotan z :=
(e)
1
sin 2 z
π
0
π
2
0
∞
0
∞
0
dθ
a+cos θ a > 1
dθ
a+sin2 |a| > 1
θ
x 2
x4 +5x2 +6 dx
cos x
x2 +a2 dx a ∈ R
cos z
sin z

π
0
π
2
0
∞
0
∞
−∞
∞
0
∞
0
∞
0
∞
0
dθ
a+cos θ
dθ
a+sin 2 θ
x 2
x4 +5x2 +6 dx
x 2 −x+2
x4 +10x2 +9 dx
a > 1
|a| > 1
x 2
(x2 +a2 ) 3 dx a ∈ R \ {0}
cos x
x2 +a2 dx a ∈ R \ {0}
log x
1+x2 dx
log(1+x 2 )
x 1+α dx 0 < α < 2
P1(z) = z 7 − 2z 5 + 6z 3 − z + 1 |z| < 1
P2(z) = z 4 − 6z + 3 1 ≤ |z| < 2
P3(z) = z 4 + z 3 + 1 {z = x + iy | x, y > 0}
P (x)
1 P (0) = −1 P (x)
P (1) = 0
Q(z) = z n + . . . + a0
(−1) n a0
fn Ω
m Ω fn
f Ω f
f m Ω
f z = 0 f ′ (0) = 0
n g 0
f(z n ) = f(0) + g(z) n .

∞
n=2
1 − 1
n2
= 1
2 .
an = n
k=2
|z| < 1
1 1 − k2
(1 + z)(1 + z 2 )(1 + z 4 )(1 + z 8 ) · . . . · (1 + z 2n
) . . . = 1
1 − z .
m
n=1
θ(z) =
∞
n=1
(1 + z 2n
2
) =
m −1
z 2n .
n=0
(1 + h 2n−1 e z )(1 + h 2n−1 e −z )
|h| < 1
θ(z + 2 log h) = h −1 e −z θ(z) .
f(z) = sin πz g(z)
{an}n ⊂ C ∗ an → ∞ m ∈ N
{an}n m m > 0
f
f(z) = z m
g(z)
e 1 −
n
z
e
an
z
an + 1
z 2+...+
1 z
mn
2 an
mn an
g mn
mn
k
n
1
< ∞ .
|an| k+1
h h
g(z)
f g

π cot πz = 1
1 1
+ + ,
z z − n n
n=0
g(z)
sin πz
sin πz = πz
∞
n=1
1 − z2
n2
.
1
n
2π
(cos θ) 2n dθ .
0
f C
f f(z) = −f(−z) z
g [0, 2π] g(0) = g(2π)
f
f(e iθ ) = g(θ)
θ ∈ [0, 2π] f
f ∈ C1 (Ω)
∂f
∂z
= 0 |z| ≤ 1
f(z) = g(Arg z) |z| = 1

Ω
∂Ω
f
f Σ + f
f(z) = f(z + 1) z
g
D ∗
g(e 2πiz ) = f(z)
z ∈ Σ +
g
f
y > 0
cn =
f =
∞
cne 2πinz
−∞
1
f(x + iy)e −2πin(x+iy) dx
0

a)
b)
c)
d)
e)
f)
g)
h)
i)
2π
0
2π
0
π
cos θ
2 + cos θ dθ
dθ
a + b sin θ
a > b > 0
sin
0
2 θ
dθ a > 1
a + cos θ
+∞
x
0
−a
dx 0 < a < 1
1 + x
+∞
dx
dx b > 1
0 1 + xb +∞
log x
0 xa dx 0 < a < 1
(x + 1)
+∞
log x
dx a, b > 0 , a = b
0 (x + a)(x + b)
1
x4 dx
x(1 − x)
0
1
−1
√
1 − x2 dx .
1 + x2

a)
b)
c)
d)
e)
f)
g)
h)
i)
2π
0
2π
0
π
cos θ
2 + cos θ dθ
dθ
a + b sin θ
a > b > 0
sin
0
2 θ
dθ a > 1
a + cos θ
+∞
x
0
−a
dx 0 < a < 1
1 + x
+∞
dx
dx b > 1
0 1 + xb +∞
log x
0 xa dx 0 < a < 1
(x + 1)
+∞
log x
dx a, b > 0 , a = b
0 (x + a)(x + b)
1
x4 dx
x(1 − x)
0
1
−1
√
1 − x2 dx .
1 + x2

R n
R n
←
→ ∃ a ∈ [0, 1]
|f(a)| > 0
|f(x) + g(x)| < |f(x)| + |g(x)|
x = (1, 0, 0, .., 0)y = (0, 1, 0, .., 0)
Γ = {γ ∈ R n : γ = γ(t), a ≤ t ≤ b }
Φ = {φ ∈ R n : φ = φ(t), a ≤ t ≤ b }
γ(b) = φ(a)
a+b
γ(2t − a) a ≤ t ≤
λ(t) ≡
2
≤ t ≤ b
φ(2t − b) a+b
2

Λ = {λ ∈ R n : λ = λ(t), a ≤ t ≤ b }
Λ = Γ ∪ Φ
P = (rP , θP ) , Q = (rQ, θQ) ∈ A
γ : t ↦→ (x = (trQ+(1−t)rP ) cos(tθQ+(1−t)θP ) , y = (trQ+(1−t)rP ) sin(tθQ+(1−t)θp)).
· Lip : (E, R m ) −→ R
• fLip ≥ 0 f ∈ (E, R m )
fLip = 0 ⇐⇒ sup |f(x)| = 0 ⇐⇒
x∈E
⇐⇒ f(x) = 0 ∀ x ∈ E.
• a ∈ R
afLip =
|af(x) − af(y)|
sup
x,y∈E |x − y|
x=y
|f(x) − f(y)|
= |a| sup
x,y∈E |x − y|
x=y
= |a| fLip .
+ sup |af(x)| =
x∈E
+ |a| sup |f(x)| =
x∈E
• f g
(E, R m )
|(f(x) + g(x)) − (f(y) + g(y))|
|x − y|
≤
+ |g(x) − g(y)|
|f(x) − f(y)|
+
|x − y|
.
|x − y|

f + gLip =
|(f(x) + g(x)) − (f(y) + g(y)|
sup
+
x,y∈E
|x − y|
x=y
+ sup |f(x) + g(x)| ≤
x∈E
≤ sup
x,y∈E
x=y
|f(x) − f(y)|
|x − y|
|g(x) − g(y)|
+ sup
x,y∈E |x − y|
x=y
= fLip + gLip .
+ sup |f(x)| +
x∈E
+ sup |g(x)| =
x∈E
((E, Rm ), fLip)
{fk}k ε > 0 N0 = N0(ε) > 0
k, h > N0
fk − fhLip =
|(fk(x) − fh(x)) − (fk(y) − fh(y))|
sup
+
x,y∈E
|x − y|
x=y
+ sup
x∈E
|fk(x) − fh(x)| ≤ ε .
{fk}k
(C(E, R m ), · ∞,E)
f ∈ C(E, R m )
N1 = N1(ε) > 0 k > N1
fk − f∞,E ≤ ε .
x, y ∈ E x = y k, h > N0
|(fk(x) − fh(x)) − (fk(y) − fh(y))|
|x − y|
h → +∞
|(fk(x) − f(x)) − (fk(y) − f(y))|
|x − y|
≤ ε
≤ ε .
· ∞,E f ∈ C(E, R m )
f∞,E ≡ sup |f(x)| .
x∈E

x, y ∈ E k > N0
|(fk(x) − f(x)) − (fk(y) − f(y))|
sup
≤ ε .
x,y∈E
|x − y|
x=y
N = N(ε) = max{N0(ε), N1(ε)}
k > N
fk − fLip ≤ 2ε
f · Lip
f ∈ (E, R m )
k > N
fLip ≤ f − fkLip + fkLip < ∞.
x RN
x = {xn}n
RN
{x (k) } x (k)
x (k) = {x (k)
n }n
(ℓ 1 , · 1)
· 1 : ℓ 1 −→ R
x ↦−→ x1 ≡
|xn|
n∈N
• x1 ≥ 0 x ∈ ℓ 1
x1 = 0 ⇐⇒
|xn| = 0 ⇐⇒
n∈N
⇐⇒ xn = 0 ∀ n ∈ N.
• a ∈ R
ax1 =
|axn| =
n∈N
= |a|
|xn| = |a| x1 .
n∈N

•
x y ℓ1
x + y1 =
|xn + yn| ≤
n∈N
≤
(|xn| + |yn|) =
n∈N
=
|xn| +
|yn| =
n∈N
n∈N
= x1 + y1 .
{x (k) }
ℓ 1 ε > 0 N0 = N0(ε) > 0
k, h > N0
x (k) − x (h) 1 ≡
n∈N
n ∈ N
|x (k)
n − x (h)
n | ≤ ε .
|x (k)
n − x (h)
n | ≤ ε
{x (k)
n }k R
k +∞
xn
x = {xn}n .
x {x (k) }k
· 1
M > 0 k, h > N0
M
n=0
|x (k)
n − x (h)
n | ≤ ε ;
h → +∞
M
n=0
|x (k)
n − xn| ≤ ε
M
ε ≥
∞
n=0
|x (k)
n − xn| = x (k) − x1

x ∈ ℓ 1 k > N0
x1 ≤ x − x (k) 1 + x (k) 1 < ∞.
(ℓ∞ , ·∞)
x ∈ ℓ 1 x∞ < ∞
n |xn|
x ∈ ℓ ∞
ℓ 1 ⊆ ℓ ∞ ;
˜x = {1, 1, . . . , 1, . . .} ;
˜x∞ = 1 ˜x1 = ∞ .
ℓ1 · ∞
ℓ1
x (k)
= 1, 1, 1
1
, . . . , , 0, . . .
2 k
{x (k) }k
· ∞
x = 1, 1, 1
1
, . . . , , . . . .
2 k
x (k) − x∞ = 1 k→+∞
−→ 0.
k + 1
x1 = ∞
ℓ1
(ℓ ∞ , · ∞),
ℓ1 ℓ1
ℓ1
ℓ1
ℓ1

n |xn|
ℓ 1
C ≡
x ∈ R N : lim
n→∞ xn = 0
C
x ∈ C {x (k) } ⊂ ℓ 1 x
· ∞
ℓ1 C
ℓ1
C
{x (k) }k ⊂ C
x ε > 0 N0 = N0(ε) > 0
k > N0
x (k) − x∞ ≤ ε
|xn| ≤ |xn − x (k)
n | + |x (k)
n | ≤
≤ x (k) − x∞ + |x (k)
n | ≤ 2ε
n x (k)
n → 0
ε
lim
n→∞ xn = 0 ⇐⇒ x ∈ C .
2 x ∈ C
{x (k) }
x (k) = {x0, x1, . . . , xk, 0, . . .} .
{x (k) } ⊂ ℓ 1 x
· ∞ xn → 0 ε > 0
N0 = N0(ε)
|xn| ≤ ε
n ≥ N0 k ≥ N0
x (k) − x∞ ≤ ε

X
X
Z ⊂ X
X
X
X
X ⇒ X ⇐ X
X
X
X
X
Ω
Ω
{x (n) }n
x (n) = (0, 0, . . . , 0,
1, 0, . . .) .
n−1
x (n) x (m)
n = m
d x (n) , x (m)
= x (n) − x (m) 1 = 2

{x (n) }n D
D R 10
i : D −→ R 10
x ↦−→ (x1, . . . , x10)
i(D) R 10 · ∞
i : D −→ i(D)
{x (n) }n
{y (n) }n i(D) y (n) = i(x (n) )
i(D) (R10 , · ∞)
{ynk }k
y ∈ i(D)
∀ ε > 0 ∃ N0 = N0(ε) t.c. se k ≥ N0 allora y (nk) − y∞ ≤ ε .
{x (nk) }k
x (nk) = i −1 (y (nk) )
(ℓ 1 , · 1)
x = i −1 (y) ∈ D k ≥ N0
x (nk) − x1 =
=
10
|x
j=1
(nk)
j
10
|y
j=1
(nk)
j
− xj| =
− yj| ≤
≤ ny (nk) − y∞ ≤ nε .
x ∈ D
R n R m
. . .
O = (0, .., 0)
x (k) , y (k) ⊂ R n \ {O}

limk→+∞ x (k) = limk→+∞ y (k) = O limk→+∞ f(x (k) ) = limk→+∞ f(y (k) )
x (k) = (0, .., 1
k
y (k) = (0, .., −1
k
, ..0) x(k
j
, ..0) y(k
j
δi,j
= k , j = 1, .., n
= −δi,j
k
, j = 1, .., n
x0 = (0, 1, 1, 2) δ(ɛ) = min{ ɛ
5α , |x0| − 1 }
x0 = (0, 0, ., 0) δ(ɛ) = ɛ 1 α
δ(ɛ) = min{ ɛ 1
38 , 2 }
δ(ɛ) = min{( π 1
4 ) n , ɛ 1
2n }
δ(ɛ) = min{ √ n
2 , n 5 2 e −4n ɛ
12
}
δ(ɛ) = ɛ
√ n
δ(ɛ) = min{ ɛ
n , ɛ
5α √ n , |x0| − 1 }
δ(ɛ) = (sin 3)ɛ
| · | R4 R2
x0 = (0, 0, 0, 0) δ =
δ(ε) > 0 |x| ≤ δ
|f(x) − f(0)| =
1
− 1, sin(x1x4) ≤ ε .
1 + |x|
•
1
− 1
1 + |x| =
1 − 1 − |x|
1 + |x| =
=
|x|
1 + |x| ≤
≤ |x| ≤ δ .
−π
π
≤ t ≤
4 4
1
cos t ≥ √
1+2 sin2 t

•
| sin(x1x4)| ≤ |x1x4| ≤ |x| 2 ≤ δ 2 ;
| sin t| ≤ |t| t ∈ R |xi| ≤ |x|
i = 1, 2, 3, 4
|f(x) − f(0)| =
1
− 1, sin(x1x4) ≤
1 + |x|
≤ √
1
2 max − 1
1 + |x| , | sin(x1x4)| ≤
≤ √ 2 max δ, δ 2 .
δ ≤ 1
|f(x) − f(0)| ≤ √ 2 max δ, δ 2 =
δ(ε) = min
= √ 2δ .
ε
√2 , 1 .
f : E ⊂ R n → R m
f(x) = (f1(x), . . . , fm(x)) x0 ∈ E
f x0 ⇐⇒ fi x0 ∀ i = 1, . . . , m .
(=⇒) f x0
∀ ε > 0 ∃ δ = δ(ε) > 0 : |x − x0| < δ =⇒ |f(x) − f(x0)| < ε .
i = 1, . . . , m
|fi(x) − fi(x0)| ≤ |f(x) − f(x0)|

(⇐=) i = 1, . . . , m fi x0
∀ ε > 0 ∃ δi = δi(ε) > 0 : |x − x0| < δ =⇒ |fi(x) − fi(x0)| < ε
√ m .
δ = δ(ε) = min{δ1(ε), . . . , δm(ε)} ;
|x − x0| ≤ δ
|f(x) − f(x0| ≤ √ m max
i=1,...,m {|fi(x) − fi(x0)|} ≤
≤ √ m ε
√ m = ε .
| · | Rn x, y ∈ B1(0)
f(x) − f(y) =
1 1
−
2 − |x| 2 − |y| =
1 1
−
2 − |x| 2 − |y| ,
≤
sin
n
i=1
xi
−
sin
n
i=1
yi
.
2 − |y| − 2 + |x|
(2
− |x|)(2 − |y|)) =
| |x| − |y| |
(2 − |x|)(2 − |y|) ≤
|x − y|
≤ |x − y| .
(2 − |x|)(2 − |y|)
m N
xm =
ym =
| |x| − |y| | ≤ |x − y| x, y ∈ R n
2 − |x| ≥ 1 x ∈ B1(0)
1 − 1
, 0, . . . , 0
2m
1 − 1
, 0, . . . , 0
m

1
2
− |xm| −
1
2 − |ym| =
=
=
=
=
=
1
2 − 1 + 1
2m
1
1 + 1
2m
2m
(1 + 2m) −
−
− 1
1 + 1
m
1
2 − 1 + 1
=
m
(1 + m) =
m
(2m + 1)(m + 1) =
2m2 1
(2m + 1)(m + 1) 2m =
m
=
2m 2
(2m + 1)(m + 1) |xm − ym| .
Lm
xm ym
Lm :=
L ≥ 1
sin
n
i=1
xi
−
sin
n
i=1
yi
2m2 m→+∞
−→ 1
(2m + 1)(m + 1)
≤
sin
n
n
xi − sin yi
i=1
i=1
≤
n n
xi − yi
i=1 i=1
≤
≤
x1
n
n
n
n
xi − x1 yi + x1 yi − y1 yi
i=2 i=2 i=2 i=2
≤
n n
≤ |x1| xi − yi
i=2 i=2
+
n
yi
i=2
|x1 − y1| ≤
n n
≤ |x1 − y1| + xi − yi
≤
≤ . . . ≤
| |x| − |y| | ≤ |x − y| x, y ∈ R n
| sin t − sin s| ≤ |t − s| t, s ∈ R
x1 ≤ √ n|x| x ∈ R n
i=2
i=2
≤ |x1 − y1| + . . . + |xn − yn| =
= x − y1 ≤
≤ √ n |x − y| .

|f(x) − f(y)| ≤
2
1 1
− + sin
2 − |x| 2 − |y|
≤ |x − y| 2 + n|x − y| 2 = √ n + 1 |x − y| .
L = √ n + 1 .
n
i=1
Ω = B1(x0)
x0 = (2, . . . , 2)
x ∈ Ω
αn := 2 −
n = 1, 2 αn, βn > 0
2 √ n − 1 ≤ |x| ≤ 2 √ n + 1
xi
−
sin
n
2 √ n + 1 ≤ 2 − |x| 1
2 ≤ 2 − 2 √ n − 1 =: βn .
i=1
yi
2
≤
n = 3, 4, 5, 6 αn < 0 < βn f
Ω n = 4
x0
n ≥ 7 αn, βn < 0
n = 3, 4, 5, 6
α 2 n ≤ (2 − |x| 1
2 )(2 − |y| 1
2 ) ≤ β 2 n .
In = (2 √ n−1, 2 √ n+1)
s, t ∈ In
| √ s − √ t| ≤
=
sup
ξ∈In
1
2 √
|s − t| =
ξ
1
2(2 √ |s − t| .
n − 1)

x, y ∈ B1(x0)
1
−
1
=
| |x| − |y| |
2 − |x| 1
2
2 − |y| 1
2
L =
|(2 − |x| 1
2 )(2 − |y| 1
2 )| ≤
≤ | |x| − |y||
≤
≤
≤
α 2 n
||x| − |y||
2α2 n(2 √ n − 1) ≤
1
2α2 n(2 √ |x − y| .
n − 1)
1
2α 2 n(2 √ n − 1) .
Ω = B1(0) .
L
m N
xm =
ym =
|f(xm) − f(ym)| =
=
=
=
=
=
=
1
, 0, . . . , 0
n2
1
, 0, . . . , 0
4n2
1
2
− |xm| −
1
2 −
|ym| =
1
1 2 − m2 1
−
1 2 − 4m2
=
1
2
− 1
1
−
m 2 − 1
2m
=
m 2m
−
2m − 1 4m − 1
=
m
(2m − 1)(4m − 1) =
4m3 3
=
3(2m − 1)(4m − 1) 4m2 4m 3
3(2m − 1)(4m − 1) |xm − ym| .
.

L
4m3 m→+∞
−→ +∞
3(2m − 1)(4m − 1)
L ≥ Lm :=
L
x, y ∈ Br(0) r > 0
|f(x) − f(y)| =
≤
e |x|2
x − e |y|2
y
≤
e |x|2
x − e |x|2
y
+ e |x|2
y − e |y|2
y
≤
≤ e |x|2
|x − y| + |y||e |x|2
− e |y|2
| ≤
≤ e r2
|x − y| + re r2
(|x| + |y|) | |x| − |y| | ≤
≤ e r2
(1 + 2r 2 )|x − y| .
L = e r2
(1 + 2r 2 ) .
∀ ξ = (ξ1 , ξ2 ) = (0, 0)
∃ δ > 0 ∀ |t| < δ (tξ1) 2 = tξ2
∀ x ∈ R n \ {0} ∀ α ∈ R ∂|x|α
∂xi (0) = α|x|α−2 xi
α β
2α − β > 0 ;
2α − β > 1 ;
2α − β > 1 ;
2α − β > 1 .
lim
h→ 0
f(x0 + h) − f(x0) − L(h)
|h|
C 1
L : R n → R m f
= 0

∂ f
(x, y) =
∂ x
∂ f
(x, y) =
∂ y
y cos(xy)
y 2 e xy2
x cos(xy)
2xye xy2
.
|f(h1, h2) − f(0, 0) − L(h)|
lim
= 0 ,
h=(h1,h2)→(0,0)
|h|
L(h) ≡
=
∂ f
(0, 0)
∂ x
0 0
.
0 0
∂ f
(0, 0)
∂ y
h1
·
h2
=
•
| sin(h1h2)| ≤
≤ |h1h2|
≤ |h| 2 .
• |h| ≤ 1
|h| → 0
e h1h2
2 − 1
≤
≤
≤ 3 h1h 2 2
≤ 3 |h| 3 ≤
≤ 3 |h| 2 .
|f(h1, h2) − f(0, 0) − L(h)|
≤
|h|
√
2 10 |h|
≤ =
|h|
= √ 10 |h| h→(0,0)
−→ 0 .

F ′ (t) =
∂ f
∂ x (g(t), 1 − g2 (t))g ′ ∂ f
(t) +
∂ y (g(t), 1 − g2 (t))(−2g(t)g ′ (t)) .
g(0) = 1
g ′ (t) =
1
cosh 2 t +
t
√
1 + t2 g ′ (0) = 1
1)
F ′ (0) =
=
∂ f
f
(1, 0) − 2∂ (1, 0) =
∂ x ∂ y
−2
.
0
∂
∂x
=
∂
∂z
=
f(x, h(x, z), z) = ∂f
∂x
∂ f
(x, h(x, z), z),
∂ x
f(x, h(x, z), z) = ∂f
∂z
∂ f
(x, h(x, z), z),
∂ z
(x, h(x, z), z) + ∂f
∂y
∂ f
(x, h(x, z), z),
∂ y
(x, h(x, z), z) + ∂f
∂y
∂ f
(x, h(x, z), z),
∂ y
∂h
(x, h(x, z), z) (x, z) =
∂x
∂ f
(x, h(x, z), z)
∂ z
∂h
(x, h(x, z), z) (x, z) =
∂z
∂ f
(x, h(x, z), z)
∂ z
⎛ ⎞
1
⎜ ∂ h ⎟
· ⎝ (x, z) ⎠
∂ x
0
⎛
⎜
· ⎝
y = f(x) C 2 x = 0
x 2 + sinh y + e xy = 1 ,
f(0) = 0
F (x) ≡ x 2 + sinh f(x) + e xf(x) − 1 ≡ 0
0
∂ h
(x, z)
∂ z
1
⎞
⎟
⎠ .

f(0) = 0
0 = F ′ (0) =
2x + (cosh f(x))f ′ (x) + e xf(x) (f(x) + xf ′ (x))
= f ′ (0)
x = 0
0 = F ′′
(0) = 2 + (sinh f(x))(f ′ (x)) 2 + (cosh f(x))f ′′ (x) +
+ e xf(x) (f(x) + xf ′ (x)) 2 + 2f ′ (x) + xf ′′ (x)
=
|x=0
= 2 + f ′′ (0) ,
f ′′ (0) = −2
∂f
∂x =
∂ F
∂ y (0, 0) = [cosh y + xexy ] |(x,y)=(0,0) =
⎛
⎜
⎝
∂f1
∂x1
∂f6
∂x1
= 1 = 0.
∂f1
∂x2
∂f6
∂x2
⎞
⎟
⎠ ∂f
∂y =
⎛
∂
⎜
f(x, g(t)) = ⎜
∂t ⎝
⎛
⎜
⎝
∂f1
∂y1
∂f6
∂y1
∂f1
∂y2
∂f6
∂y2
3 ∂f1
j=1 ∂yj g′ j (t)
3 ∂f6
j=1 ∂yj g′ j (t)
⎞
⎟
⎠
∂f1
∂y3
∂f1
∂y3
⎞
⎟
⎠
|x=0
=

∂f
∂y =
∂f
∂x =
∂f
∂t =
⎛
⎝
⎛
⎜
⎝
⎛
⎜
⎝
0 0
0 0
1
1+t
2y2x3
1+t
⎞
⎠
t cos(tx1) 0 0
x1
|x|
x2
|x|
0 0
⎞
x1 cos(tx1)
|x| ⎟
⎠
− y1+y2
2 x3
(1+t) 2
x3
|x|
y 2
2
1+t
∂
∂f
f(x, g(t), t) =
∂t ∂y (x, g(t), t), g′ (t) + ∂f
∂t
=
⎛
⎜
⎝
1
(1+t) cosh 2 t
x1 cos(tx1)
|x|
⎞
⎟
⎠
(x, g(t), t) =
2 ln(ln t)x3
+ (1+t)t ln t − tanh t+x3 ln2 (ln t)
(1+t) 2
D 1 f(0)(ξ) = ∇f(0) ξ = (1, 0, .., 0) ξ = ξ1 ;
D 3 f(0)(1, 2, .., n) 3 = 6 n
j=1 j2 = n(n + 1)(2n + 1)
∂ 5 f
∂x1∂x2...∂x5 (1, 1, .., 1) = (25 x1x2x3x4x5e |x|2
= 90 e n
∂ (1,0,..,10)
x
∂ (1,0,..,10)
x
f(0) = 30240
f((−1, 1, .., (−1) n )) = −(99050016)e
∇f(0) = (1, 0, .., 0)
⎞
⎟
⎠
(x1+x4 n)+24x2x3x4x5e |x|2)
|(1,..,1) =
n
f ∈ C(R 2 \ {(x, x) | x ∈ R \ {0} } )
δ = min{ 1
2 , ɛ
√2 e }
∂ 10
∂x10 ∂
n ∂x1 (e|x|2 (x1 + x4 n )) = ex21 +..+x2 n−1 ∂10
∂x10 (e
n
x2n(1 + 2x2 1 + 2x1x4 n )) =
c = e x21 +..+x2n−1 , a = 1 + 2x2 1 , b = 2x1x4 n , f(t) = (1 + bt4 ) g(t) = et2 .
= c 10 k=0 (
10!
k!(10−k)! Dk (f)D10−k (g)) = c 4 k=0 (
10!
k!(10−k)! Dk (f)D10−k (g))
Dkf = 0 k > 4
Dj (g) = et2Pj(t) Pj(t)
(t) + 2tPk(t)
P0(t) = 1 Pk+1(t) = P ′ k

∀ k ≥ 0 f ∈ C k ({0})
f ∈
C 2 (0)
P6(x; 0) = x1 + x1x2x3x4 + o(|x| 6 )
P3(x; 0) = x3 + x1x3 + x2x3 + x2
1 x3
o(|x| 6 )
δ = 1
4e
2
x22
+ x3
2
+ x1x2x3 +
f ∈ C(Rn sin |x|
\ {0}) lim |x|→ 0 |x| = 1
f ∈ C(Rn )
f ∈ C1 (Rn \ {0})
f ∈ C1 (Rn )
R ∗ ≡ {(x, x) | x ∈ R \ {0} }
f ∈ C ∞ (R 2 \ R ∗ ) f
R ∗
3
P1 = ( 2
3 (− 2
sup f = +∞ inf f = −∞
1 , √ ) P2 =
6
1 , − √ )
6
D
Py = (0, y) (−1 ≤ y ≤ 1)
• Py 0 < y ≤ 1
• Py 1 ≤ y < 0
• P0
∂D ≡ {x 2 + y 2 = 1}

• M1 = ( 2
√ 3 , 1
√ 3 ) M2 = (− 2
√ 3 , 1
√ 3 )
f(M1,2) = 2
3 √ 3
• m1 = ( 2
√ 3 , − 1
√ 3 ) m2 = (− 2
√ 3 , − 1
√ 3 )
f(m1,2) = − 2
3 √ 3
maxD f = 2
3 √ 3 minD f = − 2
3 √ 3
∂ ∂xi g(|x|) = g′ (|x|) ∂|x|
∂xi = g′ (|x|) xi
|x| . ∇g(|x|) = g′ (|x|) x
|x|
P1000(s, t; 1, 0) =
1000
α∈N 2 |α|=1
P1000(x, y; 1, 0) =
aβ =
(−1) α1+α2+1 (α1 + α2 − 1)!
(s − 1)
α1!α2!
α1 t α2
1000
β∈N 2 |β|=1
(−1) β1+1 (β1+ β 2
2 −1)!
β1!( β 2
2 )!
aβ(x − 1) β1 y β2
β2
0 β2
f ∈ C ∞ (R 2 \ {x = 1})
{(1, y), y = 0}
∂
x = 1
n f
∂xh ∂yk y
−( e x−1 )2 P (y,x−1)
Q(x−1)
limx→1 Q(x − 1) = 0
x → 1
Py ≡ (1, y)
f y > 0
y < 0
y = 0
(1, 0)
C1

δ = 1
100
∂f
∂x = ∇f = 2f(x) x
∂f
∂x
= 2xy
z−x 2
∂f
∂x =
∂f
= 2xi (x
∂xi
2 j) = 2xi f(x) ;
j=i
1 2x2 0 . . . 0
−x2 sin (x1x2) −x1 sin (x1x2) 0 . . . 0
aβ =
β
| (−1)
PN(x; x0) =
2 |−1 | β
2 |!
(|β|−1)! ( β
2 )!
N
β∈N n |β|=1
aβ(x) β
βi ∀i = 1..n
0
G(x, y, t) ≡ t3 −2xy +y
z = z(x, y) (1, 1)
G(x, y, z(x, y)) ≡ 0 z(1, 1) = 1
(x, y)
z
0 = ∂G(x,y,z(x,y))
(−2y + 3z(x, y)
∂G
∂G
∂z
∂x = ( |(1,1) ∂x (x, y, z(x, y)) + ∂t (x, y, z(x, y)) ∂x (x, y)) |(1,1) =
2 ∂z
∂x (x, y)) |(1,1) = −2 + 3 ∂z
∂x (1, 1)
2 (1, 1) = 3 .
∂z
∂x
P2(x, y; 1, 1) = 1 + 2
1
4
3 (x − 1) + 3 (y − 1) − 9 (x − 1)2 − 1
9 (y − 1)2 + 2
9 (x − 1)(y − 1).
G(x, y, t) = t3 − 2xy + t

s(x, y) ≡ x2s(x) c(x, y) ≡ y2s(y)
x = 0
y = 0
f(x, y) = s(x, y) + c(x, y)
f f(x, y) = x2s(x) + y2c(y)
D
Py = (0, y) (−1 ≤ y ≤ 1)
• Py 0 < y ≤ 1
• Py 1 ≤ y < 0
• P0
∂D ≡ {x 2 + y 2 = 1}
• M1 = ( 2
√ 3 , 1
√ 3 ) M2 = (− 2
√ 3 , 1
√ 3 )
f(M1,2) = 2
3 √ 3
• m1 = ( 2
√ 3 , − 1
√ 3 ) m2 = (− 2
√ 3 , − 1
√ 3 )
f(m1,2) = − 2
3 √ 3
maxD f = 2
3 √ 3 minD f = − 2
3 √ 3
x y (x, y)
Π + ≡ {x ≥ 0 , y ≥ 0 }
f(x, y) = 4xy .

(x, y)
• x ≥ 0 y ≥ 0
(x, y) ∈ Π +
• R
x 2 + y 2 = R 2 .
V ≡ {(x, y) ∈ R 2 : x ≥ 0, y ≥ 0, x 2 + y 2 = R 2 } .
R
Π + x = 0
y = 0 f
F (x, y, λ) = 4xy − λ(x 2 + y 2 − R 2 )
∂xF (x, y, λ) = 4y + 2λx = 0
∂yF (x, y, λ) = 4x + 2λy = 0
∂λF (x, y, λ) = x 2 + y 2 − R 2 = 0
x > 0 y > 0
R√2
P = , R
√ , λ = −2 .
2
P1 =
R√2 , R √ 2
A = R2
2
, P2 = − R √ ,
2 R
√ , P3 = −
2
R √ , −
2 R
R√2
√ , P4 = , −
2
R
√
2
R 2
f y = x
lim f(x, x) = lim
x→±∞ x→±∞ x − x3 = ±∞ ;
sup
R 2
f = +∞ e inf f = −∞ .
R2

f
∂xf(x, y) = 2x − y 2 = 0
∂yf(x, y) = −2xy = 0
⇐⇒
x = 0
y = 0 .
f(0, 0) = 0
Bδ(0, 0) 0 < δ < 1
P1 =
δ 2 , δ
2
e P2 = − δ
δ
,
2 2
f(P1) = − 3
4 δ2 < 0 e f(P2) = δ2 δ3
+ > 0 .
4 8
K x = ± 1
2
x 2 + y 2 = 1
A =
− 1
2 ,
√
3
, B =
2
1
2 ,
√
3
, C = −
2
1
√ √
3 1 3
, − , D = , − .
2 2
2 2
f(A) = f(C) = 5
8
f(B) = f(D) = − 1
8 .
• L+
L± =
x = ± 1
√
3
, −
2 2
f+(y) = f(x, y) |L+
f ′ +(y) = −y
√
3
< y < .
2
1 1
= f(1 , y) = −
2 4 2 y2
E = ( 1
1
2 , 0) f(E) =
• L−
f−(y) = f(x, y) |L−
f ′ −(y) = y
1 1
= f(−1 , y) = +
2 4 2 y2
G = (− 1
1
2 , 0) f(G) =
;
4
4

F (x, y, λ) = x 2 − xy 2 − λ(x 2 + y 2 − 1) .
⎧ ⎨
2x − y 2 = 2λx
−2xy = 2λy
x 2 + y 2 = 1 ;
⎩
y = 0 x = ±1
x = −λ 3x2 + 2x − 1 = 0
x = −1
x = 1 3 y = √ 8
3
H =
f(H) = − 5
27
1
3 ,
√
8
• K 5 8 A B
• K − 5
27 H
D = {x ∈ R 4 : xi ≥ 0, 4
i=1 xi = 1}
f(x) ≥ 0 x ∈ D
f 0
xm = (0, 0, 0, 1) ∈ D
F (x, λ) =
4
x i
4
i − λ xi − 1
i=1
⎧
⎨
⎩
∂xj
F (x, λ) = j
4
i=1 xi = 1 .
4
i=1 xi i
xj
3
i=1
− λ = 0 j = 1, 2, 3, 4
xj = 0
xj = 0 0
4 i=1 λ = j
xii xj
∀ j = 1, 2, 3, 4.

2
3
4
4
i=1 xi i
x2
4 i=1 xii x3
4 i=1 xii x4
=
=
=
4
i=1 xi i
x1
4 i=1 xii x1
4 i=1 xii x1
⇐⇒ x2 = 2x1
⇐⇒ x3 = 3x1
⇐⇒ x4 = 4x1 ,
x1(1 + 2 + 3 + 4) = 1 ⇐⇒ x1 = 1
10 .
f(xM ) = 27648
10 10
xM =
1 2 3 4
, , ,
10 10 10 10
f(x, y) ≥ 0 ∀ (x, y) ∈ R 2 \ {(0, 0)}.
{( 1
n , 2n)}n A
lim
n→0 f
1
, 2n
n
inf
A f = 0 .
= 0
g(t) = t + 1
2 sin t limt→+∞ g(t) = +∞
limt→−∞ g(t) = −∞ ∃! c g(t) > 1 t > c g(t) < 1
t <
A = {(x, y) y > c
x }
∂A
P1 = ( √ c, √ c) P2 = (− √ c, − √ c)
1
2c
sup A f = 1
2c
(x0, y0) ∈ A f(x0, y0) > 1
2c
(x0, y0) ∈ A =⇒ x0y0 > c ,

1
f(x0, y0) = x2 0 + y 2 0 < 2c < 2xy ⇐⇒ x 2 0 + y 2 0 − 2x0y0 < 0
⇐⇒ (x0 − y0) 2 < 0
c
0.68403 < c < 0.68404
2 ˙y = 2y
= 1 .
y 1
2
y
1
dy
= 2
y2 t
1
2
t dt ⇐⇒ y(t) =
˙x = − π
π
cos =
4t2 4t
= d
sin
dt
π
4t
1
2(1 − t) .
π
x(t) = sin + C ;
4t
C
C = 0 .
⎧
⎪⎨
x(t) = sin
π
4t
⎪⎩
1
y(t) =
2(1 − t) .
t ∈ (0, 1)

dist
lim dist (x(t), y(t)), y =
t↓0 1
= 0
2
(x(t), y(t)),
y = 1
=
2
1
y(t) −
2
.
lim dist (x(t), y(t)), y =
t↓0 1
=
2
= lim
1
t↓0 y(t) −
2
=
= lim
1 1
t↓0 −
2(1 − t) 2
= 0 .
lim |(x(t), y(t))| ≥ lim |y(t)| =
t↑1 t↑1
= lim
1
t↑1 2(1
− t) = +∞
P ∈ [−1, 1] ×
1
2
t ↓ 0 P = x0, 1
2
γ0 ∈ π 3π
2 , 2
x0 = sin γ0 = sin(γ0 + 2πn) ,
n ∈ N
γn ≡ γ0 + 2πn ∈ [2πn, 2π(n + 1)], γn
tn ≡ π
4γn
n→+∞
−→ 0 .
n→+∞
−→ +∞
tn ∈ (0, 1) n ∈ N
0 ≤ tn = π π
=
4γn 4(γ0 + 2πn) ≤
≤
π
4γ0
≤ π 2
4 π =
= 1
< 1 .
2

(x(tn), y(tn)) =
=
=
π 1
sin ,
=
4tn 2(1 − tn)
1
sin γn,
=
2(1 − tn)
1 n→∞
x0,
−→ x0,
2(1 − tn)
1
.
2
K
A ≡ [−1, 1] ×
1
, ∞
2
,
A
K
R 2
m = min y M = max
(x,y)∈K (x,y)∈K y
m > 1 2
lim y(t) =
t↓0 1
2
lim y(t) = +∞ ,
t↑1
t0 t1 0 < δ < 1
•
x
0
1 +
⎧
⎨
⎩
1
β
2 dx =
x
β
π
4
˙x = β π
1 +
4
x2
β2
x(0) = 0 .
t
0
t dt ⇐⇒ arctg
x(t, β) = β tg
π
4 t
.
x
=
β
π
4 t

•
y(t, β) = e βt .
⎧
⎨
x(t, β) = β tg π
4 t
⎩
y(t, β) = eβt .
Iβ = (−2, 2) ,
β
K R \ {0}
β, β ′ ∈ K C ∩β∈KIβ = (−2, 2) t0 ∈ C
T ≡ sup tg
t∈C
π
4 t
< ∞
M ≡ sup
(β,t)∈K×C
e βt < ∞
B ≡ sup |t| < ∞ .
t∈C
|x(t0, β) − x(t0, β ′ )| ≤
βtg
π
4 t0
≤ T |β − β ′ |
|y(t0, β) − y(t0, β ′ )| ≤
− β ′ tg
e βt0 − e β′
t0
≤
≤ M |βt0 − β ′ t0| ≤
≤ M B |β − β ′ | .
L = T 2 + M 2 B 2 .
π
4 t0
≤
∀ε > 0 ∃N0 ∀m, n >
N0

sup x∈(a,b) |um(x) − un(x)| < ε
sup x∈(a,b) |um(x) − un(x)| = sup x∈[a,b] |um(x) − un(x)|
{xk}k≥2 = {a + b−a
k }k≥2 ⊆ (a, b)
limk→∞ xk = a un
limk→∞ |un(xk) − um(xk)| = |un(a) − um(a)|
|un(a) − um(a)| ≤ sup x∈(a,b) |um(x) − un(x)|
b
(a, b)
[a, b]
{un(a)}
⎧
⎨
fn(x) =
⎩
2nx x ∈ [0, 1
−2nx + 2 x ∈ [ 1
2n ]
1
2n , n ]
0 x ∈ [0, 1
n ]
x ≤ 0 fn(x) ≡ 0 ∀n x > 0 n0 = [ 1
x ] + 1
∀ n ≥ n0 fn(x) = 0
∀ n ≥ 1 sup [0,1] |fn(x)| = 1
lim 1
0 fn = lim 1
2n = 0 = 1
0 f
⎧
⎨
fn(x) =
⎩
4n2x x ∈ [0, 1
2n ]
−4n2x + 4n x ∈ [ 1 1
2n , n ]
0 x ∈ [ 1
n , 1]
1 [x(
gn(x) =
n − x)]k+1 x ∈ [0, 1
n ]
0 fn = gn
1
0 gn
(−e α , e α )
K ⊂ (−e α , e α )
α ≤ 0
• (1, +∞)
• [a, +∞) a > 1

α > 0
• [0, 1)
• [0, a] 0 < a < 1
A ≡ { 1
n 2 | n ≥ 1}
[−1, 1] \ A
([−1, a] ∪
[0, 1]) ∩ A c −1 < a < 0
(−e −1 , e −1 )
K ⊂ (−e −1 , e −1 )
(0, +∞)
[a, +∞) a > 0
∀x ∈ (−1, 1),
n≥1 n3xn = x3 +4x 2 +x
(1−x) 4
x = 1
2
n≥1 n3
2n = 26
x |x| < 1
1
1−x
k≥0 xk = 1
1
1−x j 1−x
dj
dx j ( 1
1−x
) =
j!
(1−x) j+1
j!
|x| < 1 (1−x) j+1 = (k+j)!
k≥0 k! xk 1
∀ |x| < 1 :
k≥0
k + n − 1
n − 1
(1−x) j+1 =
x m
(1+x) n m 1 = x (1−x) n
m k + n − 1
= x k≥0
n − 1
xm+k =
n − 1 − m + k
k≥m
x
j
k
k + j
j
xk =
e x = +∞
k=0 xk
k!
∀ x ∈ R e x = +∞
k=0 xk
k!
e x
|e x −
N
k=0
xk |x|N+1
| ≤
k! (N + 1)! eξ N→∞
−→ 0 ∃ξ
limn→∞
k≥0
√
n!
2πn( n ) n = 1
e
(xe) n
√ 2πn+x
x k

log(1 + x) = x
0
1
1+tdt |t| < 1
1 1+t = ∞ k=0 (−1)ktk
−t
|x| < 1
log(1+x) =
x
0
1
dt =
1 + t
+∞
x
k=0
0
(−1) k t k ∞
k xk+1
dt = (−1)
k + 1 =
∞
(−1)
log(1−x) = − ∞ k=1 xk
k
x −x
log 1+x
1−x =
log (1 + x) − log (1 − x)
log
1 + x
1 − x =
∞
(−1)
k=1
k+1 xk
k +
∞
k=1
k=1
k=0
j=0
k=1
xk k =
∞
[(−1) k+1 +1] xk
∞ x
= 2
k 2j+1
2j + 1 .
+∞ α
k=0
( α ∈ R \ Z )
lim
k→∞
k
x k
k+1 xk
α
k + 1
α
= lim
k!α(α − 1) . . . (α − k)
k→∞ (k
+ 1)!α(α − 1) . . . (α − k + 1) = lim
α − k
k→∞ k + 1 = 1
k
lim sup α
k
1
k
= 1
R = 1
Dk (1+x) α |x=0 = α(α−1) . . . (α−k+1)(1+x)α−k |x=0 = α(α−1) . . . (α−k+1)
PN (x, 0) =
N α
k=0 x
k
k
∀ |x| < 1 (1 + x) α =
+∞ α
k=0 x
k
k
|(1 + x) α − PN−1(x, 0)| N→∞
−→ 0
x ∈ (−1, 1) |x| < θ < 1 ∃ ɛ > 0 θ(1 + ɛ) < 1
1
θ < 1 lim sup α k
= 1
N0 ∀ N ≥ N0 α
N
lim sup
1
N
≤ 1 + ɛ
k
k .

N ≥ N0
|(1+x) α − N−1
k=0
|x| N | 1
0
|x| N | 1
0
N|x| N
α
k
xk | = | x (x−s)
0
N−1
(N−1)! f (N) (s)ds| = | 1 (x−xt)
0
N−1
(N−1)! f (N) (tx)xdt| =
(1−t) N−1
(N−1)! α(α − 1) . . . (α − N + 1)(1 + tx)α−N dt| =
(1−t)N−1 α
N (N−1)! (1 + tx)
N
α−N dt| =
α
|
N
1
N−1
(1−t)
0 1+tx
N((1 + ɛ)θ) N (1 − θ) −|α−1| 1
0
N((1 + ɛ)θ) N −|α−1| N→∞
(1 − θ) −→ 0 .
(1 + tx) α−1 dt| ≤
N−1 (1−t)
1+tx dt ≤
+∞ x2k+1
k=0 (−1)k (2k+1)!
∀ x ∈ R sin x = +∞ x2k+1
k=0 (−1)k (2k+1)!
| sin x −
N
k=0
k x2k+1 |x|2N+3
(−1) | ≤
(2k + 1)! (2N + 3)! |f 2N+3 (ξ)| ≤ |x|2N+3 N→∞
−→ 0 .
(2N + 3)!
arcsin x = x
√ 1 dt
1−t2 ∀ |t| < 1
√ 1
1−t2 = ∞ k=0
arcsin x =
x
0
1
√ dt =
1 − t2 0 −1
∞
k=0
2
k
(−1) k t 2k = ∞
k=0
(2k − 1)!!
(2k)!!
x
0
t 2k dt =
(2k−1)!!
(2k)!!
t2k
∞
k=0
(2k − 1)!!
(2k)!!
x2k+1 2k + 1
arccos x = π
2 − arcsin x
arctan x = x
0
k=0
1
1+t2 dt
sinh x = −i sin(ix)
∞
k (ix)2k+1
sinh x = −i sin(ix) = −i (−1)
(2k + 1)! =
∞
2k+2 (x)2k+1
(−1)
(2k + 1)! =
∞ (x) 2k+1
(2k + 1)!
supt∈[0,1] (1 + tx) α−1 = supξ∈[−θ,θ](1 + ξ) α−1 ≤ (1 − θ) −|α−1|
1−t
t x 0 ≤ ≤ 1
1+tx
(−1)!! ≡ 1 , 0!! ≡ 0 , 1!! ≡ 1 n ≥ 2 n!! ≡
n(n − 2)!!
k=0
k=0

cosh x = cos(ix)
x = −i arcsin(ix)
x = −i arctan(ix)
x = 1
2
fn(x) = n
= n
=
x + 1
n − √
x
x + 1
n − √ x
1
x + 1
n + √ x
f(x) = 1
2 √ x
=
log 1+x
1−x
x + 1
n + √
x
=
x + 1
n + √ x
n→∞
−→ 1
2 √ x .
∀ x > 0
fn f E ⊆ (0, +∞) ⇔ lim
|fn(x) − f(x)| =
1
x + 1
n + √ x
− 1
2 √
=
x
n→∞ sup
E
1
2n √
x x + 1
n + √ 2
x
|fn(x)−f(x)| = 0 .
√
1
1 x+
−→ 0
(0, +∞) sup (0,+∞)
2n √ x
n +√x
[a, +∞) a > 0
sup
[a,+∞)
1
2n √
x x + 1
n + √ x
n≥1 un(x) :
2
=
1
2 = +∞ n→+∞
2n √
a a + 1
n + √ 2
a
n→+∞
−→ 0 .

• [0, +∞)
• [0, +∞)
• [0, +∞)
n≥1 u′ n(x) :
• (0, +∞)
• [a, +∞) a > 0
• [a, +∞) a > 0
u ∈ C 1 ((0, +∞)) u ′ (x) = v(x)
•
exp z ≡
∞
k=0
z k
k!
∀ z ∈ C
•
∞
∞
k z2k
k z2k+1
cos z ≡ (−1) sin z ≡ (−1)
(2k)! (2k + 1)!
k=0
k=0
∀ z ∈ C
• π A
A ≡ {x > 0 : cos x = 0}.
cos 0 = 1
cos 2 < − 1
3
π = 2α1 α1 ≡ inf A
exp(iπ) = −1
∀ z ∈ C e iz = cos z + i sin z
e iz =
∞
k=0
(iz) k
k! =
4k+3 z4k+3
i
(4k + 3)!
∞
k=0
k zk
i
k! =
∞
k=0
∞
4k z
=
(4k)!
k=0
4k z4k z4k+1
z4k+2
i + i4k+1 + i4k+2
(4k)! (4k + 1)! (4k + 2)! +
z4k+1 z4k+2 z4k+3
+ i − − i =
(4k + 1)! (4k + 2)! (4k + 3)!

∞
4k z
k=0
(4k)!
k=0
z4k+2
−
(4k + 2)!
k=0
+ i
∞
4k+1 z
k=0
(4k + 1)!
z4k+3
− =
(4k + 3)!
∞
∞
k z2k
k z2k+1
= (−1) + i (−1) ≡ cos z + i sin z.
(2k)! (2k + 1)!
∀ z ∈ C cos z = cos(−z) sin z = − sin(−z)
cos sin
∞
k=0 (−1)k (−z) 2k
(2k)! = ∞ z2k
k=0 (−1)k (2k)!
∞ k=0 (−1)k (−z) 2k+1
(2k+1)! = − ∞ z2k+1
k=0 (−1)k (2k+1)!
∀ z, w ∈ C exp(z + w) = exp(z) exp(w)
∀ z ∈ C cos2 z + sin 2 z = 1
cos sin
cos 2 z+sin 2 z = (cos z+i sin z)(cos z−i sin z) = (cos z+i sin z)(cos (−z)+i sin (−z)) =
= e iz e −iz = e 0 = 1 .
sin(2z) = 2 sin z cos z cos(2z) = cos 2 z − sin 2 z ∀ z ∈ C
sin cos
sin 2z = ∞ k=0 (−1)k 2k+1 z2k+1 2
2 sin z cos z
∞
2 sin z cos z = 2
k=0
k z2k+1
(−1)
(2k + 1)!
∞
(2k+1)!
k=0
k z2k
(−1)
(2k)!
α2k+1 =
k
n=0
(−1) n
(2n)!
= 2
∞
k=0
α2k+1z 2k+1
(−1) k−n (−1)k
=
(2(k − n) + 1)! (2k + 1)!
(∗)
k
n=0
=
(2k + 1)!
(2n)!(2k + 1 − 2n)! =

k
2k + 1
2n
= (−1)k
(2k + 1)!
n=0
2k + 1 2k + 1
=
2n 2k + 1 − 2n
= (−1)k 1
(2k + 1)! 2
2k+1
n=0
2k + 1
n
=
1
(2k + 1)!
= (−1) k
2 22k+1
e iπ = cos π + i sin π = cos 2 π
π π π π
+ i sin 2π = cos2 − sin2 + 2i cos sin
2 2 2 2 2 2 =
π cos π
2 = 0
2 π π
= − sin = cos2 − 1 = −1
2 2
φɛ R
f ∈ L 1 (R) C ∞ f ∈ C ∞ 0
+∞
φɛdx =
−∞
ɛ
γ
0
γ(x) ≡
φɛdx ≤ φɛ∞ɛ < ∞ .
x
−∞ φɛdx
+∞ .
φɛdx −∞
φɛ C ∞ φɛ ∀k ≥
1 γ (k) (x) = φ(k−1)
ɛ (x)
+∞
−∞ φɛdx
2k+1 2k + 1
n=0
= (1 + 1)
n
2k+1 = 22k+1
φɛ [0, ɛ]

γ φɛ ≥ 0
x
−∞ x ≤ 0 γ(x) =
φɛdx
+∞
−∞ φɛdx = (−∞, x) ∩ φɛ = ∅
0
+∞
φɛdx −∞
= 0
x
−∞ x ≥ ɛ γ(x) =
φɛdx
+∞
−∞
+∞ =
φɛdx −∞ φɛdx
+∞ = 1
φɛdx −∞
(−∞, x] ⊇ φɛ x
−∞ φɛdx = +∞
• x = 0 γ(0) = 0 ∀ k ≥ 1 γ (k) (0) = 0 φɛ C∞
−∞ φɛdx
[0, ɛ] x = 0
• x = ɛ γ(ɛ) = 1 ∀ k ≥ 1 γ (k) (0) = 0 φɛ C ∞
[0, ɛ] x = ɛ
≡ 0 ≡ 1
(⇒) : M(t) ∀ i, j Mij(t)
∀ ɛ > 0 , ∃ δij |Mij(t) − Mij(t0)| < ɛ , ∀ |t − t0| <
δij δ = mini,j δij
(⇐) : ∀ i, j |Mij(t) − Mij(t0)| < M(t) − M(t0)
f ∈ C ∞ ({y0}, R 2 )
f ′ (y0) =
1 + 2y1 cos y2 −y 2 1 sin y2
2y1
1
|y0=(0,0)
=
1 0
0 1
det f ′ (y0) = 1 = 0 ⇒ ∃! f
C ∞ ({f(y0)}, R 2 )

I − T f ′ (y)∞,∞ = max {|2y1 cos y2| + |y 2 1 sin y2| , |2y1|} ≤ 3ρ
ρ = 1
6 r = ρ 1
2T = 12
f(x0, g(x0)) = 0 ⇐⇒ g(x0) 2 − 6g(x0) − 16 = 0
⇐⇒ g(x0) = −2 g(x0) = 8
P+ = (1, −2, 8) e P− = (1, −2, −2).
∂ f
∂ y (P+) = (2y − 6) |y=8 = 10 = 0
∂ f
∂ y (P−) = (2y − 6) |y=−2 = −10 = 0
g±
C∞ x0 f(x, g±(x)) = 0
g+(x0) = 8 e g−(x0) = −2 .
x0 g+
x0
∇g(x0) = 0
x0
f(x, g+(x)) = |x| 2 + g(x) 2 − 2x1 + 4x2 − 6g(x) − 11 ≡ 0 ,
0 ≡ ∇(f(x0, g+(x0))) =
= 2(x0) T
−2
+ 2g+(x0) ∇g+(x0) + − 6∇g+(x0) =
4
1
−2
= 2 + 16 ∇g+(x0) + − 6∇g+(x0) =
−2
4
= 10 ∇g+(x0)

∇g+(x0) = 0 x0
Hg+ (x0)
0 ≡ ∂2
f(x, g(x)) =
∂ x2
2 0
=
+ (2g+(x0) − 6)Hg+ 0 2
(x0) +
∂x1g+(x0)∂x1g+(x0) ∂x1g+(x0)∂x2g+(x0)
+ 2
=
∂x2g+(x0)∂x1g+(x0) ∂x2g+(x0)∂x2g+(x0)
2 0
=
+ 10 Hg+(x0) ,
0 2
Hg+ (x0) =
⎛
⎜
⎝
− 1
0
5
0 − 1
5
f
f ∈ C ∞ (R n , R n )
∂f
∂x (x) = In + 2(cos |x| 2 )A Aij ≡ (vixj)
∂f
∂x (0) = In ,
det ∂f
∂x (0) = det In = 1 .
g f g ∈ C ∞ (Br(0))
r > 0 r
ρ
sup
In
∂f
− In
∂x
|x|≤ρ
≤ 1
2
∂fi
= δi,j + 2vixj cos |x|
∂xj
2 δi,j
1 se i = j
δi,j =
0 se i = j
⎞
⎟
⎠

g Br(0)
r ≡ ρ ρ
=
2 In 2 .
ρ
In
∂f
− In
∂x
= In − In − 2(cos |x| 2 )A ≤
≤ 2A
A = max {|v1|x1, . . . , |vn|x1} ≤
≤ v∞ x1 ≤ √ n v∞ |x| .
sup
In
∂f
− In
∂x
|x|≤ρ
≤ sup 2
|x|≤ρ
√ n (v∞ |x|) ≤
≤ 2 √ n v∞ ρ
1
ρ ≤
4 √ nv∞
1
r ≤
8 √ .
nv∞
r → +∞ v → 0
v → 0 f
F (x, y) = f(x) − y
R n
A = sup
x∈Rn x∞=1
Ax∞
x∞
=
= sup
x∈Rn x∞=1
Ax∞.
⎧ ⎫
⎨ n ⎬
A = max |aij|
k=1, ..., n ⎩ ⎭
j=1

(x, y) =
(0, 0)
F (0, 0) = f(0) − 0 = 0
∂ F
∂ x (0, 0) = In + 2(cos |x| 2 )A
= In
(x,y)=(0,0)
(0, 0)
g y = 0
F (g(y), y) = 0 ⇐⇒ f(g(y)) = y
g f x = 0
r > 0 g Br(0)
r, ρ > 0
sup |F (0, y)| ≤
Br(0)
ρ
con T ≡
2T
sup
In
∂ F
− In (x, y)
1
∂ x ≤
2 .
Bρ(0)×Br(0)
sup |F (0, y)| = sup |y| = r ,
Br(0)
Br(0)
−1 ∂ F
(0, 0) = In
∂ x
r ≤ ρ
2
A
sup
Bρ(0)×Br(0)
In
∂ F
− In (x, y)
∂ x
= sup
Bρ(0)×Br(0)
In − In − 2(cos |x| 2 )A =
= sup
Bρ(0)×Br(0)
(2A) ≤
≤ sup
Bρ(0)×Br(0)
√
2 nv∞|x| ≤
≤ 2 √ nv∞ρ
1
ρ ≤
4 √ nv∞
g
1
r ≤
8 √ .
nv∞

g(x, y) = e x2 +y 2
− x 2 − 2y 2 + 2 sin y − 1 .
g(0, 0) = 0
∂ g
(x, y) = 2y e
∂ y x2 +y 2
− 2 + 2 cos y
∂ g
(0, 0) = 2 = 0 .
∂ y
y = f(x)
x = 0 Br(0) r > 0
g(x, f(x)) = 0 ∀ x ∈ Br(0) e f(0) = 0 .
f r
r, ρ > 0
sup |g(x, 0)| ≤
Br(0)
ρ
con T ≡
2|T |
sup
∂ g
1 − T (x, y)
1
∂ y ≤
2 .
Br(0)×Bρ(0)
−1 ∂ g
(0, 0) =
∂ y 1
2
r < 1
sup |g(x, 0)| = sup |e
Br(0)
Br(0)
x2
− x 2 − 1| ≤
≤
sup |e x2
− 1| + |x| 2
≤
Br(0)
≤ sup
Br(0)
≤ sup
Br(0)
e r2
|x| 2 + |x| 2
≤
e r2
+ 1 |x| 2 ≤
th.
2
≤ 4 r ≤ ρ .

ρ < 1
sup
Br(0)×Bρ(0)
∂ g
1 − T (x, y)
∂ y
ρ ≤ 1
14
r 2 ≤ 1 1
=
4 · 14 56
≤ sup
≤ sup
Br(0)×Bρ(0)
+ 2y − cos y
≤
1 − ye
Br(0)×Bρ(0)
x2 +y 2
|1 − cos y| + |y| |e x2 +y 2
− 2| ≤
≤ sup
Br(0)×Bρ(0)
(|y| + 6|y|) ≤
≤ sup
Br(0)×Bρ(0)
7|y| ≤
≤ 7ρ th.
≤ 1
2 ,
⇐⇒ r ≤ 1
√ 56 .
f ′ ∂ g
(x, f(x))
(x) = − ∂ x =
∂ g
(x, f(x))
∂ y
= −
2x
f(x)
lim
x→0 x2 f
= lim
x→0
′ (x)
2x =
= − lim
x→0
= 0 .
e x2 +f 2 (x) − 1
2f(x) e x2 +f 2 (x) − 2 + 2 cos f(x)
e x2 +f 2 (x) − 1
2f(x) e x2 +f 2 (x) − 2 + 2 cos f(x) =

R n
R n
• q = m
n 0 <
m ≤ n mn f q
{αn} ⊂ [0, 1]\Q αn
n→+∞
−→ q
[0, 1]
lim
n→+∞ f(αn) = 0 = 1
= f(q).
n
• f q = 0
n→+∞
{βn} ⊂ [0, 1] βn −→ 0 ∀ε > 0, ∃N : n >
N, 0 < βn < ε βn
f(βn) = 0 βn = mn
n > N ε >
kn
βn = mn 1 ≥ = f(βn)
kn kn
∀ε > 0, ∃N : n > N, 0 ≤ f(βn) < ε
• (0, 1] \ Q
α ∈ (0, 1] \ Q {βn} ⊂ [0, 1] βn
q = m
n
n→+∞
−→ α
∀ε > 0 qj
f(qj) ≥ ε. ∃N : n > N, 0 < f(βn) < ε
m, n
f(q) = 1
1
≥ ε ⇐⇒ n ≤
n ε
1
Φ = ϕ(1) + ϕ(2) + . . . + ϕ < ∞
ε
ϕ(n) n n

limn→+∞ f(βn) = 0 = f(α)
f ∈ R([0, 1]) ∀ε > 0, ∃f1, f2
f1(x) ≤ f(x) ≤ f2(x) ∀x ∈ [0, 1]
[0,1] (f2 − f1) < ε
• f1 ≡ 0 f1(x) ≤ f(x) ∀x ∈ [0, 1]
• f2
N ≥ 3 qj = mj
nj f(qj) >
1
N Φ δ ≡ inf1≤i 0 bj ≡ min{δ, 1
2hjNΦ
1
nj
f2(x) =
1
N
x ∈ (qj − bj, qj + bj)
}
bj f2
(f2 − f1) =
[0,1]
1
0
f2 = 1
N +
Φ
j=1
N
hj2bj ≤ 1
N +
Φ 1
hj2
2hjNΦ
j=1
= 2
N .
A ⇐⇒ χA
⇐⇒ ∀ ε > 0, ∃f1, f2
f1(x) ≤ χA(x) ≤ f2(x)
Rn(f2 − f1) < ε
f1(x) = N1
n=1 χ R 1 n (x) {R1 n} N1
n=1
f2(x) = N2
n=1 χ R 2 n (x) {R2 n} N2
n=1
f1 f2 {0, 1} f1 E1 = ∪ N1
n=1 R1 n
f2 E2 = ∪ N2
n=1 R2 n E1, E2
f1(x) ≤ χA(x)
x ∈ E1 =⇒ f1(x) = 1 =⇒ χA(x) = 1 =⇒ x ∈ A
E1 ⊂ A
A ⊂ E2
nE2 −nE1 =
Rn(f2 − f1) < ε

∂A
∂A
χA
∀ ε > 0, ∃ {En}n≥1
∂A
n≥1nEn < ε A =⇒ ∂A
{E ′ n} N n=1
N n=1nE ′ n ≤
n≥1nEn < ε ∂A
A = A ∪ ∂A o
A= A \ ∂A
Qn ∩ E
Qn ∩ E E
Qn ∩ E = {qj}j≥1 qj = (q1 j , . . . , qn j )
∀ ε > 0, {Qj}j≥1
Qj = (q 1 j − 1
ε
2 2j 1
n
, q 1 j + 1
ε
2 2j 1
n
) × . . . × (q n j − 1
ε
2 2j 1
n
, q n j + 1
Qj n nQj =
ε
2j {Qj}j≥1 Qn ∩ E
+∞
+∞
nQj =
j=1
j=1
ε
= ε.
2j 2
2 1
ε
2 2j ε
2 j
1
n
).
1 n
n =
E1 E1 ⊂ Qn ∩ E Qn ∩ E
E1
nE1 = 0
sup{nE1 : E1 ⊂ Q n ∩ E } = 0.
E2 ⊃ Qn ∩ E o
E⊂ E2
p ∈ o
E p ∈ E2 =⇒ (E2) c ∃ Dn r (p) ⊂ o
E⊂ E
Dn r (p) ∩ E2 = ∅
F ⊂ P(R n )
∅ ∈ F
A ∈ F =⇒ Ac ∈ F
{An} N n=1 ⊂ F =⇒ ∪N n=1An ∈ F

D n r (p) Q n ∩ E
Q n E E2 ⊃ Q n ∩ E
nE2 = nE2 ≥ n
o
E= nE > 0
inf{nE2 : E2 ⊃ Q n ∩ E } ≥ nE > 0.
Qn ∩ E
∀ E1 ⊂ Qn ∩ E ⊂ E2
nE2 − nE1 ≥ nE > 0
o
X= ∅
x0 ∈ X Dr(x0) ⊂ X
0 = (X) ≥ (Dr(x0)) > 0
X = [0, 1] \ Q
o
X= ∅ X
1 = ([0, 1]) = (X ∪ ([0, 1] ∩ Q)) = (X) +([0, 1] ∩ Q) = 0
∀ ɛ > 0
{Xn}n , {Ym}m ⊂ R
Qx ⊂ ∪nXn nXn
< ɛ
Q y ⊂ ∪mYm
mYm < ɛ.
Qx × Qy
{Xn × Ym}n,m
Qx×Q y ⊂ ∪n,mXn × Ym
(Xn × Ym) =
((Xn)·(Ym)) =
n,m
(Xn) ·
(Ym) < ɛ
n
m
2
n
m

Q = {0} × R y
x = 0 Q0 = R
l =
limn xn
ɛ > 0
{Rn} N n=1 N n=1(Rn) < ɛ
∀ ɛ > 0, ∃ N > 0 xn ∈ [l − ɛ, l + ɛ] n ≥ N
Rk = {xk} k < N
RN = [l − ɛ, l + ɛ]
{xn = n}n [0, 1]
9 i) 4
π ii) 24
e 1 iii) 4 − 2
1 iv) 12
D ≡ {0 ≤ y ≤ 1, y2 ≤ x ≤ 1} = {0 ≤ x ≤ 1, 0 ≤ y ≤ √ x}
D ≡ {0 ≤ x ≤ 1, 1 − x ≤ y ≤ √ 1 − x2 }
1 1
1
x − y
dy
dx = −
(x + y) 3
0
0
0
1
dy
0
y − x
dx =
(x + y) 3
1
0
dx ′
1
0
x ′ − y ′
(x ′ + y ′ dy′
) 3
x ′ = y
y ′ = x

D
x2 dx dy =
y2 2 x
x
dx
1
1
x
2 2
dy = x
y2 1
2 x
1
dx dy =
1 y2 x
2
= x
1
2
− 1
x 2
dx = x
y 1
1
x
2
x − 1
dx =
x
2
= (x
1
3 2
4 x x2
− x) dx = − =
4 2 1
= 9
4 .
D
x
D = (x, y) ∈ R 2 : 0 ≤ y ≤ 1, y 2 ≤ x ≤ 1 ;
y
D
3 e x dx dy =
=
1 1
dy
0
1
0
= e
4 −
y2 y 3 e x dy =
3
y e − e y2
dy =
y 2
2 ey2
= e e 1
− +
4 2 2
1
0
1
+
e y2 1
0
0 =
= e e e 1
− + −
4 2 2 2 =
= e 1
−
4 2 .
1
y 3 1
dy
0
y2 y 3 e x dy =
1
y 3 1
e dy −
0
e y2
y dy =
0
y 3 e y2
D
y
D =
(x, y) ∈ R 2 : 0 ≤ x ≤ 1, 1 − x ≤ y ≤ 1 − x 2
;
=

D
xy dx dy =
=
=
=
1
dx
√ 1−x2 xy dy =
0
1−x
1
x dx
√ 1−x2 y dy =
0
1
0
1
0
x
2
= 1 1
−
3 4 =
= 1
12 .
1−x
2 2
1 − x − (1 − x) dx =
(x 2 − x 3 ) dx =
0
y = x
a
y = a 2 x 2
=⇒ Aa =
1 1
,
a3 a4
y = ax
y = a2x2
1
=⇒ Ba = , 1 .
a
Ra
O Aa Ba
A(a) := Area(Ra) =
=
=
=
1
a 3
0
1
a3 0
1
6a
dx
ax
x
a
Ra
dy +
ax − x
dx +
a
1 − 1
a6
.
dx dy =
1
a
1
a 3
dx
ax
dy =
1
a3 a2x2 1
a 2 2
ax − a x dx =
lim
a→1 +
A(a) = 0 e lim A(a) = 0
a→+∞
a > 1
A ′ (a) = 1
6
− 1 7
+
a2 a8

a = ± 6√ 7
a = 6√ 7
Amax =
=
=
1
6 6√
1 −
7
1
=
7
1
6 6√ 6
7 7 =
1
7 6√ 7 .
⎧
⎨
⎩
x = aρ sin ϕ cos θ
y = bρ sin ϕ sin θ
z = cρ cos ϕ
(ϕ, θ) ∈ (0, π) × (0, 2π) 0 < ρ < 1
∂(x, y, z)
∂(ρ, ϕ, θ) = abcρ2 sin ϕ.
D
x 2 dx dy dz = 4
15 πa3 bc.
• B (1)
3 (0, 1) ≡ {x ∈ R3 : x1 = |x1|+|x2|+
|x3| ≤ 1}
B (1)
3 (0, 1) ∩ {xi ≥ 0, i = 1, 2, 3} = {x ∈ R3 : x1 + x2 + x3 ≤ 1} = {x ∈
R3 : 0 ≤ x3 ≤ 1, 0 ≤ x2 ≤ 1 − x3, 0 ≤ x1 ≤ 1 − x3 − x2}.
(B (1)
3 (0, 1)) =
B (1)
3 (0,1)
dx1dx2dx3 = 8
1 1−x3 1−x3−x2
= 8 dx3 dx2
0
0
0
B (1)
dx1dx2dx3 =
3 (0,1)∩{xi≥0, i=1,2,3}
dx1 = 4 23
=
3 3! .
• 4
B (1)
4 (0, 1) ≡ {x ∈ R4 : x1 = |x1| + |x2| + |x3| + |x4| ≤ 1} = {x ∈ R 4 :
−1 ≤ x4 ≤ 1, (x1, x2, x3) ∈ B (1)
3
(B (1)
4 (0, 1)) =
(0, 1 − |x4|)}
B (1)
4 (0,1)
dx1dx2dx3dx4 =
1
dx4
−1
B (1)
3 (0,1−|x4)|)
dx1dx2dx3 =

x ′ xi
i = 1−|x4| i = 1, 2, 3
1
= dx4
−1
B (1)
3 (0,1)
(1 − |x4|) 3 dx ′ 1dx ′ 2dx ′ 3 =
1
−1
4
3 (1 − |x4|) 3 dx4 = 24
4! .
• n (B (1)
n (0, 1)) = 2n
n!
D2 = {0 ≤ x2 ≤ 1, 0 ≤ x1 ≤ x2}
1 x2
I2 = dx2 x1x2 dx1 dx1 =
0 0
1 1
=
8 4!! .
D2(r) ≡ {0 ≤ x1 ≤ x2 ≤ r}
D3 = {0 ≤ x3 ≤ 1, (x1, x2) ∈ D2(x3)}.
x ′ i
I3 =
1
0
x3 dx3
x1x2 dx1 dx2 =
D2(x3)
1
0
= xi
x3 i = 1, 2
1
0
x3 dx3
x 5 3I2 dx3 = 1 1
=
48 6!! .
D2
x 4 3 x ′ 1 x ′ 2 dx ′ 1 dx ′ 2 =
In = 1
(2n)!!
(x, y) ∈ D D
R 2
x 2 + y 2 − 2 ≤ −1 e 4 − (x + y) ≥ 2 ,
x 2 + y 2 − 2 ≤ 4 − (x + y) ,
z = 4 − (x + y)
z = x 2 + y 2 − 2
R
R = {(x, y, z) ∈ R 3 : x 2 + y 2 ≤ 1, x 2 + y 2 − 2 ≤ z ≤ 4 − (x + y)}.
(2n)!! ≡ (2n) · 2(n − 1) · . . . · 2 = 2 n n!

Vol(R) =
=
=
dx dy dz =
4−(x+y)
dx dy dz =
R
D x2 +y2−2 2 2
4 − (x + y) − (x + y − 2) dx dy =
D
2 2
6 − (x + y) − (x + y ) dx dy = (∗)
D
2π
0 ,
(∗) =
0 cos θ dθ = 2π
1 2π
dρ ρ 6 − (ρ cos θ + ρ sin θ) − ρ 2 dθ =
0
0
1
= 2πρ
0
6 − ρ 2 dρ =
= 2π 6 1
1
− =
2 4
= 11
π .
2
0 sin θ dθ =
E (ρ, θ)
x ≥ 0 θ
− π
2
x = ρ cos θ y = ρ sin θ
≤ θ ≤ π
2 (x2 + y 2 ) 2 ≤ (x 2 − y 2 )
ρ 4 ≤ ρ 2 (cos 2 θ − sin 2 θ) ⇐⇒ ρ 2 ≤ cos 2θ .
θ
cos 2θ ≥ 0 − π π
≤ θ ≤ E
4 4
E (pol)
= (ρ, θ) : − π π
≤ θ ≤
4 4 , ρ2
≤ cos 2θ .
I IV x
(0, 0) (1, 0)
E

Area(E) =
=
= 1
2
= 1
4
E
π
4
− π
4
= 1
2 .
dx dy =
dθ
π
4
− π
4
E (pol)
√ cos 2θ
0
[sin 2θ] π
4
− π
4
cos 2θ dθ =
=
ρ dρ =
ρ dρ dθ =
Ek
E (pol)
k
=
(ρ, θ) : − π π
≤ θ ≤
4 4 , ρ2 ≤ k 2 cos 2θ
Ek = {(x, y) ∈ R 2 : x ≥ 0, (x 2 + y 2 ) 2 ≤ k 2 (x 2 − y 2 )} .
E
Area(Ek) =
=
= 1
2
Ek
π
4
− π
4
dx dy =
dθ
π
4
− π
4
E (pol)
k
√ k cos 2θ
0
= 1
4 [k2 sin 2θ] π
4
k 2 cos 2θ dθ =
− π
4
ρ dρ =
ρ dρ dθ =
= k2
2 .
G G
G = {(x, y, z) ∈ R 3 : x ≥ 0, (x 2 + y 2 ) 2 ≤ (1 − z 2 )(x 2 − y 2 ), |z| ≤ 1}
= {(x, y, z) ∈ R 3 : |z| ≤ 1, (x, y) ∈ E√ 1−z2} .
=
,

Vol(G) =
=
=
=
G
1
dz
−1
1
−1
1
−1
= 2
3 .
dx dy dz =
E√ 1−z 2
dx dy =
Area(E√ 1−z2)dz =
1 − z2 dz =
2
3 u = y − x
Φ(x, y) ≡
v = y + x 3 ⇐⇒ Φ −1 (u, v) ≡
∂(x, y)
∂(u,
v)
2
|v − u|− 3
=
3 3√ .
2
x = 3
v−u
2
y = v+u
2 .
T
Φ −1 (T ) ≡ {(u, v) : 0 ≤ u ≤ 2, 2 + u ≤ v ≤ 6 − u}.
x
T
2 (y − x 3 )e y+x3
dx dy = 1
6
= 1
6
= 1
6
= e6
6
Φ −1 (T )
u e v du dv =
2 6−u
du u e v dv =
0
2
0
− 2e4
3
2+u
u e 6−u − e 2+u du =
− e2
6 .
∆ D
z
∆ = {(ρ, θ, z) : θ ∈ (0, 2π), (ρ, z) ∈ D}.

(∆) =
2π
dx dy dz = dθ
ρ dρ dz = 2π x dx dz =
∆
0
D
D
D (xb, zb) =
1 x dx dz, dx dz D D
D z dx dz
x dx dz
D = 2π
dx dz = 2πxb(D).
dx dz
D
D
D
4
3πr3 2π2ar2 πr2h 1
3 πr2 h
πh
3 (R2 + rR + r 2 )
Φ −1 (D) ≡ {(ρ, θ, ϕ) : 0 ≤ ρ ≤ 1, 0 ≤ θ ≤ 2π, ρ cos ϕ ≥ ρ sin ϕ} =
= {(ρ, θ, ϕ) : 0 ≤ ρ ≤ 1, 0 ≤ θ ≤ 2π, 0 ≤ ϕ ≤ π
4 }.
ρ 2 sin ϕ π
3 (2 − √ 2)
D ≡ {(x, y, z) : x 2 + y 2 − 2y ≤ 0, 0 ≤ z ≤ 1
4 (x2 + y 2 )}
A ≡ {(x, y) : x 2 + y 2 − 2y ≤ 0}
x, y C = (0, 1, 0) 1
D
x
|yz| dx dy dz =
= 2
1
3
4 3
2
A
A
1
4 (x2 +y 2 )
0
x |y| √ z
x |y|(x 2 + y 2 ) 3
2 dx dy.
dx dy =
A T −1 (A) = {(ρ, θ) : 0 ≤ ρ ≤ 2 sin θ, 0 ≤
θ ≤ π} 0

Dk = {(x, y) : 0 < x < k, x < y < k}
f Dk k
xe −xy
dx dy = lim
k→∞
xe −xy √
π
dx dy =
2 .
D
Fp z0 ∈ (0, 1)
Fp z = z0
z = z0
x 2 + y 2 = z 2p
0
Cz0 = (0, 0, z0) z p
Dk
0
z0
z p
z = 1 1 p
• 0 < p < 1 z p z
Fp
p = 1 2
• p = 1
• p > 1 z p xy
Fp
z α Fp
p > 0 α ≥ 0
Fp
α Fp
{Kn}n
Fp Fp
Fp
n > 1
lim
n→∞
Kn ≡ {(x, y, z) ∈ R 3 :
Kn
z α dx dy dz .
1
n < z < 1 , x2 + y 2 ≤ z 2p } ;

Kn ⊂ Kn+1
n > 1 n Fp
n > 1
Kn
z α dx dy dz =
=
=
1
dz
1
n
1
1
n
1
1
n
x 2 +y 2 ≤z 2p
z α dx dy =
z α Area (B 2 zp(0)) dz =
z α πz 2p dz =
1
= π z α+2p dz =
=
⎧
⎨
⎩
1
n
π [log z] 1 1
π
n
α+2p+1
z
α+2p+1
1
1
n
se α + 2p = −1
se α + 2p = −1
π log n se α + 2p = −1
=
π
1
α+2p+1 1 − nα+2p+1
se α + 2p = −1 .
n → +∞
z
Kn
α ⎧
⎨ +∞ se α + 2p = −1
n→∞
dx dy dz −→ +∞ se α + 2p < −1
⎩ π
α+2p+1 se α + 2p > −1 .
zα Fp a > −1 − 2p
π
α+2p+1
R n
Γ
⎧
⎨
Φ(t) =
⎩
x(t) = t
y(t) = t 2
z(t) = t 3
t ∈ I = (1, 2).
Γ = Φ(I) Φ ∈ C 1 (I) ˙ Φ(t) = (1, 2t, 3t 2 ) =
(0, 0, 0) ∀ t ∈ I Γ
fdσ1 =
Γ
2
1
log t3
√ 1 + 4t2 + 9t4 dt = 3(log 4 − 1).
1 + 4t2 + 9t4

S = Φ(A) A = (a, b) × (0, α) Φ(t, θ) =
(u(t) cos θ, u(t) sin θ, v(t)) ∈ C 1 (A)
Γ (0, ∞) × R
γ(t) = (u(t), v(t))
∂Φ
∂t
A(S) =
dσ2 =
∂Φ
∂t
∧ ∂Φ
∂θ
dt dθ =
S
A
∂Φ ∧
∂θ = u(t) ˙u(t) 2 + ˙v(t) 2 u(t) > 0
b
= α u(t)
a
˙u(t) 2 + ˙v(t) 2 dt.
A(S) = α
Γ
x dσ1.
Γ l(γ)
A(S) = α l(γ) 1
x dσ1 = α l(γ) xb
l(γ) Γ
xb Γ
α Γ
Γ
γ
x(θ) = ρ(θ) cos θ = a (1 + cos θ) cos θ
θ ∈ (0, 2π) .
y(θ) = ρ(θ) sin θ = a (1 + cos θ) sin θ
γ
2π
lungh (γ) = ds := x
γ
0
′2
+ y ′2
dθ .
′ ′ ′ ′ x (θ) = ρ (θ) cos θ − ρ(θ) sin θy (θ) = ρ (θ) sin θ + ρ(θ) cos θ ,

x ′2 + y ′2 = (ρ ′ (θ) cos θ − ρ(θ) sin θ) 2 + (ρ ′ (θ) sin θ + ρ(θ) cos θ) 2 =
= ρ ′ (θ) 2 + ρ(θ) 2 =
= a2 sin 2 θ + a2 (1 + cos2 θ + 2 cos θ) =
= a √ 2 + 2 cos θ =
1 + cos θ
= 2a
=
2
= 2a
θ
cos
2
.
lungh (γ) =
γ
ds :=
2π
2π
= 2a
θ
cos
0 2
dθ =
π
= 4a cos
0
θ
dθ =
2
= 8a − sin θ
π =
2 0
= 8a .
0
x ′2 + y ′2 dθ =
∂T3 x, y
x = ρ cos θ y = ρ sin θ ρ ρ
z ρ, z
ρ = R + r cos ϕɛ, z = r sin ϕɛ ϕɛ ∈ (0, 2π) .
⎧
⎨ x = (R + r cos ϕɛ) cos th
X(θ, ϕɛ) ≡ y = (R + r cos ϕɛ) sin th
⎩
z = r sin ϕɛ
θ, ϕɛ ∈ (0, 2π) .
X C1
θ ϕɛ

θ = 0 ϕɛ = 0
Area (∂T 3 ) =
dprT 3
dσ :=
2π 2π
dθ Xθ ∧ Xϕɛ dϕɛ .
Xθ = (−(R + r cos ϕɛ) sin θ, −(R + r cos ϕɛ) cos θ, 0)
Xϕɛ = (−r sin ϕɛ cos θ, −r sin ϕɛ sin θ, −r cos ϕɛ)
Xθ ∧ Xϕɛ =
⎛
î
⎝ −(R + r cos ϕɛ) sin θ
−r sin ϕɛ cos θ
ˆj
−(R + r cos ϕɛ) cos θ
−r sin ϕɛ sin θ
kˆ
0
−r cos ϕɛ
= r (R + r cos ϕɛ) · (− cos θ cos ϕɛ, − sin θ cos ϕɛ, sin ϕɛ) .
Area (∂T 3 ) =
Xθ ∧ Xϕɛ = r (R + r cos ϕɛ) .
=
0
2π 2π
dθ Xθ ∧ Xϕɛ dϕɛ =
0
0
2π 2π
dθ r (R + r cos ϕɛ) dϕɛ =
0
0
0
⎞
⎠ =
= 4π 2 rR .
R2
A
f dx dy =
∂A
f · ν dσ1 ,
ν ∂ A .

(2, 0)
A
f dx dy =
=
=
A
A
π
= 1
3
= 2
3 .
[∂x(1 + xy) + ∂y(x)] dx dy =
y dx dy =
1
dθ ρ 2 sin θ dρ =
0
π
0
0
sin θdθ =
∂A
∂A1 = {(t, 0) 1 < t < 3} e ∂A2 = {(2 + cos θ, sin θ) 0 < θ < π} ;
∂A
∂A1 ∂A2
ν1 ν2
f · ν dσ1 =
∂A
3
= (−t) dt +
1
ν1 = (0, −1) e ν2 = (cos θ, sin θ) .
∂A1
π
f · ν1 dσ1 +
= 2
3 .
0
∂A2
f · ν2 dσ1 =
(1 + (2 + cos θ) sin θ, 2 + cos θ) · (cos θ, sin θ) dθ =
Φ +
∂Ω (F ) F
∂Ω
Φ +
∂Ω (F ) =
∂Ω
F · ν dσ2 ,
ν ∂Ω
F = 3
(Ω)
Φ +
∂Ω (F ) =
Ω
F dx = 3(Ω) = 3.

ω R3
f(x, y, z) = x4
4
+ y3
3
+ z2
2
ω R2 \{(0, 0)} ∂y
−4xy
(x 2 +y 2 ) 2 = ∂x
2y
x 2 +y 2
x
x2 +y2
= −2xy
(x2 +y2 ) 2 =
ω R 2
f(x, y) = x
1+y 2
ω ⇐⇒
A = D
B = −C .
ω R2 \ {(0, 0)}
=⇒
γ
ω = 0 ω
γ
γ
C 0 1
γ
ω =
C
ω .
ω ⇐⇒
C
ω = 0 .
ω ⇐⇒
A = D
B = C = 0
fA(x, y) = A log x 2 + y 2
ϕ
1 ω = 3
.

T
R3 ·1
(±1, 0, 0), (0, ±1, 0), (0, 0, ±1)
T
T xy
z T T (z) = {(x, y) : |x| + |y| ≤ 1 − |z|}
√ 2(1 − |z|)
|z| γ 1
dx dy dz = |z| γ
1
dz dx dy = |z| γ 2(1 − |z|) 2 dz =
T
−1
T (z)
1 γ γ+1 γ+2
= 4 z − 2z + z
0
dz =
γ > −1
1 2 1
8
= 4 − + =
γ + 1 γ + 2 γ + 3 (γ + 1)(γ + 2)(γ + 3) .
T |x|α dx dy dz
T |y|β dx dy dz
α > −1, β > −1 8
(α+1)(α+2)(α+3)
8
(β+1)(β+2)(β+3)
α > −1, β > −1
γ > −1
8
(α + 1)(α + 2)(α + 3) +
−1
8
(β + 1)(β + 2)(β + 3) +
8
(γ + 1)(γ + 2)(γ + 3) .
F F = 3
T 3 dx dy dz = 33(T ) = 4
T 4
3 1)
γ = 0
Σ
Σ = {(x, y, 1−x−y), (x, y) ∈ D} D ≡ {(x, y) : x, y ≥ 0 x+y ≤ 1}.
Σ ν = 1
√ 3 (1, 1, 1)
Σ
F × ν dσ =
D
(1, 1, 1) × 1
√ 3 (1, 1, 1) √ 3 dx dy = 3(D) = 3
2 .

∂S = ∂S1 ∪
∂S2
∂S1 = {(cos t, sin t, 0), 0 ≤ t < 2π}
∂S2 = {(cos t, sin t, 1), 0 ≤ t < 2π} .
∂S
∂S1
∂S2
+∂S
ω =
=
=
+∂S1
ω +
ω −
−∂S2
+∂S1 +∂S2
2 2
cos t + sin t
2π
0
= π.
1
ω =
ω =
− cos2 t + sin 2
t
dt =
2
F (x, y, z) = (F1, F2, F3) =
−y
x2 + y2 ,
+ z2 F3 = 0
F = (−∂zF2, ∂zF1, ∂xF2 − ∂yF1).
S
x
x2 + y2 , 0
+ z2
.
S = {Φ(t, z) = (cos t, sin t, z) : t ∈ [0, 2π), z ∈ [0, 1]} ;
ν = (cos t, sin t, 0)
∂tΦ ∧ ∂zΦ = 1
∂S
ω =
=
F × ν dσ =
S
2π 1
dt
0
= π .
0
2z
(1 + z2 =
) 2
C
C = A ∪ B A ≡ D × [1, 4]
z = 1, z = 4 D R 2 B
{(x, y, z) ∈ R 3 : (x, y) ∈ D, x 2 + y 2 ≤ z ≤ 1}

xy z ∈ [0, 1] z C
C(z) ≡ {(x, y) ∈ R2 : x2 + y2 √
≤ z}
z πz z ∈ [1, 4] C(z)
D R2 π
4
(C) =
0
(C(z)) dz = 7
π ;
2
F = 1 C
C
F xy
C z = 4
F D × 1 A
C A
3π
π 2
∂C ∩{z = 4} π ∂C ∩{x2 +y2 = 1}
6π
1 3 ∂C ∩{z = x2 +y 2 }
(x, y) → (x, y, x2 +y2 )
(x, y) ∈ D 1 + |∇f(x, y)| 2dxdy
D
1 + |∇f(x, y)| 2 dxdy =
=
D
1 + 4x 2 + 4y 2 dxdy =
[0,1]×[0,2π]
7π + π
6 (5√ 5 − 1)
∂E = A ∪ B
A = ∂E ∩ {x 2 + y 2 + z 2 = 1}
r 1 + 4r 2 drdθ = π
6 (5√ 5 − 1) .
B = ∂E ∩ {z = x 2 + y 2 } .
A = {(cos θ sin ϕɛ, sin θ sin ϕɛ, cos ϕɛ) : θ ∈ (0, 2π), ϕɛ ∈ (0, π
) }
4
B = {(z cos θ, z sin θ, z) : θ ∈ (0, 2π), z ∈ (0, 1
√ ) } .
2

|A|2 =
|B|2 =
A
B
2π
dσ =
dσ =
0
√2
1
0
π
4
dθ
0
sin ϕɛ dϕɛ = 2π(1 − 1
√ 2 )
2π √ π
dz 2 z dθ = √2 ,
|∂E|2 = |A|2 + |B|2 = π(2 − 1
√ 2 ) .
F
B B
0 A
A
Φ +
A (F ) =
=
0
F × νe dσ = (x, y, z) × (x, y, z) dσ =
A
A
1 dσ = |A|2 = 2π(1 − 1
√ ) .
2
A
F
∂E
Φ +
∂E
(F ) = Φ+
A
(F ) + Φ+
B
(F ) = 2π(1 − 1
√ 2 ) .
2π(1 − 1
√ 2 ) = Φ +
∂E
=
E
E
(E) = 2π 1
3 (1 − √ ) .
2
(F ) = F × νe dσ =
∂E
F dxdydz = 3 dxdydz = 3(E) ;
E E
• x2 + y2 + z2 ≤ 1
1
• 0 ≤ z ≤ x2 + y2 z ≥ 0
z = x2 + y2
z

(ρ, ϕɛ, θ)
P = (x, y, z)
⎧
⎨
⎩
x = ρ sin ϕɛ cos θ
y = ρ sin ϕɛ sin θ
z = ρ cos ϕɛ ,
OP
ρ P −→
ϕɛ −→
OP z
ϕɛ 0 π
θ −→
OP xy
x θ 0 2π
x 2 + y 2 + z 2 ≤ 1 ⇐⇒ 0 < ρ ≤ 1
z ≥ 0 ⇐⇒ 0 ≤ ϕɛ ≤ π
z ≤ x 2 + y 2 ⇐⇒ cos ϕɛ ≤
2
⇐⇒ π
4 ≤ ϕɛ ≤ π
2 .
E =
sin 2 ϕɛ = | sin ϕɛ| = sin ϕɛ (0 ≤ ϕɛ ≤ π
2 )
(r, ϕɛ, θ) : 0 < ρ ≤ 1, π
4 ≤ ϕɛ ≤ π
, 0 ≤ θ ≤ 2π .
2
∂E
∂E1
∂E2
∂E3)
∂E1 1
xy π

∂E1
Ψ 1 (u, v) =
⎧
⎨
⎩
x = u cos v
y = u sin v
z = 0
0 < u < 1 0 < v < 2π
Ψ 1 u ∧ Ψ 1 v =
î
cos v
−u sin v
ˆj
sin v
u cos v
kˆ
0
0
Area (∂E1) =
= (0, 0, u) .
=
=
∂E1
dσ =
1 2π
du
0
0
Ψ 1 u ∧ Ψ 1 v
1 2π
du u dv =
0
0
1
= 2π u du =
= π .
0
=
dv =
∂E2 1
π
4 ≤ ϕɛ ≤ π
Ψ 2 (u, v) =
2 ∂E2
⎧
⎨
⎩
π
π
4 < u < 2 0 < v < 2π
Ψ 2 u ∧ Ψ 2 v
=
x = sin u cos v
y = sin u sin v
z = cos u
î ˆj ˆ k
cos u cos v cos u sin v − sin u
− sin u sin v sin u cos v 0
= (sin 2 u cos v, sin 2 u sin v, sin u cos u) .
=

Area (∂E2) =
=
=
=
=
=
∂E2
π
2
π
4
π
2
π
4
π
2
π
4
π
2
π
4
π
2
π
4
= 2π
= 2π
dσ =
2π
du
0
2π
du
0
2π
du
0
2π
du
0
Ψ 2 u ∧ Ψ 2 v
dv =
sin 4 u + sin 2 u cos 2 u dv =
sin 2 u sin 2 u + sin 2 u cos 2 u dv =
sin 2 u dv =
2π
du | sin u| dv =
π
2
π
4
π
2
0
| sin u| du =
sin u du =
π
4
= 2π [− cos u] π
2
π
4
= √ 2 π .
∂E3
x 2 + y 2 + z 2 = 1
z = x 2 + y 2
⇐⇒
⇐⇒
⇐⇒
=
x 2 + y 2 + ( x 2 + y 2 ) 2 = 1
z2 = x2 + y2 2 2 1
x + y = 2
z = x2 + y2 2 2 1
x + y = 2
z = 1 √
2
1
√ 2
∂E3
Ψ 3 (u, v) =
⎧
⎨
⎩
x = u cos v
y = u sin v
z = u

0 < u < 1
√ 2 0 < v < 2π
Ψ 3 u ∧ Ψ 3 v
Area (∂E3) =
=
=
=
=
∂E3
√2
1
0
√2
1
0
π
2
π
4
= 2 √ 2 π
î ˆj ˆ k
cos v sin v 1
−u sin v u cos v 0
= (−u cos v, −u sin v, u) .
dσ =
2π
du
0
2π
du
= √ 2 π u
= √ 2 π
2 =
= π √ 2 .
0
Ψ 3 u ∧ Ψ 3 v
dv =
=
u 2 cos 2 v + u 2 sin 2 v + u 2 dv =
2π √
du 2 u dv =
0
√2
1
0
√
1
2 2
0
∂E
u du =
Area (∂E) = Area (∂E1) + Area (∂E2) + Area (∂E3) =
=
= π + √ 2 π + π √ 2 =
= 3 + √ 2
√ 2
π .
F (x, y, z) = (x, y, z)
∂E Φ +
F (∂E)
Φ +
F (∂E) = Φ+
F (∂E1 ∪ ∂E2 ∪ ∂E3) =
= Φ +
F (∂E1) + Φ +
F (∂E2) + Φ +
F (∂E3) ;

Φ +
F (∂E1)
Ψ 1 u ∧ Ψ 1 v
Ψ 1 u ∧ Ψ 1 v
1
= (0, 0, u) =
u
= (0, 0, 1) .
ˆn1 = (0, 0, −1)
Φ +
F (∂E1)
= F · ˆn1 dσ =
∂E1
= (x, y, z) · (0, 0, −1) dσ =
∂E1
= − z dσ =
∂E1
= 0 ,
z ∂E1
Φ +
F (∂E2)
Ψ 2 u ∧ Ψ 2 v
Ψ 2 u ∧ Ψ 2 v
=
1
sin u (sin2 u cos v, sin 2 u sin v, sin u cos u) =
= (sin u cos v, sin u sin v, cos u) ,
ˆn2 = (sin u cos v, sin u sin v, cos u) = (x, y, z) .
Φ +
F (∂E2) =
=
=
=
∂E2
∂E2
∂E2
∂E2
F · ˆn2 dσ =
(x, y, z) · (x, y, z) dσ =
x 2 + y 2 + x 2 dσ =
1 dσ =
= Area (∂E2) =
= √ 2 π .
∂E1
xy z

Φ +
F (∂E3)
Ψ 3 u ∧ Ψ 3 v
Ψ 3 u ∧ Ψ 3 v
1
= (−u cos v, −u sin v, u) =
u
= (− cos v, − sin v, 1) .
ˆn3 = (− cos v, − sin v, 1) .
∂E3
F (x, y, z) · ˆn3 = (u cos v, u sin v, u) · (− cos v, − sin v, 1) =
= −u cos 2 v − u sin 2 v + u =
= −u + u =
= 0 .
F (x, y, z)
F
∂E3
Φ +
F (∂E3) =
= 0 .
=
∂E3
∂E2
F · ˆn3 dσ =
0 dσ =
Φ +
F (∂E) = Φ+
F (∂E1 ∪ ∂E2 ∪ ∂E3) =
= Φ +
F (∂E1) + Φ +
F (∂E2) + Φ +
F (∂E3)
=
=
0 + √ 2 π + 0 =
= √ 2 π .
E

F
Φ +
F (∂E) =
=
=
Vol (E) = 1
F · ˆn dσ =
∂E
F dx dy dz =
E
3 dx dy dz =
E
= 3 Vol (E) .
3 Φ+
F
√
2
(∂E) = π .
3
1
ω(x, y) = A(x, y) dx + B(x, y) dy ≡
≡ (y3 − x 2 y) dx + (x 3 − y 2 x) dy
(x 2 + y 2 ) 2
R 2 \ {0} 1
R 2
lim
n→+∞ A
0, 1
n
= lim
n→+∞
= lim
n→+∞
= lim
n→+∞
= lim
n→+∞
3 2 y − x y
(x2 + y2 ,
) 2
1
n 3
1
n 4
n 4
=
=
n3 n =
x=0
y= 1 n
= +∞ .
ω
∂y A(x, y) = ∂x B(x, y) .
,
=

∂y A(x, y) = (3y2 − x 2 )(x 2 + y 2 ) 2 − (y 3 − x 2 y) 2(x 2 + y 2 ) 2y
(x 2 + y 2 ) 4
= (3y2 − x 2 )(x 2 + y 2 ) − (y 3 − x 2 y) 4y
(x 2 + y 2 ) 3
= 6x2 y 2 − x 4 − y 4
(x 2 + y 2 ) 3
∂x B(x, y) = (3x2 − y 2 )(x 2 + y 2 ) 2 − (x 3 − y 2 x) 2(x 2 + y 2 ) 2x
(x 2 + y 2 ) 4
= (3x2 − y 2 )(x 2 + y 2 ) − (x 3 − y 2 x) 4x
(x 2 + y 2 ) 3
= 6x2 y 2 − x 4 − y 4
(x 2 + y 2 ) 3
.
ω
ω 1 R2 \ {0}
α > 0 γα
Φα(θ) =
x = α cos θ
y = α sin θ ,
=
=
=
=

0 ≤ α < 2π
2π
ω =
γα
=
=
=
=
=
=
=
0
(α 3 sin 3 θ − α 3 cos 2 θ sin θ)
α 4
+ (α3 cos3 θ − α3 sin 2 θ cos θ)
α4 3 3 3 2 (α sin θ − α cos θ sin θ)
2π
0
α 4
+ (α3 cos 3 θ − α 3 sin 2 θ cos θ)
2π
0
2π
0
2π
0
2π
0
2π
0
α 4
(cos 4 θ − sin 4 θ) dθ =
d(α cos θ) +
d(α sin θ) =
(cos 2 θ cos 2 θ − sin 2 θ sin 2 θ) dθ =
(−α sin θ) +
(α cos θ) dθ =
[cos 2 θ cos 2 θ − (1 − cos 2 θ) sin 2 θ] dθ =
[cos 2 θ cos 2 θ − sin 2 θ + cos 2 θ sin 2 θ] dθ =
[cos 2 θ − sin 2 θ] dθ =
2π
cos 2θ dθ =
0
= 1
[sin 2θ]2π 0 2 =
= 0 .
γ
R
γ 0 ∈ R
α
R Bα(0) ⊂ R
∆ = R \ Bα(0) ⊂ R 2 \ {0} ,
∂∆
ω =
∂∆ = γ ∪ ∂Bα(0) = γ ∪ γα .
∆
[∂yA(x, y) − ∂xB(x, y)] dx dy ;

ω
∆
[∂yA(x, y) − ∂xB(x, y)] dx dy =
+∂∆
ω =
+γ
ω −
= 0
+γα
∆
ω .
0 dx dy =
+γ
ω =
+γα
ω = 0 .
γ
ω = 0 γ
γ1 γ2
ω =
γ1
γ2 ω
γ
ω = 0
γ
1
R 2 \{0}
ω 1
γ
R 2
Γ
0 ∞
R 2 \ Γ γ

ω 1 R 2 \ {0}
f(x, y) ω = df
⎧
⎪⎨
⎪⎩
fx(x, y) = A(x, y) = y3 − x 2 y
(x 2 + y 2 ) 2
fy(x, y) = B(x, y) = x3 − y2x (x2 + y2 .
) 2
f(x, y)
g(ρ, θ) ≡ f(ρ cos θ, ρ sin θ) ,
f
g
gρ ≡ fx(ρ cos θ, ρ sin θ) cos θ + fy(ρ cos θ, ρ sin θ) sin θ =
= r3 sin 3 θ − r 3 cos 2 θ sin θ
r 4
+ r3 cos 3 θ − r 3 sin 2 θ cos θ
r 4
cos θ +
sin θ =
= sin3 θ cos θ − cos 3 θ sin θ + sin θ cos 3 θ − sin 3 θ cos θ
= 0
gθ ≡ fx(ρ cos θ, ρ sin θ) (−ρ sin θ) + fy(ρ cos θ, ρ sin θ) (ρ cos θ) =
= r3 sin 3 θ − r 3 cos 2 θ sin θ
r 4
+ r3 cos 3 θ − r 3 sin 2 θ cos θ
r 4
r
(−ρ sin θ) +
(ρ cos θ) =
= (sin 3 θ − cos 2 θ sin θ) (− sin θ) + (cos 3 θ − sin 2 θ cos θ) (cos θ) =
= − sin 4 θ − cos 2 θ sin 2 θ + cos 4 θ − sin 2 θ cos 2 θ =
= cos 4 θ − sin 4 θ =
= (cos 2 θ − sin 2 θ)(cos 2 θ + sin 2 θ) =
= cos 2 θ − sin 2 θ =
= cos 2θ .
g ρ
g(ρ, θ) =
sin 2θ
2
.
=

f
g(ρ, θ) =
sin 2θ
2
= 2 sin θ cos θ
2
=
= sin θ cos θ = r2 sin θ cos θ
r2 =
= (r sin θ) (r cos θ)
r2 ;
f(x, y) = xy
x2 .
+ y2
ω
c0 = 1
π 2|x|
2π (1 − −π π )dx = 0
cn = 1
π 2|x|
2π (1 − −π π )e−inxdx = 2
π2n2 (1 − (−1) n ) .
f C1 R
f
2
n=0 π2 (2n+1) 2 < ∞
f(x) =
cne inx =
=
n=0
∞
n=0
n∈Z
4
π 2 (2n + 1) 2 einx =
8
π2 cos (2n + 1)x .
(2n + 1) 2
f
•
f
1 = f(0) =
∞
n=0
8
π 2 (2n + 1) 2
=⇒
∞
n=1
1 π2
=
(2n + 1) 2 8 ;

•
•
•
∞
n=1
1
n 2
=
∞
n=1
= 1
4
1
+
(2n) 2
∞
n=1
∞
n=1 1
n 2 = π2
6
∞
n=0
1 π2
+
n2 8
1
=
(2n + 1) 2
f 2 2 = 1
π
(1 −
2π −π
2|x|
π )2dx = 1
3
|cn| 2 =
∞
n=1
n
1
n 4
n≥0
n≥0
32
π4 ;
(2n + 1) 4
1 π4
=
(2n + 1) 4 96 .
=
∞
∞ 1
1
+
=
(2n) 4 (2n + 1) 4
n=1
n=0
= 1
∞ 1 π4
+
16 n4 96
n=1
∞
n=1 1
n 4 = π4
90
u(0, t) = u(π, t) = 0
u(x, t) =
cn(t) sin nx .
n≥1
cn
f(x) = x [−π, π]
ˆ fn {sin nx}n
ut − uxx = 0 ⇐⇒ c ′ n(t) + n 2 cn(t) = 0 ∀ n ≥ 1

cn(t) cn(t) = Ane −n2 t
u(x, t) =
n≥1
Ane −n2 t sin nx
An
u(x, 0) = f(x) =
ˆfn sin nx ⇐⇒ An = ˆ fn .
n≥1
u(x, t) =
n≥1
ˆfne −n2 t sin nx .
ˆ fn
ˆfn = 2
π
π
0
x sin nx dx = 2
n (−1)n+1 .
u(x, t) =
n≥1
2
n (−1)n+1 e −n2 t sin nx .
x t
u(x, t) = X(x)T (t)
X(x)T ′ (t) − X ′′ ⇐⇒
(x)T (t) = 0
T ′ (t)
T (t) − X′′ (x)
= 0
X(x)
⇐⇒ T ′ (t)
T (t) = X′′ (x)
X(x)
−µ
′
T (t)
T (t)
X ′′ (x)
X(x)
= −µ
= −µ .

X ′′ (x)
X(x)
= −µ
X(0) = X(π) = 0 ;
µ = 0 µ < 0
µ > 0
Xµ(x) = Aµ cos √ µx + Bµ sin √ µx
Xµ(0) = 0 =⇒ Aµ = 0
Xµ(π) = 0 =⇒ Bµ sin √ µπ = 0 =⇒ √ µ = n =⇒ µ = n 2 .
µ
Tn(t) = Ane −n2 t .
u
XnTn n
u(x, t) =
n≥1
Cne −n2 t sin nx
•
∆ u1, . . . , un
∆u = 0 v = u1 + . . . un

⎧
⎪⎨
a)
⎪⎩
⎧
⎪⎨
b)
⎪⎩
⎧
⎪⎨
c)
⎪⎩
∆u1 = 0 0 < x < π, 0 < y < π
u1(x, 0) = x 2 0 ≤ x ≤ π
u1(x, π) = 0 0 ≤ x ≤ π
u1(0, y) = 0 0 ≤ y ≤ π
u1(π, y) = 0 0 ≤ y ≤ π
∆u2 = 0 0 < x < π, 0 < y < π
u2(x, 0) = 0 0 ≤ x ≤ π
u2(x, π) = x 2 0 ≤ x ≤ π
u2(0, y) = 0 0 ≤ y ≤ π
u2(π, y) = 0 0 ≤ y ≤ π
∆u3 = 0 0 < x < π, 0 < y < π
u3(x, 0) = 0 0 ≤ x ≤ π
u3(x, π) = 0 0 ≤ x ≤ π
u3(0, y) = 0 0 ≤ y ≤ π
u3(π, y) = π 2 0 ≤ y ≤ π.
•
u = πx − v
v u
πx
∆u = −∆v v = πx − u
⎧
∆v = 0
⎪⎨ v(x, 0) = x(π − x)
0 < x < π, 0 < y < π
0 ≤ x ≤ π
v(x, π) = x(π − x)
⎪⎩
v(0, y) = 0
v(π, y) = 0
0 ≤ x ≤ π
0 ≤ y ≤ π
0 ≤ y ≤ π .
v(0, y) = v(π, y) = 0
v(x, y) =
cn(y) sin nx .
n≥1
cn
f(x) = x(π − x)
[−π, π] ˆ fn

{sin nx}n
∆v = 0 ⇐⇒ c ′′ n(y) − n 2 cn(y) = 0 ∀ n ≥ 1
v(x, 0) = f(x) =
n≥1 ˆ fn sin nx ⇐⇒ cn(0) = ˆ fn
v(x, π) = f(x) =
n≥1 ˆ fn sin nx ⇐⇒ cn(π) = ˆ fn
⎧
⎨
⎩
c ′′ n(y) − n 2 cn(y) = 0 ∀ n ≥ 1
cn(0) = ˆ fn
cn(π) = ˆ fn
cn(y) = an sinh ny + bn cosh ny ;
bn = ˆ fn
an sinh nπ + bn cosh nπ = ˆ fn
bn = ˆ fn
an = ˆ fn
u(x, y) = πx −
n≥1
1 − cosh nπ
sinh nπ
ˆfn(
1 − cosh nπ
sinh ny + cosh ny) sin nx
sinh nπ
ˆ fn
ˆfn = 2
π
π
ˆf2n = 0
0
x(π − x) sin nx dx = 4
πn 3 (1 − (−1)n ) .
ˆf2n+1 =
8
π(2n+1) 3
u(x, y) = πx −
n≥0
8 − cosh(2n + 1)π
(1 sinh(2n + 1)y +
π(2n + 1) 3 sinh(2n + 1)π
+ cosh(2n + 1)y) sin(2n + 1)x .
.

(1 + i) n + (1 − i) n =
=
n
j=0
n
j=0
[ n
2 ]
n
j
n
j
n
= 2
2j
j=0
i j +
n
j=0
n
j
(−i) j =
[ n
2 ]
i j (1 + (−1) j
n
) = 2
2j
(−1) j .
γ = (1 + i) n + (1 − i) n ∈ R Im γ = 0
[ n
2 ]
Re {(1 + i) n + (1 − i) n
n
} = 2
2j
j=0
j=0
(−1) j .
i i = e i log i π −(
= {e 2 +2πn) : n ∈ Z} =
(2n+1)π
−
= {e 2 : n ∈ Z}.
(−1) 2i = e 2i log(−1) = {e −2(π+2πn) : n ∈ Z} =
= {e −2(2n+1)π : n ∈ Z}.
(−1) 2i ⊂ ((−1) 2 ) i
i 2j =

• inf | sin z| = 0
4√
i(
i = {e π kπ
8 + 2 ) , k = 0, 1, 2, 3} =
π 5π 9π 13π
i
= {e 8 i
, e 8 i
, e 8 i
, e 8 }.
• sup | sin z| = +∞ | sin(in)| = | sinh(−n)| n→∞
−→ ∞
• infD | sin z| = 0
• sup D | sin z| = cosh R
| sin(x + iy)| = cosh 2 y − sinh 2 x
z → π
2 + iR
• infD z−i
z+i = 0 z = i
• supD z−i
z+i
{|z| < 1}
= 1
• infD |e z−i
z+i | = e −1
• sup D |e z−i
z+i | = e .
f Ω =⇒ f(z) :=
f(z) Ω
f = f f
z0 ∈ Ω
f(z0 + w) − f(z0)
lim
=
w→0 w
=
f(z0 + w) − f(z0)
lim
=
w→0 w
f(z0 + w) − f(z0)
= lim
w→0 w
=
= f ′ (z0)
f z0

f(z) = u(x, y) + iv(x, y)
Ω |f| 2 = u 2 + v 2 ≡ C
∂x(u 2 + v 2 ) = 0
∂y(u 2 + v 2 ) = 0
⇐⇒
uux + vvx = 0
uuy + vvy = 0
uux − vuy = 0
uuy + vux = 0
(u 2 + v 2 )(u 2 x + u 2 y) = C(u 2 x + u 2 y) = 0.
C = 0 f(z) ≡ 0
C = 0 u2 x + u2 y = 0 ux = uy = 0
u
vx = vy = 0 v
Re f = f v(x, y) ≡ 0 vx = vy = 0
ux = uy = 0 u
∆P (x, y) = 0
c = −3a
P
b = −3d
P (x, y) = ax 3 − 3dx 2 y − 3axy 2 + dy 3 .
vy = Px
vx = −Py
v(x, y) = dx 3 + 3ax 2 y − 3dxy 2 − ay 3 + cost.
f(z) = f(x + iy) = (a + id)z 3 + i cost.

2z + 3
z + 1
1
=
2
= ((z − 1) + 5
2 )
∞
= 2(z − 1) + 5
(z − 1) + 2
= 5
2 +
∞
n=1
n=0
2(z − 1) + 5
1 + z−1
2
(−1) n
(−1) n
2 n+1 (z − 1)n ;
=
2 n (z − 1)n = . . . =
1
1 − z
= z n |z| < 1 .
n≥0
(m − 1) |z| < 1
D m−1
D m−1
1
1 − z
= (m − 1)!
(z − 1) m
⎛
⎝
z n
⎞
⎠ =
z n−(m−1) n(n − 1) . . . (n − (m − 2)) .
n≥0
n≥m−1
1
=
(1 − z) m
n≥0
z n
n + m − 1
m − 1
.
R = ∞
R = 0
R = 1
R = 1
R = 1
∆ = {|z + i| < √ 2};
f(z) = 1
1−z

|z| ≤ 1 |z| > 1
|z| ≤ 1 |z| > 1
• R 2 R = ∞
R 2 = ∞)
• √ R
•
R
|z|=R x dz = iπR2
|z|=2
σ(0,1+i)
x dz = 1+i
2
dz
z2 1
=
− 1 2 |z|=2
dz
z − 1 +
|z|=2
dz
;
z + 1
• n ≤ 0
• n > 0
(n − 1)
f (n−1) (z) =
(n − 1)!
2πi
|z|=1
|z|=1
ez 2πi
dz =
zn (n − 1)! .
f(ζ
dζ ,
(ζ − z) n
|z|=2
dz
z2 +1 = 0

1
=
|z − a| 2
1
ρ 2 − |a| 2
2πia
• |a| < ρ |ρ2−|a| 2 |
2πiρ
• |a| > ρ 2
a|ρ2−|a| 2 |
a ρ2
+
z − a ρ2 − az
• n ≤ 0
|z|=1
• n > 0 n
• n > 0 n ≡ 1 4
• n > 0 n ≡ 3 4
• n ≥ 0 m ≥ 0
• n ≥ 0m < 0
• n < 0m ≥ 0
• n < 0 m < 0
|z|=2
z n (1−z) m dz = 2πi
sin z
zn dz = 0
sin z
|z|=1 zn dz = 0
sin z
|z|=1 zn dz = 2πi
(n−1)!
sin z
|z|=1 zn dz = − 2πi
(n−1)!
|z|=2 zn (1 − z) m dz = 0
|z|=2 zn (1−z) m dz = 2πi(−1) m
|z|=2 zn (1−z) m dz = 2πi(−1) n+1
|m| + |n| − 2
|n| − 1
n
|m| − 1
m
|n| − 1
|m| + |n| − 2
+
|m| − 1
z |z| ≤ ρ δ = R − ρ > 0
Bδ(z) ⊂ BR(0)
f
f
f n (γ) = n!
2πi
∂Bδ(z)
f(ζ)
(ζ − z) n+1
x = 1
2

|γ − z| < δ
|f (n) (z)| ≤ n!M
2π
=
∂Bδ(z)
1
δ n+1
n!M
2δ n+1 π |∂Bδ(z)| = n!Mδ −n = n!M(R − ρ) −n
sup |f
Bρ(0)
(n) (z)| ≤ n!M(R − ρ) −n .
δ
Bδ(z) ⊂ Ω Ω
f
n!n n ≤ |f (n) (z)| sup |f(ζ)|n!δ
Bδ(z)
−n ,
sup |f(ζ)| ≥ n
Bδ(z)
n δ n n→+∞
−→ +∞
|f| Bδ(z)
f
z ≥ 0 z ∈ C
g(z) = e −f(z) .
g C |g(z)| = e −f(z) ≤
1
f
• f(z) = sin 1
1−z
• f g
|z| ≤ 1
g ≡ 0 {|z| ≤ 1} g f

f
R = {aω1 + bω2 : 0 ≤ a, b ≤ 1}
z ∈ C z = r + nω1 + mω2
r ∈ R f
f(z) = f(r)
sup |f(z)| = sup |f(r)| < ∞
C
R
R |f| f
f
f
f(z)
g(z) = zm z = 0
f (m) (0)
m! z = 0
g {|z| < 1}
0 < r < 1 g
1
r m |z| ≤ r
r → 1− |g| < 1 |z| < 1
g
|f(z)| < |z| m
g
f Ω B1(1) =
{|z − 1| < 1}
log∗ z
γ
f ′
f =
γ
d
dz (log ∗ z) dz = 0
ζ(z) =
az + b
cz + d

ζ −1 (w) =
dw − b
a − cw
z(0) = b
d ∈ R ζ−1 (0) = −b
∈ R .
a
b, d ∈ R a ∈ R ζ(1) = a+b
c+d
∈ R
c ∈ R
b, d ∈ iR a ∈ iR
ζ i
•
•

•
z R
a z ∗ (z ∗ − a)(z − a) = R 2
•
z0 → 0 z0 → ∞ ;
1
1
ζ(z) =
z − z0
.
z − z0
−2 → 0 0 → i
i
−1+i
2
ζ(z) =
i + i
2
·
z + 2
2z + (1 − i) .
C1 C2
p 0
C1 C2 ∞
p
∗ 1 (p − 4
)(p − 1
4
) = 1
16
p ∗ p = 1
p = 2 + √ 3
2 + √ 3 → 0
1
2 + √ → ∞ 1 → 1
3

ζ(z) = (2 + √ 3) − z
(2 + √ 3)z − 1 .
1 → ∞ 0 → 0 − 1 → 1
ζ(z) = 2z
z − 1 .
{0 < Rez < 1}
eiπw
f(z) = e iπζ(z) = e 2πiz
z−1 .
0 → −1 i → −i − i → i
1 c = 1
c = 1
R(z) =
2 + iy
iy
= 1 + 2
iy .
c = 1 + ε c = 1 − ε
Rez = 1 c < 1
∞ 1
c+1
c−1
1
c+1
2 c−1 + 1
1
2 1 − c+1
c−1
c → ∞ c → 1
Rez = 1

c > 1
c = 1 Rez = 1
c < 1 Pc = 1
2 1 + c+1
c−1
Rc = 1
2 1 − c+1
c−1
c > 1 Pc = 1
2 1 + c+1
c−1
Rc = 1
2
c+1
c−1 − 1
f
z0 Imz0 > 0
z0
ζ(z) =
z − z0
z − z0
g = f ◦ ζ−1 : B1(0) → C
z = 1
g
g(0) = 1
2πi
S 1
g(ξ)
ξ
dξ = 1
2πi
S 1
f ◦ ζ −1 (ξ)
ξ
g(0) = f(z0)
z = ζ −1 ξ → ξ = ζ(z)
dξ
ξ = ζ′ (z) z0 − z0
dz =
ζ(z) (z − z0)(z − z0) .
1
z0 − z0
f(z0) = f(z)
dz =
2πi R (z − z0)(z − z0)
= Imz0
+infty
f(z)
dz .
π |z − z0| 2
−∞
f z0
a−1 z0
f z0
dξ .

f z0 h > 0
a−1 z0
(z − z0) h f(z)
z0 bh−1 z0
a−1
1
Resz0f =
(h − 1)! D(h−1) z [(z − z0) h f(z)] |z=z0 .
f z0 g
fg = (a0 + O(z)) · (b−1(z − z0) −1 + b0 + O(z)) =
= a0
(z − z0)
b−1
−1 + a0b0 + a1b−1 + O(z) .
Resz0fg = a0b−1 = f(z0)Resz0g .
•
z = −2, −3
1
z 2 + 5z + 6 =
Res−2 = 1 Res−3 = −1
•
1
1 1
= −
(z + 2)(z + 3) z + 2 z + 3 .
2 z = −1, 1
Res−1 = 1
4 Res1 = − 1 4
•
π
z = kπ k ∈ Z
2π
0 π
0
1 z 1
= ·
sin z sin z z ⇒ Res0 = 1
π 1
sin z =
−1 −(z − π)
=
sin(z − π) sin(z − π) ·
1
(z − π)

Resπ = −1 .
Reskπ = (−1) k
•
Reskπ = cos(kπ) · (−1) k = 1 .
•
π
2 π
z = 0
1
sin 2 z =
=
= 1
z 2
1
z − z 3
6 + O(z5 )
z 2 1 − z2
2 =
1
6 + O(z4 1 1
2 = ·
) z2
1 + z2
3 + O(z4
) = 1 1
+
z2 3 + O(z2 ) .
1 − z2
3 + O(z4 ) =
Res0 = 0 Reskπ = 0
π
0
cos θ = 1
2 (eiθ + e −iθ ) = 1
2
dθ
a + cos θ
z + 1
z
.
2π
1 dθ
=
2 a + cos θ =
= 1
2
= 1
i
0
|z|=1
|z|=1
dz (iz) −1
a + 1
2
1 =
(z + z )
dz
z 2 + 2az + 1
α± = −a± √ a2 − 1
Resα+ =
1
α+ − α−
= −Resα− .
α+
=
π
√ α 2 − 1 .

sin 2 z (0, π
2
( π
2
)
, π) π
1
4
2π
0
dx
a + sin 2 x
1
=
4
= 1
i
|z|=1
|z|=1
dz (iz) −1
a + 1
2i (z − z−1 ) 2 −z dz
z4 − (4a + 2)z2 + 1
± β± β± = (2a+1)± (2a + 1) 2 − 1
α++, α+−, α−+, α−−
• a > 0 β− α−+ α−−
Resα−+ =
α−+
α−−
Resα−− =
(β− − β+)(α−+ − α−−) (β+ − β−)(α−− − α−+) .
=
π
(2a + 1) 2 − 1 .
• a < 0 β+ α++ α+−
Resα++ =
α++ Resα+−
(β+ − β−)(α++ − α+−) =
α++
(β+ − β−)(α+− − α++) .
=
−π
(2a + 1) 2 − 1 .
f(z) =
sgn(a) π
(2a + 1) 2 − 1 .
z 2
z4 +5z2 +6
γR = CR ∪ σR = {|z| = R, Imz ≥ 0} ∪ {−R ≤ x ≤ R} .
R
f(z) z = ±i √ 2 z = ±i √ 3
R

Res √
i 2 = i√2 Res √
i 3 =
2 −i√3 .
2
γR
f dz =
CR
f dz +
σR
f dz = π( √ 3 − √ 2) .
R → ∞
z 2
CR z4 + 5z2 + 6 dz
σR
≤
f(z) dz R→∞
−→
≤
CR
CR
= πR
∞
−∞
R2 |z4 + 5z2 |dz| ≤
+ 6|
R2 R4 − 5R2 |dz| =
− 6
R 2
R 4 − 5R 2 − 6
x2 x4 + 5x2 dx .
+ 6
R→∞
−→ 0 ;
π( √ 3 − √
∞
2) = lim f(z) dz = f(x) dx
R→∞ γR
−∞
π
2 (√3 − √ ∞
2) =
0
x 2
x 4 + 5x 2 + 6 .
a = 0
0
f(z) = eiz
z 2 + a 2
γR
z = ±i|a|
R z = i|a|
Resi|a| = e−|a|
2i|a| .
•
lim f(z) dz =
R→∞ γR
π
|a| .

•
•
CR
e iz
z 2 + a
+∞
f(x) dx =
−∞
=
dz
2 ≤
+∞
−∞
+∞
−∞
≤
1
R2 − a2
|e
Cr
iz | |dz| ≤
πR
R2 − a2 R→0
−→ 0 .
cos x
x2 +∞
sin x
dx + i
+ a2 −∞ x2 dx =
+ a2 cos x
x2 dx .
+ a2
+∞
0
cos x
x2 1
dx =
+ a2 2
+∞
−∞
cos x
x2 π
dx =
+ a2 2|a| e−|a| .
|z| = 1
π
0
cos θ = 1
2 (eiθ + e −iθ ) = 1
2
dθ
a + cos θ
z + 1
z
.
2π
1 dθ
=
2 a + cos θ =
= 1
2
= 1
i
0
|z|=1
|z|=1
dz (iz) −1
a + 1
2
1 =
(z + z )
dz
z 2 + 2az + 1
α± = −a± √ a 2 − 1
Resα+ =
1
α+ − α−
= −Resα− .
α+
=
π
√ a 2 − 1 .

sin 2 z (0, π
2
( π
2
)
, π) π
1
4
2π
0
dx
a + sin 2 x
1
=
4
= 1
i
|z|=1
|z|=1
dz (iz) −1
a + 1
2i (z − z−1 ) 2 −z dz
z4 − (4a + 2)z2 + 1
± β± β± = (2a+1)± (2a + 1) 2 − 1
α++, α+−, α−+, α−−
• a > 0 β− α−+ α−−
Resα−+ =
α−+
α−−
Resα−− =
(β− − β+)(α−+ − α−−) (β+ − β−)(α−− − α−+) .
=
π
(2a + 1) 2 − 1 .
• a < 0 β+ α++ α+−
Resα++ =
α++ Resα+−
(β+ − β−)(α++ − α+−) =
α++
(β+ − β−)(α+− − α++) .
=
−π
(2a + 1) 2 − 1 .
f(z) =
sgn(a) π
(2a + 1) 2 − 1 .
z 2
z4 +5z2 +6
γR = CR ∪ σR = {|z| = R, Imz ≥ 0} ∪ {−R ≤ x ≤ R} .
R
f(z) z = ±i √ 2 z = ±i √ 3
R

Res √
i 2 = i√2 Res √
i 3 =
2 −i√3 .
2
γR
f dz =
CR
f dz +
σR
f dz = π
2 (√ 3 − √ 2) .
R → ∞
z 2
CR z4 + 5z2 + 6 dz
σR
≤
f(z) dz R→∞
−→
≤
CR
CR
= πR
∞
−∞
R2 |z4 + 5z2 |dz| ≤
+ 6|
R2 R4 − 5R2 |dz| =
− 6
R 2
R 4 − 5R 2 − 6
x2 x4 + 5x2 dx .
+ 6
R→∞
−→ 0 ;
π( √ 3 − √
∞
2) = lim f(z) dz = f(x) dx
R→∞ γR
−∞
π
2 (√3 − √ ∞
2) =
0
x2 x4 + 5x2 dx .
+ 6
f(z) = z2 −z+2
z 4 +10z 2 +9
γR = CR ∪ σR = {|z| = R, Imz ≥ 0} ∪ {−R ≤ x ≤ R} .
R
f(z) z = ±3i z = ±2i
R
Res3i =
7 + 3i
1 − i Res2i = .
48i 16i
γR
f dz =
CR
f dz +
σR
f dz = 5π
12 .

R → ∞
z 2 − z + 2
CR z4 + 10z2 + 9 dz
σR
5π
12
≤
f(z) dz R→∞
−→
≤
CR
CR
R2 + R + 2
|z4 + 10z2 |dz| ≤
+ 9|
R2 + R + 2
R4 − 10R2 |dz| =
− 9
= πR R2 + R + 2
R4 − 10R2 R→∞
−→ 0 ;
− 9
∞
−∞
x2 − x + 2
x4 + 10x2 dx .
+ 9
∞
= lim
R→∞
f(z) dz =
γR
f(x) dx
−∞
5π
12 =
∞
x
0
2 − x + 2
x4 + 10x2 dx .
+ 9
a = 0
f(z) =
z 2
(z2 +a2 ) 3
γR = CR ∪ σR = {|z| = R, Imz ≥ 0} ∪ {−R ≤ x ≤ R} .
R
f(z) z = ±i|a|
R
Res i|a| = 4a2
2 6 |a| 5 i .
∞
0
x2 (x2 + a2 dx =
) 3
1
.
16|a| 3
a = 0
0
f(z) = eiz
z 2 + a 2
γR
z = ±i|a|

R z = i|a|
Res i|a| = e−|a|
2i|a| .
•
•
•
CR
e iz
z 2 + a
+∞
f(x) dx =
−∞
=
lim f(z) dz =
R→∞ γR
π
|a| .
dz
2 ≤
+∞
−∞
+∞
−∞
≤
1
R2 − a2
|e
Cr
iz | |dz| ≤
πR
R2 − a2 R→0
−→ 0 .
cos x
x2 +∞
sin x
dx + i
+ a2 −∞ x2 dx =
+ a2 cos x
x2 dx .
+ a2
+∞
0
cos x
x2 1
dx =
+ a2 2
+∞
−∞
cos x
x2 π
dx =
+ a2 2|a| e−|a| .
Ω = C \{z = iy | y ≤ 0}
log z log 1 = 0 log(−1) = iπ
f(z) =
Ω \ {i} z = i Resi =
log i
2i
= π
4
γR,ε = CR ∪ Cε ∪ σ+ ∪ σ− = {|z| = R, Imz ≥ 0} ∪
log z
1+z2
∪ {|z| = ε, Imz ≥ 0} ∪ {−R ≤ x ≤ −ε} ∪ {ε ≤ x ≤ R} .

π 2 i
2
∞
= lim f(z) dz =
ε → 0
R → ∞
γR,ε
=
0
log |x| + iπ
−∞ 1 + x2 =
=
∞
log x
dx +
dx =
0 1 + x2 ∞
∞
log x
1
2
dx + iπ
dx =
0 1 + x2 0 1 + x2 ∞
log x
2
dx + iπ2
1 + x2 2
0
0
log x
dx = 0 .
1 + x2
R
ε
0 < α ≤ 1
1 < α < 2
π
α sin πα
2
γ = ∂D1(0)
g(z) = 6z 3
|g(z)| = 6 γ
|P1(z)−g(z)| ≤ 5 γ P1
g D1(0)
P1 3
D1(0)
γ = ∂D1(0) g(z) = −6z
γ
|P2(z) − g(z)| ≤ 4 < 6 = |g(z)|
P2
D2(0) γ = ∂D2(0)
g(z) = z 4
|P2(z) − g(z)| ≤ 15 < 16 = |g(z)|
.

4
P2 1 ≤ |z| < 2.
γR
[0, R] , [0, iR] 0 R
R iR R
γR
g(z) = z4 + 1
γR |P3(z) −
g(z)| ≤ |g(z)| z ∈ γR
|P3(x) − g(x)| = |x| 3 < x4 + 1 = |g(x)| x ∈ R
|P3(iy) − g(iy)| = |y| 3 < y4 + 1 = |g(iy)| y ∈ R
|P3(z) − g(z)| = |z| 3 = R3 < R4 − 1 ≤ |g(z)| |z| = R R ≥ 2
R4 − 1 > 2R3 − R3 = R3
n
(z − α1) · . . . · (z − αn)
P (x) ±1
1
|z| = 1 P (0) = −1 < 0
limx→∞ P (x) = +∞
P (1) = 0
f
m Ω Ω ′ Ω
1
Ω ′ l > m f
Ω ′
m < l =
=
=
1 f
2πi
′
f
1
2πi
1
2πi lim
n→∞
dz =
∂Ω ′
∂Ω ′
f
lim
n→∞
′ n
fn
f ′ n
∂Ω ′ fn
dz =
dz ≤ m

f(0) =
0 g(z) = f(z) − f(0)
f ′ (0) = 0 f 0
f(z) = zf1(z) f ′ 1(0) = 0
f2(z) = f1(z n ) Dρ(0)
f2(z) = 0
log f2 n f2(z) =: h(z)
h(z) n = f2(z)
z ∈ Dρ(0)
f(z n ) = z n f1(z n ) = z n f2(z) = (zh(z)) n =: g(z) n
g
a2 = 3 4 n ≥ 3
an =
= (n + 1)(n − 1)
= n + 1
1 − 1
n 2
n2 n→∞
−→
n
1
2 .
an−1 = n2 − 1
n 2 an−1 =
n(n − 2)
(n − 1) 2 an−2 = . . . =
(n + 1)(n − 1)
an−1 =
n
n + 1 4
n 3 a3 =
(1 + z)
∞
n=1
(1 + z 2n
) = (1 + z) lim
m→∞
(1 + z2n
= lim
m→∞
2 m −1
z
n=0
2n + z 2n+1 =
2
) = (1 + z) lim
m→∞
m −1
n=0
∞
n=1
z n = 1
1 − z .
z 2n =
θ C
C
∞
n=1
|h 2n−1 e z + h 2n−1 e −z + h 4n−2 |
DR(0) R > 0 |z| ≤ R
∞
n=1
|h 2n−1 e z + h 2n−1 e −z + h 4n−2 | ≤
∞
n=1
|h| 2n−1 (2e R + 1) ≤ C
∞
|h| n < ∞ .
n=1

h −1 e −z θ(z) = h −1 e
−z
(1 + h 2n−1 e z )(1 + h 2n−1 e −z ) =
n≥1
= h −1 e −z (1 + he z )(1 + he −z )
(1 + h 2n−1 e z )(1 + h 2n−1 e −z ) =
n≥2
= (1 + h −1 e −z )(1 + he z )
(1 + h 2n−1 e z )(1 + h 2n−1 e −z ) =
n≥2
= (1 + h −1 e −z )
(1 + h 2n−1 e −z )(1 + he z )
(1 + h 2n−1 e z ) =
n≥2
n = m − 1 n = m + 1
n≥2
h −1 e −z θ(z) = . . . =
(1 + h 2m−3 e −z )
(1 + h 2m+1 e z ) =
=
m≥1
m≥1
m≥1
(1 + h 2m−3 e −z )(1 + h 2m+1 e z ) = θ(z + log h 2 ) .
z =
±n
n 1
n
n 1
n2
h = 1
g(z)
sin πz = ze 1 − z
e
n
z
n
g
π cot πz =
d(sin πz)
sin πz
n=0
1
=
z + g′ (z) +
1 1
+ ;
z − n n
g ′ (z) = 0 g(z) ≡ c
sin πz
lim = π
z→0 z
eg(z) = ec = π
sin πz = πz
1 − z
n
n=0
n=0
e z
n

1
n −n
∞
sin πz = πz
.
n≥2
1 − 1
n2
n=1
1 − z2
n 2
sin πz
= lim
z→1 πz(1 − z)(1 + z) =
= 1
2 lim
sin π(1 − z) 1
=
z→1 π(1 − z) 2 .
cos θ = 1
1
2 z + z |z| =
1
2π
(cos θ) 2n dθ =
0
=
1
22n
1
i2 2n
S 1
S 1
z + 1
2n dz =
z
(z 2 + 1) 2n
z 2n+1
0 2n + 1
Res0 = D2n z (z2 + 1) 2n
.
2n! |z=0
(1 + z 2 ) 2n =
Res0 =
2n
j=0
2n
n
2n
j
.
2π
(cos θ)
0
2n dθ = . . . = 1
i22n
= 2πi
22ni =
2n
n
S 1
z 2j
dz .
(z 2 + 1) 2n
z2n+1 dz =
(2n − 1)!!
= 2π
2n!!

f+
f+(z) = f+(z) ;
f(z) = f(z) z ∈ C
f
g(z) = if(iz)
C
g
if(iz) = g(z) = g(z) = −if(iz) .
z ∈ C
f(iz) = −f(iz) = −f(−iz) = −f(−iz)
f
1
f(0) =
2πi S1 f(z
dz =
z
= 1
2π
g(θ) dθ ;
2π 0
z0
z0
z0 |z0| < 1
z0 0
S(z) =
z − z0
.
1 − zz0

g = f ◦ S−1
f(z0) = g(0) = 1
2πi S1 f ◦ S−1 =
(z)
dz =
z
1
2πi S1 f(ξ) 1 − |z0|
ξ
2
=
dξ =
|ξ − z0| 2 1
2π
2 1 − |z0|
g(θ)
2π 0 |eiθ dθ
− z0| 2
1−|z0|2
|eiθ−z0| 2
Ω
g(z) = f( 1
log z) .
2πi
D ∗ 1− f

z ∈ Σ +
g
cn =
g(e 2πiz ) = f(z)
f(z) =
∞
−∞
cnz n
1
2πi CR
f(ξ)
ξ n+1
0 < r < 1
g e 2πiz
z ∈ Σ +
a) 2π 1 − 2
√
3
b)
2π
√ a 2 − b 2
c) π(a − a2 − 1)
d)
π
sin(πa)
e)
π
b sin
π
b
f)
π 2 cos(πa)
sin 2 (πa)
log 2 a − log 2 b
g)
2(a − b)
h)
35
128 π
i) π( √ 2 − 1) .

un [a, b]
un (a, b)
[a, b]
un (a, b) un(a)
un(b) un
(a, b)
{x > 0}
n≥1
x + 1
n
n+ x
n
+∞
n=1
+∞
e −xy2
dy .
(a)
n≥1 x√ n (b)
n≥1 log n 1 + x
n
n
(c)
n≥1
log n
n 4 +x 2
sin nx
n≥0 n! R
e iθ = cos θ + i sin θ

(a)
n≥1 (23n + 3 2n ) x n (b)
n≥1 n3 x 2n+1 (c)
n≥1
(d)
n≥1
1 − 4n2 +7n+1
n 3 +18n+2
n
xn (e)
x+2
n≥1 x2 n +1
⎧
⎨
x2 + y2 1 − e
f(x) =
⎩
− x2 +y 2
|x| x = 0
0 x = 0
• f (0, 0)
• f (0, 0)
• f (0, 0)
3n−2
n+1
n
(f)
π
n≥1 sin n 4 x2n
x
(a) lim (x,y)→0 2 −y 2
x2 +y2 1 −
(b) lim (x,y)→0 e x2 +y2 1−cos(x
(c) lim (x,y)→0 2 +y 2 )
(x2 +y2 ) 2 sin(xy)
(d) lim (x,y)→0 y
x + y
f(x, y) =
x + ye
x > 0
−x2 x ≤ 0
f(x, y) = x 2 + y 2
f(x, y) = [(y + 1)] x+1
f(x, y) = log y+1(x + 1)
z = x 2 + y 2 − 1
z
x 2n

f(x) =
sin 2 (xy)
x 2 +y 2
(x, y) = (0, 0)
0 (x, y) = (0, 0)
f (0, 0) (0, 0)
z = x 2 + y 2 A = (0, 1, 1) B = (3, 4, 5) C = (0, 0, 0)
z = x 2 + y 2
2x + 4y − z = 0
z
x = 1 y = 1
z = x y x = cos t y = sin t dz(t)
dt
z = x 2 y x = 2u + v
y = ue v
∂z(u,v)
∂u
∂z(u,v)
∂v
f(x, y) = x 2 − y 2
D ≡ {(x, y) ∈ R 2 : x 2
a 2 + y2
b 2 ≤ 1}
R 2
f(x, y) = x 2 + xy 2 + y 4 ;
f(x, y) = (ax 2 + by 2 )e −(x2 +y 2 ) a, b > 0
f(x, y) = ye y2 −x 2
D ≡ {(x, y) ∈ R 2 : |x| + |y| ≤ 1}
.

f(x, y) = x 2 y + 3y − 2 (x − 1)
(y + 2)
f(x, y) =
y
x
e −t2
dt
K ≡ {(x, y) ∈ R 2 : 0 ≤ x ≤ y ≤ 2x}
f(x, y) = (x + y)
D ≡ {(x, y) ∈ R2 x : 2
4 + y2 ≤ 4}
f(x, y, z) = xye −z2
D ≡ {(x, y, z) ∈ R 3 : 4x 2 + y 2 − z 2 ≥ 1}
T : C([−1, 1]) −→ C([−1, 1])
T u(x) ≡ x
2 u(x ) + g(x) .
2
g T
T T
∃! u u = T u
T : C([−a, a]) −→ C([−a, a]) a > 0
f : R 2 −→ R
(x, y) −→ f(x, y) = −xe y + 2y − 1 .
P0 = (x0, y0) x0 ≤ 0 f(x0, y0) = 0
P0 y

2 y(x) (0, 1
2 )
{(x, y) : f(x, y) = 0}
y(x)
E ≡ {(x, y, z) ∈ R 3 : z 2 = xy + 1}.
E
E
f(x, y, z) = x 2 + y 2 + z 2 E
E f(E)
E (−1, 1, 0)
Γ ≡ {(x, y, z) ∈ R 3 : z = x 2 − y 2 x 2 + y 2 + z 2 = 1}.
Γ
f Γ f(x, y, z) = x
Γ (x, y) P
Γ P
Γ P
i)
ii)
iii)
iv)
x
D
2
y2 dxdy D ≡ {(x, y) ∈ R2 : 1 ≤ x ≤ 2, 1
x ≤ y ≤ x}
D x2y2 dxdy D ≡ {(x, y) ∈ R2 : x2 + y2 ≤ 1}
D y3ex dxdy D ≡ {(x, y) ∈ R2 : y ≥ 0, x ≤ 1, x ≥ y2 }
D xy dxdy D ≡ {(x, y) ∈ R2 : x + y ≥ 1, x2 + y2 ≤ 1}
1 1
1 1
x − y 1
dx
dy = dy
0 0 (x + y) 3 2
0 0
x − y
dx = −1
(x + y) 3 2 .

C1 ≡ {x 2 + y 2 ≤ 1} C2 ≡ {x 2 + z 2 ≤ 1} .
l ρ(x, y)
k
P ≡ {(x, y, z) ∈ R 3 :
x2 + y2 z2
+
a 2
b
2 ≤ 1} 0 < a < b .
r
y 2 = 4x + 4 y 2 = −2x + 4 .
D
D x2
a2 + y2
b2 + z2
c2 ≤ 1
x 2 dxdydz
x 2 + y 2 = 1 z = x 2 + y 2 − 2
x + y + z = 4
R3 .1
R4 .1
n

Ac 1 + 3x2 dxdy
+ y2 Ac ≡ {(x, y) ∈ R2 : 3x2 + y2 ≤ c2 }, c > 0
z
A 1 + 3x2 dxdydz
+ y2 A ≡ {(x, y, z) ∈ R3 : 3x2 + y2 ≤ (z − 2) 2 , 0 < z < 1}
1
R2
S0 ≡ [0, 1][0, 1]
1 3 S1
1
3
S1 1
32 {Sn}
Sn
1 3n−1 1
3n S ≡ ∩nSn
S
S
S
S
J2 =
J3 =
Jn =
Dn ≡ {(x1, . . . , xn) ∈ R n : 0 ≤ x1 ≤ x2 ≤ . . . ≤ xn ≤ 1};
D2
D3
xy dxdy
xyz dxdydz
Dn x1 . . . xn dx1 . . . dxn

T x 2 (y − x 3 )e y+x3
dxdy
T ≡ {(x, y) ∈ R 2 : x 3 ≤ y ≤ 3, x ≥ 1}.
E ≡ {(x, y) ∈ R 2 : y 2 ≤ x 2 (1 − |x|)}
E
E
E
{x > 0}
Sx Sy E
x y
fn : [0, 1] ↦−→ R
fn(x) −→ f(x) ∀x ∈ [0, 1] .
Γn fn Γ f
• L(Γn) ≤ M L(Γ) ≤ M
• L(Γn) ≥ M L(Γ) ≥ M
∃c ∈ R
• e −λ(x4 +y 4 ) c
dxdy = √λ ∀λ > 0 .
R 2
• π√π 2 ≤ c ≤ π√2π 2
lim
r↦→+∞ e−r
B(0,r)
e |x|+|y| dxdy .
R 2
D ≡ {(x, y) ∈ R 2 : x > 0, | log x| ≤ 1, |y − x log x| ≤ 1} .

ϕ : R ↦→ R
1 1
ϕ f(x)dx ≤ ϕ(f(x)dx .
0
M ≥ 1 α
Dα (x2 + y2 dxdy < +∞
) M
Dα = {(x, y) ∈ R2 : x ≥ 0, 0 ≤ y ≤ xα } .
1
Γ = {(x, y, z) ∈ R 3 : y 2 + z 2 = 1} ∩ {(x, y, z) ∈ R 3 :
x 2 + z 2 = a 2 } α
Γ
Γ
Γ
∞
0
0
dx
(1 + x 2 ) 2
U, V ⊂ R n ω U
V ω U ∩ V U ∪ V
U ∪ V U ∩ V
ω = (2xy 3 − y 2 cos x)dx + (1 − 2y sin x + 3x 2 y 2 )dy
R 2
ω1 = x 3 dx + y 2 dy + zdz

ω2 = x
x 2 +y 2 dx + 2y
x 2 +y 2 dy
ω3 = y
x 2 +y 2 dx + x
x 2 +y 2 dy
ω4 = x
x 2 +y 2 dx + y
x 2 +y 2 dy + z
x 2 +y 2 dz
|A(x, y)| ≤ k |B(x, y)| ≤ k ∀x, y ∈ Ω
l(Γ) Γ
Γ
A(x, y)dx + B(x, y)dy
≤ √ 2kl(Γ)
ϕ(x, y) y
i) x 2 ydx + ϕ(x, y)dy ; ii) sin ydx + ϕ(x, y)dy .
x = a(t − sin t)
y = a(1 − cos t)
0 ≤ t ≤ 2π
x
r(θ) 2 = 2a 2 cos 2θ − π π
≤ θ ≤
4 4
ω Ω ⊂ Rn
γ
ω ∈ Q
γ
ω
ω(x, y, z) = P (x, y, z)dx + (x 2 + 2yz)dy + (y 2 − z 2 )dz
P ∈ C 1 (R 3 , R) ω

P x P
A, B, C, D ∈ R R 2 \ {(0, 0)}
ω(x, y) =
Ax + By
x2 Cx + Dy
dx +
+ y2 x2 dy
+ y2
ω(x, y) = (3yx 2 −1)dx+2x 3 dy
ϕ ∈ C 1 ((0, +∞), R) ∼ ω (x, y) = ϕ(x)ω
(0, +∞)R ϕ ∼ ω
ω = 2(x2 − y 2 − 1)dy − 4xydx
(x 2 + y 2 − 1) 2 + 4y 2
R2 \ {(1, 0) ∪ (−1, 0)}
γ1 γ2 (1, 0) (−1, 0)
γ1 (−1, 0) γ2
(1, 0)
1
2π
ω = − 1
2π
ω = 1
γ1
ω R n
ω α
γ2
dω = ω ∧ α.
R 3
ω = dy − mdx
ω = Adx + Bdy + Cdz
ω ⇔ (A, B, C) ⊥ (A, B, C).

ω = xdy − ydx.
ϕ(x, y) = 1
xy ψ(x, y) = 1
x 2 +y 2
γ(t) ω(γ(t), · γ (t)) = 0 ϕ(γ(t))
ψ(γ(t))
= .
ω R n
Ω, Ω ′ ⊂ R n
ϕ : Ω −→ Ω ′
γ ∈ C 1 ([a, b], Ω ′ ) ˜γ(t) = ϕ −1 (γ(t))
d
˜γ(t) =
dt
ϕ ′ | ˜γ(t)
−1 ˙γ(t) .
V Ω ′
V : Ω ′ −→ R n .
V
ϕ ∗ V : Ω −→ R n .
ω Ω ′
(ϕ ∗ ω) · (ϕ ∗ V ) = ω · V .
Q = [0, 1] × [0, 1]
ϕ : Q −→ R 3
(u, v) −→ (u + v, u − v, uv) .
ω = xdy ∧ dz + ydx ∧ dz .
ϕ(Q)
ω =
Q
ϕ∗ω .

f : R 3 → R
µf (x, r) = 1
4
3 πr3
B(x,r)
f(y) dy .
limr→0 µf (x, r) = f(x) .
f ∈ C(R 3 , R)
µf (x, r) = xyz + r .
F : R 3 −→ R 3 C 1
B(x, r)
7xr 3 + xyzr 4 .
F
(2, 2, 2)
(x0, y0), (x1, y1), . . . , (xm, ym) = (x0, y0)
A = 1
m
(xn−1yn − xnyn−1)
2
n=1
=
= 1
m
yn(xn+1 − xn)
2
n=1
=
= 1
m
xn(yn+1 − yn)
2
.
n=1
α
γ γ
r < R A = 4π 2 Rr

x = a(t − sin t)
t ∈ [0, 2π]
y = a(1 − cos t)
x
S = ∂B(0, 1)
S
x 2 dσ
Σ
z dσ
Σ z = xy
U = {(x, y) : x 2 + y 2 ≤ 1, 0 ≤ y ≤ √ 3x} .
x ds
ϕ
ϕ y = x2 0 ≤ x ≤ a
ρ = √ z 2 + 1
x 2 + y 2 − z 2 = 1 .
z = 0 z = 1
D
x
dx dy
y
D = {(x, y) : 1 x2 1 x
≤ ≤ 1, ≤ ≤ 1}.
2 y 2 y

D = {(x, y, z) : x 2 + y 2 + z 2 ≤ 2z, z ≤ 2(x 2 + y 2 )} .
f(x) =
sin 3 (nx)
n!
n≥1
f ∈ C∞ (S1 ) f
x(π − x) [0, π]
n≥0
1
(2n + 1) 6
.
n≥1
1
.
n6 f ∈ C m (S 1 , C)
|cn| ≤ M
.
|n| m
m ≥ 2 f ∈ C m−2 (S 1 , C) .
f ∈ C 2 (S 1 , R)
2π
f(t) dt = 0 .
f f ′
0
2π
|f(t)| 2 2π
dt ≤
0
0
|f ′ (t)| 2 dt .
(i − √ 3) 14
1 + cos θ − i sin θ
1 + cos θ + i sin θ

(1 + 2i) 5 − (1 − 2i) 5
(2 + i) 7 + (2 − i) 7
(1 + i) n
(1 − i) n−2
1
2i (i5 − i −5 ) ;
5√ 3 − i .
cos nθ cos θ n
|z| = 1 m ∈ N
m 1 + iw
= z
1 − iw
(z0, z1, z2, z3) (w0, w1, w2, w3)
az + b
T (z) =
cz + d zi wi
z0 − z2 z1 − z2 w0 − w2 w1 − w2
/
=
/
.
z0 − z3
z1 − z3
w0 − w3
w1 − w3
(z0, z1, z2, z3)
|z| = 1 |z − 1| = 4
az + b
T z =
cz + d . (z1, z2, z3) (a, b, c, d)
z − z2 z1 − z2
T z = [z, z1, z2, z3] := /
.
z − z3 z1 − z3

az + b
T z = . T (R∪∞) a, b, c, d
cz + d
R
T (S1 ) = S1
G
G
G = {z : 0 < |z| < 1} .
G
|z|=1
|z|=2
e z
z
dz .
dz
z 2 + 1 .
ρ > 0 a ∈ C ρ = |a|
|z|=ρ
|dz|
.
|z − a| 2
zz = ρ 2 |dz| = −iρ dz
z
f |z| < 1
|f(z)| ≤
1
1 − |z| .
|f (n) (0)|
f DR ≡ {|z| ≤ R}
|f(z)| ≤ M
z ∈ DR |f (n) (z)|
Dρ ρ < R

f ∈ H(C)
f
|f(z)| ≤ A + B|z| n ,
|f(z)| ≤ 1 per |z| < 1 ,
|f ′ (0)| ≤ 1 f(0)
f
Π + ≡ {z ∈ C : Imz > 0}
f(i) = i
|f ′ (i)|
Ω ⊂ C |f|
Ω
f ∈ H(Π + ) |f| ≤ 1
|f ′ (i)|
f(z) =
f(z) =
f(z) =
sin z
z
cos z − 1
z
log (1 + z)
z 2
f(z) = z2 + 1
z(z − 1)
f(z) = z sin 1
z

cos z f(z) =
z
f(z) = e 1
z
f(z) = 1 1
cos
z z
f(z) = 1
1 − ez f(z) = zn sin 1
z
f(z) =
1
z(z − 1)(z − 2) .
f
(a) B(0, 1) \ {0} (b) B(0, 2) \ B(0, 1) (c) C \ B(0, 2)
f(z) = tan z C
+ nπ n ∈ Z
zn = π
2
f
G ⊂ C f : G → C
G
d
d(z, w) = log
|1 − zw| + |z − w|
|1 − zw| − |z − w|
d
d(z, w) = log
|1 − zw| + |z − w|
|1 − zw| − |z − w|
d(z, w) = log
(P − w)(z − Q)
(P − z)(Q − w)
,
,

a)
b)
c)
d)
e)
f)
g)
+∞
−∞
+∞
−∞
+∞
−∞
+∞
−∞
+∞
−∞
+∞
−∞
+∞
−∞
d x
x 2 + a 2
d x
(x 2 + a 2 ) 2
x2 (x2 dx
+ 1) 2
d x
x 4 + 1
cos(ax)
x 4 + 1
a > 0
a > 0
dx a ∈ R
cos(ax)
x2 dx a ∈ R , b > 0
+ b2 sin 2 x
x2 dx .
+ 1