A f : A f ( f (α, x)
× J → → R h : A : A → R
J α
∈ A
J h(α) =
⊆R
x
→
f ( f (α, x) dx
J
h x
→ f ( f (α, x)
α
h
h h A x f ( f (α, x)
≤ f ( f (β, x)
h A
⊆R
α <
≤ β ≥>
x
R
α
h J
f
h
A J
×
f (α, x) → f ( h
J = [a, b] b
h(α) =
⊆R
f (α, x) dx.
a
A
lim f (α, x) = g(x)
x
α→α0
∈ [a, b]
b
lim h(α) =
α→α0
A
int K
⊆
g(x) dx;
a
k R
f : A
× [a, b](⊆
h
k+1 R )
α0 A K [a, b]
×
K
∈
→
R
⊆A
α0 f
∈
K
× [a, b] ∀ε > 0 ∃δ > 0 ∀α , α ∈ K ∀x, x ∈ [a, b] ||(α , x) − (α, x)|| < δ ⇒ |f (α, x ) − f (α, x )| < ε. α0 ∈ int K K α0 ∀α ∈ K ∀x ∈ [a, b] ||(α, x) − (α0 , x)|| < δ ⇒ |f (α, x) − f (α0, x)| < ε, f (α, x) ⇒ f (α0 , x) x ∈ [a, b] f (α, x) → f (α0 , x)
A f A
α
× [a, b]
⊆R
x
h b
h (α) =
a
f : A [a, b] ∂f (α, x) ∂α
×
→R
∂f (α, x) dx. ∂α θ
h(α +
b
α) − h(α) = 1 α α 1 = α
α, x) − f (α, x)) dx = b ∂f (α + θ α, x) ∂f (α + θ α, x) α dx = dx.
(f (α +
a b
a
∂α
b ∂f (α, x) dx a ∂α
a
∂α
∂f ∂α
x
∈ (0, 1)
∂f ∂α
α0
ϕ
ψ
ψ(α)
h(α) =
f (α, x) dx,
ϕ(α)
f ψ(α)
h (α) =
ϕ(α)
∂f (α, x) dx + f (α, ψ(α))ψ (α) ∂α
− f (α, ϕ(α))ϕ(α). {(α, x) | α ∈ A, ϕ(α) ≤ x ≤ ψ(α)} α ∈ (α0 − δ, α0 + δ )
f α0
∈ A
ϕ(α0 )
h(α) =
f (α, x) dx +
ϕ(α)
g(α) = g(α + 1 = α
ψ(α) ψ(α0 )
h
ψ(α0 )
ψ(α)
f (α, x) dx +
ϕ(α0 )
f (α, x) dx.
ψ(α0 )
f (α, x) dx
α) − g(α) = 1 α α
ψ(α+α)
ψ(α)
f (α +
ψ(α0 )
ψ(α)
−
ψ(α)
S 1
→
ψ(α0 )
f (α, x) dx =
ψ(α0 )
ψ(α+α)
1 (f (α+ α, x) f (α, x)) dx+ α ψ(α0 )
α, x) dx
−
f (α+ α, x) dx = S 1 +S 2 .
ψ(α)
∂f (α, x) dx, ∂α
ψ(α+α) 1 S 2 = f (α + α, x) dx = α ψ(α) 1 f (α + α, ξ )(ψ(α + α) ψ(α)) = α f (α, ψ(α))ψ (α)
→
−
−f (α, ϕ(α))ϕ(α) f : [c, d] × [a, b] → [c, d]
A = γ
c
c
b
≤ γ ≤ d
a
b
f (α, x) dx dα =
a
f (α, x) dα dx,
c
γ γ
γ
→
γ f (α, x) dα = ψ(α, x) c
∂ψ (γ, x) = f (γ, x), ∂γ
R
b
∂ ∂γ ∂ ∂γ
b
ψ(γ, x) dx =
a
b
a
∂ ψ(γ, x) dx, ∂γ
γ
a
b
f (α, x) dα dx =
c
f (α, x) dx,
a
α
[c, d]
× [a, b]
f
[a, b) f (α, x) α A F (α, u) =
∈
F : A
u f (α, x) dx a
b− a
F (α, u) ⇒ h(α)
u
f (α, x) dx b α B
→ −
α α
∈
b−
×
∈B
f (α, x) dx
g(α, x) dx
a
B b−
(λf (α, x) + µg(α, x)) dx,
a
B ε > 0
λ, µ
∈R
b− f (α, x) dx a u0 [a, b)
∈
α v
v
b− a
f (α, x) dx
u
≥ u0
∈B
α
f (α, x) dx < ε
u
α
≥
|f (α, x)| ≤ ϕ(x)
b− ϕ(x) dx a
→R
∈ B ⊆ A
F (α, u) ⇒ h(α)
b−
a
[a, b) [a, b)
< +
∞
b− a f (α, x) dx
λF (α, x) + µG(α, x) ⇒ λF (α, b ) + µG(α, b );
−
−
F (α, u)
f : A α
f (α, x) b− a f (α, x) dx K K
α
→ →
α α
∈
× [a, b) → R
α α
→ F (α, u) =
x [a, b) b− a f (α, x) dx
∈
u f (α, x) dx a
u
f, g : A x
→ g(α, x)
α
b− a f (α, x) dx
b− a
u a f (α, x) dx
f (α, x)g(α, x) dx
≤
M
A
× [a, b) → R
|g(α, x)| ≤ M g(α, x) ⇒ 0 x → b−
g
g
v
ξ
f (α, x)g(α, x) dx = g(α, u)
u
v
f (α, x) dx + g(α, v)
u
∈ (u0, b)
f (α, x) dx.
ξ
α > 0 v, u
u0
v
≤
b
f (α, x) dx < ε/2M,
u
v
ξ
f (α, x)g(α, x) dx
u
u
∈ (u0 , b)
M
f (α, x) dx +
u
α
ε > 0 g(α, u)
u0 ε/4M
|
v
f (α, x) dx
ξ
≤
≤ M (ε/2M + ε/2M ) = ε. b
|≤ v ε ε f (α, x)g(α, x) dx ≤ 2M + 2M = ε, 4M 4M
u
ξ f (α, x) dx u
=
− ≤ ξ a
u a
M + M = 2M
f (α, x) x α α0
→ α0
∈ [a, b)
[a, u]
b− f (α, x) dx a
α b−
lim
α→α0
b−
f (α, x) dx =
a
lim
α→α0
u
f (α, x) dx = lim lim
α→α0 u→b−
a
= lim lim
u→b− α→α0 u
= lim
u→b−
lim
u→b−
f 0 (x) dx.
a
b−
⊆
f 0 (x) [a, b)
a
f (α, x) dx =
u
f (α, x) dx =
a
b−
f 0 (x) dx =
a
f 0 (x) dx.
a
b− f a
α
α0
f (α, x) f (α, x) dx b− h(α) = a f (α, x) dx b− a
f (α, x) ⇒ f (α0 , x) 1◦ f : A ∂f ∂α
2◦ 3◦
α
f
→ α0
× [a, b) → R
α
A [a, b) b− f (α, x) dx a b− ∂f ∂α (α, x) dx a b− h(α) = a f (α, x) dx
×
h (α) =
b−
u a f (α, x) dx
u
h(α) b α ∂F (α, u) = ∂α
1
→ −
u
a
∈ [a, b)
α
∈A
α
∂f (α, x) dx. ∂α
a
x
◦
∂f (α, x) dx. ∂α
F (α, u) = F
2◦ 3◦ ∂f ∂α (α, x) dx
α b− a
F (α, u) ∂F ∂α (α, u)
u
∂F h (α) = lim (α, x) = u→b− ∂α
b−
∂f (α, x) dx. ∂α
a
f : [c, d] b− f (α, x) dx a
d
b−
c
f : [c, d) f
f (α, x) dx dα =
b− a
b−
f (α, x) dα dx;
c
f (α, x) dx
α [c, γ ] [c, d) x [a, b)
⊆
∈
d−
d−
d−
b−
f (α, x) dx dα =
a
c
a
b−
d−
a
d− f (α, x) dα c
b−
|f (α, x)| dα dx
c
c
× [a, b) α ∈ [c, d]
d
a
× [a, b) → R
a
b−
a
→ b−
|f (α, x)| dx dα f (α, x) dα dx.
c
β
β
c
β
f (α, x) dx dα =
a
d
a
b− d a c
f (α, x) dα dx.
c
d β c a β→b
lim
β→b
lim
d c
γ
b−
c
b−
f (α, x) dx dα =
a
γ
a
lim
f (α, x) dα dx.
c
γ →d− d−
b−
c
b−
f (α, x) dx dα = lim
γ →d−
a
lim
γ →d−
b− a
γ
a
c
f (α, x) dα dx
∈ [c, d)
γ
Φ(x, γ ) =
d−
f (α, x) dα
Φ0 (x) =
c
x
∈
Φ(x, γ ) γ [a, b)
f (α, x) dα.
c
Φ0 (x)
→bd−−
a
Φ(x, γ ) dx
γ
γ
|Φ(x, γ )| =
d−
f (α, x) dα
c
|f (α, x)| dα,
c
+∞
0
1 x
b > 0
≤
γ →d−
b− a
sin bx dx. x x 2/ b
0
−π/2
lim
b < 0
||
→ +∞
u 0
sin bx dx =
1−cos bu b
π/2
a > 0 b
∈R
+∞
ϕa (b) =
0
f a
sin bx e−ax dx = x
f a (b, x) =
∂f a (b, x) = ∂b b
x
∂f a ∂b
e−ax b,
+∞
f a (b, x) dx,
0
sin bx , x=0 . x x = 0
e−ax cos bx, 1,
x=0 . x = 0
1◦
f a b
2◦
b x
→ e−ax ∂f a ∂b
ϕa (b)
∂f a = e−ax cos bx ∂b
+∞
=
e−ax ,
e−ax cos bx dx =
0
ϕa (b) =
a , a2 + b2
a db b = arctg + C. a2 + b2 a
C C = 0
0
≤
b = 0
ϕa (0) =
b f a (b) = arctg . a a = 0 a > 0 a
→ 0+
π/ 2sgn b
h(a)
sin bx x
a
sin bx e−ax x
a
−
→ 0+ a
a
→ 0+
e−ax − 1 −ax | sin bx| | − e | |x| ≤ −ax · |a| ⇒ 0,
sin bx = 1 x
−ax
a
e
1 +∞
π/2sgn b = lim h(a) = a→0+
0
a
+∞ sin bx x 0
sin bx dx. x
dx
(a, b) f : (a, b) t = t 0
→R
b
G(x) =
√ 2πe−xf (t )
t0
0
−xf (t)
e
dt
∼
xf (t0 )
,
x
−xf (t)
→ +∞.
((1
e
f f (t)
∞
f (t0 )
a
t0
b+
−∞
f (t) > f (t0 )
a
⊆R
f ε)t0 , (1 + ε)t0 )
−
2
≈ f (t0 ) + (1/2)f (t0 )(t − t0) (1+ε)t G(x) ∼
0
e−x(f (t0 )+f
(t0 )(t−t0 )2 /2)
dt.
(1−ε)t0
e−xf (t0 )
x(t
t0 = f (t0 ) = 0
a < 0 < b x/2 = y t = s/y
− t0)/2t = s
f (t0 )
G(x) = G(2y2 ) =
yb
e−2y
2
ya
1 f (s/y) 1 ds = y y
yb
− exp
ya
f (s/y) 2s2 (s/y)2
ds,
+∞
2
yG(2y ) =
ψ(s, y) ds,
−∞
ψ(s, y) =
−
2s2 f (s/y) (s/y)2
exp 0,
,
ya < s < yb
.
+ y f (0) = f (0) = 0 f (u) = (1/2)f (0)u2 + o(u2 ) u f (u)/u2 f (0)/2 > 0 u δ >0 0 0 2 f (s/y)/(s/y) > f (0)/4 s/y < δ
→ ∞
→
→
ψ(s, y) f u f (u)/u f (0) = 0 s/y > δ y > 1
γ >0
→
→
f ( ε)/ε
−
ψ(s, y)
≤ exp ψ
ψ(s, y)
2
−
−f (0)s /2
≤e
s/y < δ.
|u| > δ
f (u) f
≥ γ |u|
γ f (s/y)/ s/y
2s2
+ e
2
≤ e−f (0)s /2 ,
|
γ s/y (s/y)2
|
−2|s|
f (ε)/ε
| | ≥ γ = e −2|sy| ≤ e−2|s| , s/y < δ
+∞
−∞
(e−f
(0)s2 /2
s/y > δ. s/y > δ
+ e−2|s| ) ds < +
∞,
y +∞
lim yG(2y ) =
y→+∞
f (s/y)/(s/y)2
+∞
→ 2
lim ψ(s, t) =
−∞ y→+∞
f (0)/2 ya f (0)s = u y =
→ −∞
−f (0)s2
e
ds =
yb
y
→ +∞
π , f (0)
→ +∞
x/2
2π . xf (0)
∼
t0 g(t) = f (t0 + t)
∈R
(a
g(0) = 0 b
G(x) =
−∞
G(x)
f (t0 ) f (t0 )
∈ [1, +∞)
g (0) = f (t0 ) −xf (t)
e
dt =
a
− t0 , b − t0 )
t = t 0 + s
b−t0
−
e−xf (t0 +s) ds =
a−t0
= e−xf (t0 )
b−t0
e−xg(s) ds
a−t0
∼ e−xf (t ) 0
2π , xf (t0 )
x
→ +∞.
Γ(x)
√
2πx x−1/2 e−x ,
∼
x
→ +∞.
Γ +∞
xΓ(x) =
x −t
t e
+∞
dt =
0
e−t+x log t dt.
0
t = xs +∞
xΓ(x) =
e−xs+x log s+x log x x ds,
0
Γ(x) = x
x
+∞
e−xh(s) ds,
h(s) = s
0
h (s) = 1
− 1/s Γ(x)
− log s.
h (s) = 1/s2
∼ xx
s = 1
√ 2πe−xh(1)
xh (1)
h h(1) = 1
h (1) = 1
√ 2πe−x = x x √ , x
n+1/2
log n! = log 1 + log 2 +
· ·· + log n =
log x dx + Rn =
1/2
= (n + 1/2)log(n + 1/2) = (n + 1/2)log n C 1 =
− n + C 1 + Rn =
− n + (n + 1/2) log(1 + 1/(2n)) + C 1 + Rn,
Rn (n + 1 /2) log(1 + 1/(2n))
−(1/2) log(1/2)
n!
∼ eC √ nnne−n,
→ 1/2
C = 1/2 + C 1 + lim Rn C eC =
√ 2π Γ
Γ (n)
Γ
+∞
(x) =
tx−1 e−t logn t dt.
0
∈ [δ, ∆] |tx−1e−t logn t| ≤ t∆−1 e−t| logn t|, x
t > 1
|tx−1e−t logn t| ≤ tδ−1e−t| logn t|,
t< 1
2 x F (x) = 0 e−t dt F (x) = G (x)
2
2
2
1 e−x (t +1) dt 0 t2 +1 F (x) = π4 + G(x) +∞ −t2 e dt 0
G(x) =
−
π
J n (x) = 0 cos(nϕ x sin ϕ) dϕ 2 x y + xy + (x2 n2 )y = 0 R f : (0, + ) f (0+) = lim f (x) f (+ ) = lim f (x) x→0+ x→+∞ +∞ f (bx) f (ax) dx = (f (+ x 0
∞ → ∞
−
−
a, b > 0
∞) − f (0+)) log ab .
−
0 < u < v < v f (bx)−f (ax) x u
b f (vt)−f (ut) t a
+
∞
=
f : (0, 1]
dt
× (0, 1] → R
x2 y 2 f (x, y) = 2 . (x + y 2 )2
−
f 1
0
D
1
1
f (x, y) dx dy =
0
⊆ (0, 1] × (0, 1] lim
a→+∞
0
1
f (x, y) dy dx.
0
|f (x, y)| dx dy = ∞ D
0 < ρ < 1 δ
≤ |θ − π/4| ≤ π/4
a
+ 0
1+x 1+x2a
∞
1+xa 1+x2a
+ 0
dx = 1
∞
+ 0
arctg αx arctg βx x2
+
f (y) =
∞ 0
a
log(a2 +x2 ) b2 +x2
+ 0
∞ ∞
dx
dx
dx
sin xy dx, x(a2 + x2 )
a = 0.
2
f (y)
− a f (y) = −π/2 J 1 =
−1
a, b >
−
1 0
sin(log(1/x)) =
J 2 1
b
a
x −x log x
dx a , b >
αx sinβ x x
+ 0
∞ e− ∞ I (α) = ∞ I (a) =
∈R
dx
−1
α > 0
√
log(αx+ α2 x2 −1) + 1/α x(1+x2 )
1 xb −xa 0 log x
dx a,b > = cos(log(1/x)) − dx 1 0
J 3
xb xa log x
β
∈ R
dx α > 0
I (1)
+ +∞ eax eax J (a) = 0 0
−
1 sin αx 0 x x
√ ∞ J (α) =
J (α) =
+ 0
M ∞ π/2 = ∞ dx = π/2 dx + 0
+ 0 sin x x
sin ax x(1+x2 )
a > 0
dx
cos ax 1+x2
|
|≤
∈
J (a) +
∈
J (0) ∞
J (0) lim J (a)
a→∞
J (a)
J (a) = π/2
∈R
dx a R J (a) M a R J (a) = J (a) a > 0
J (a)
J (a) =
α
−
+ 0
∞
sin ax dx x(1+x4 )
a > 0
lim
a→+∞
a 0
+ + 0
cos ax 1+x2
dx da
J (iv)(a) +
+ 0
∞ +
I (x, y) =
∞ e−
xt
x, y
∈R
x, y
−
−e
bx2
x2
dx a, b
∈ R
+
yt
cos at dt
t
0
a
− e−
2
e−ax
xt
∞ e−
J (x, y) =
0
I (x, y) J (x, y) lim I (x, y) lim J (x, y) a→+∞ a→+∞ x = 0 y > 0 lim y →+∞
≥ 0
yt
t
sin at dt,
≥ 0
I (α) = J (α) = J (α)
I (α) = + 0
arctg αx x2
√
x sin2 αx x2
+ 0
∞
dx
x2 −1
√
α
sin t t
dt
∈R
≤ α < 1
√
log(α2 +x2 ) 1+x2
+ 0
∞
∞ e−
+ 1
∞
1 log(1−α2 x2 ) dx 0 0 x 1−x2 1 dJ 1 dα = 0 x 1−x2
J (α)
α ∈∞ − ∞ xe − ∞ x e− x e
− e−
dx α
dx
∈R
α
d dα log(1
2 2
− α x ) dx
0 < α < 1
∈R
+ 0
∞
sin2 x x2
dx
R
1 x2 2
2 π
+ 0
2 π
+ 0
2
1 x2 2
cos αx dx
2 π
+ 0
3
1 x2 2
sin αx dx
sin αx dx
2
1 e−αx x2
+ 0
∞ − dx α ≥ 0 I (α) = (log(1/x)) log(log(1/x)) dx ∞ I (a, b) = b ≥ 0 I (α) =
1 0
xb dx (xa +1)(x4 +1)
+ 0
+ 0
− ∞
α
xa+b log x dx (xa +1)2 (x4 +1)
I a (a, b) =
I (a, b)
I (a, 1)
I (a, a + 1)
α 1 +∞ x α 0 < α < 1 g(α) = 0 x1+x dx h(α) = 0 1+x dx n (−1) ∞ (−1)n 1 h(α) = g(α) + g(1 α) g(α) = ∞ n=1 α−n h(α) = α + 2α n=1 α2 −n2 ∞ x α log x h(α) dx 1+x 0
−
−
+ 0
∞
arctg ax dx x(1+x2 )
f : R2
α < 0
f (x, α) =
−
a
≥0
→x, α ≥ 00 ≤ x ≤ √ α √ √ √ f (x, α) = −x + 2 α, α
f 1 h (0) = −1
−
R
h :
R
2
∂f ∂α (x, 0) dx
R
→R
∂f ∂α
1
h(α) = −1 f (x, α) dx
