✶

▼Ö❈ ▲Ö❈
❚r❛♥❣

▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶
▲❮■ ◆➶■ ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷
❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✶ ❍➔♠ ●❛♠♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✷ P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ L1 (R) ✈➔ L2 (R) ✳ ✶✵
❈❤÷ì♥❣ ✷✳ ●✐↔✐ ♠ët ❜➔✐ t♦→♥ ❦❤✉➳❝❤ t→♥ ♥❣÷ñ❝ ❜➟❝ ♣❤➙♥ ❜➡♥❣
♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛ ♣❤ê ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✶ ✣↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✷ ●✐î✐ t❤✐➺✉ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✸ ❈❤➾♥❤ ❤â❛ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷


▲❮■ ◆➶■ ✣❺❯
●➛♥ ✤➙②✱ ♣❤➨♣ t➼♥❤ ✈✐ ♣❤➙♥ ❜➟❝ ♣❤➙♥ ✈➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠
r✐➯♥❣ ❜➟❝ ♣❤➙♥ ✤➣ ✤÷ñ❝ sû ❞ö♥❣ ✤➸ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ tr♦♥❣ ❝→❝ ❧➽♥❤
✈ü❝ ✈➟t ❧þ✱ ❤â❛ ❤å❝✱ s✐♥❤ ❤å❝✱ ❝ì ❦❤➼✱ ①û ❧þ t➼♥ ❤✐➺✉✱ ✤✐➺♥ tû✱ ✤✐➲✉ ❦❤✐➸♥
tè✐ ÷✉ ✈➔ t➔✐ ❝❤➼♥❤ ✭①❡♠ ❬✻❪✮✳
P❤÷ì♥❣ tr➻♥❤ ❦❤✉➳❝❤ t→♥ ❜➟❝ ♣❤➙♥ ①✉➜t ❤✐➺♥ ❦❤✐ t❛ t❤❛② ✤↕♦ ❤➔♠
❜➟❝ ♥❣✉②➯♥ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❦❤✉➳❝❤ t→♥ ❜➡♥❣ ♠ët ✤↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥✳
❑✐➸✉ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ✤÷ñ❝ sû ❞ö♥❣ ✤➸ ♠æ t↔ ❝→❝ q✉→ tr➻♥❤ ❦❤✉➳❝❤ t→♥
❜➜t t❤÷í♥❣ ✭❛♥♦♠❛❧♦✉s ❞✐❢❢✉s✐♦♥✮ ♥❤÷ ❦❤✉➳❝❤ t→♥ tr➯♥ ✭s✉♣❡r❞✐❢❢✉s✐♦♥✮✱
❦❤✉➳❝❤ t→♥ ❞÷î✐ ✭s✉❜❞✐❢❢✉s✐♦♥✮✳ ❈→❝ q✉→ tr➻♥❤ ❦❤✉➳❝❤ t→♥ ♥➔② ❦❤æ♥❣ t✉➙♥
t❤❡♦ ✤à♥❤ ❧✉➟t ❋✐❝❦ ❝ê ✤✐➸♥✳
❇➔✐ t♦→♥ ❦❤✉➳❝❤ t→♥ ♥❣÷ñ❝ ❜➟❝ ♣❤➙♥ t❤÷í♥❣ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤ t❤❡♦




tr õ 0 Dt u t ợ (0 <

1) ữủ

x


t
1
g (s)
=
ds, 0
(1 ) 0 (t s)
dg(t)

, = 1.
0 Dt g(t) =
dt

0 Dt g(t)








õ ú tổ ợ t t t ữủ
tr t qừ õ t tr ụ ữ
t ự ởt t qừ ợ

t

ử tr t ởt số t t ỡ
ừ õ

ợ t

ử ợ t t t ữủ

õ t

ử tr ữỡ õ t t
ữủ t qừ tố ở ở tử ừ ữỡ
tr ụ ữ t ởt t qừ ợ
ữủ tỹ t rữớ ồ ữợ sỹ ữợ
ừ t ự tọ ỏ t
ỡ s s ừ t t
ỡ Pỏ t ồ ừ ữ P
ồ ỡ t ổ tr ở ổ t ữ P
ồ t t ú ù t tr sốt tớ
ồ t t ố ũ t ỡ
ỗ t tr ợ ồ t


✭✶✳✶✮

0

✈î✐ z t❤✉ë❝ ♥û❛ ♠➦t ♣❤➥♥❣ ❜➯♥ ♣❤↔✐ ❝õ❛ ♠➦t ♣❤➥♥❣ ♣❤ù❝ ❘❡z > 0✳

✶✳✶✳✷ ◆❤➟♥ ①➨t✳ ❚➼❝❤ ♣❤➙♥ ✭✶✳✶✮ ❤ë✐ tö ✈î✐ ♠å✐ z ∈ C t❤ä❛ ♠➣♥ ❘❡z >
0✳ ❚❤➟t ✈➟②✱ ✈î✐ z ∈ C t❤ä❛ ♠➣♥ ❘❡z > 0✱ t❛ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ z = x + iy
✈î✐ x, y ∈ R ✈➔ x > 0✳ ❑❤✐ ✤â t❛ ❝â
∞

Γ(z) = Γ(x + iy) =
∞

=
0

=

∞

e−t tx−1+iy dt

0

e−t tx−1 eiy ln t dt
e−t tx−1 (cos(y ln t) + i sin(y ln t)) dt.

✭✶✳✷✮


e−t tz dt

0
−t z

= −e t

∞

t=∞

+z
t=0

e−t tx−1+iy dt

0

= zΓ(z).
2) ❚❛ ❝â
∞

−t 0

−t

e t dt = −e

Γ(1) =


◆❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✭✶✳✸✮ ✈î✐ e−u ✈➔ ❧➜② t➼❝❤ ♣❤➙♥ tø 0 ✤➳♥ ∞ t❛ ✤÷ñ❝
∞
2

I =

e

−u2

0

u

2 2

e−u t dt du

0
∞

∞

=
0

1
=
2

2
ờ t = u2 t s t ữủ

õ I =

x2
dx
0 e

=



(z) = 2

2

eu u2z1 du.



0

t z =

1
t ữủ
2



1
n
n
= n
2
2
2
3
3 1
1
1
n
ã ã ã . .
= n
2
2
2 2
2
(2n)!
.
= 2n
2 n!
=

n

ỵ ợ ồ z C tọ z > 0 t õ
n!nz
.
n z(z + 1) ã ã ã (z + n)


t ữủ
1
z

(1 )n z1 d

fn (z) = n

0
1

nz

=

z
= ããã

(1 )n1 z d

n
0

n!nz
=
z(z + 1) ã ã ã (z + n 1)
n!nz
=
.

n 0
n

t
=
lim 1
n
0 n

n

tz1 dt
n


z1

t

dt =

et tz1 dt = (z).

0



t ữủ ử ú t ữủ




> 0 tũ ỵ ứ sỹ ở tử ừ t ợ ồ z C tọ
z > 0 t s r tỗ t số tỹ n0 s ợ ồ n N

n

n0 t õ


t z1

e t
n



dt
n


et tx1 dt < , (x = z).
3




✾

❱î✐ ♠å✐ n ∈ N∗ ♠➔ n > n0 ✱ t❛ ✈✐➳t ∆ t❤➔♥❤ tê♥❣ ❝õ❛ ❜❛ t➼❝❤ ♣❤➙♥ s❛✉
N

∞
z−1

t

dt +

e−t tz−1 dt.

✭✶✳✶✶✮

n

❚❛ ❝â
n

e
N

−t

n

t
− 1−
n

n

n

n

t2


e dτ = e
.
n
2n
ττ

t

❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✸✮ t❛ ❝â ✤→♥❤ ❣✐→ s❛✉ ✈î✐ n ✤õ ❧î♥
N

e
0

−t

t
− 1−
n

n

t

z−1

1
dt

✶✳✶✳✻ ◆❤➟♥ ①➨t✳ ❈æ♥❣ t❤ù❝ ✭✶✳✺✮ ❦❤æ♥❣ ❝❤➾ ✤ó♥❣ ✈î✐ ♠å✐ z ∈ C t❤ä❛
♠➣♥ ❘❡z > 0 ♠➔ ❝á♥ ✤ó♥❣ ✈î✐ ♠å✐ z ∈ C ♠➔ z = 0, −1, −2, · · · ❚❤➟t
✈➟②✱ tø ❝æ♥❣ t❤ù❝ ✭✶✳✶✺✮ ✈➔ ✣à♥❤ ❧þ ✶✳✶✳✹ t❛ ❝â

Γ(z + m)
z(z + 1) · · · (z + m − 1)
1
nz+m n!
lim
=
z(z + 1) · · · (z + m − 1) n→∞ (z + m) · · · (z + m + n)
1
nz n!
=
lim
z(z + 1) · · · (z + m − 1) n→∞ (z + m)(z + m + 1) · · · (z + n)
nm
× lim
n→∞ (z + n)(z + n + 1) · · · (z + n + m)

Γ(z) =

1
nz n!
=
lim
z(z + 1) · · · (z + m − 1) n→∞ (z + m)(z + m + 1) · · · (z + n)
1
nz n!
= lim

f (ξ) := √
2π
∨

+∞

eix.ξ f (x)dx (ξ ∈ R).
−∞

✭✶✳✶✼✮


✶✶

❱➻ e±ixξ = 1 ✈➔ f ∈ L1 (R) ♥➯♥ ❝→❝ t➼❝❤ ♣❤➙♥ tr➯♥ ❤ë✐ tö ✈î✐ ♠é✐ ξ ∈ R.
❙❛✉ ✤➙② ❧➔ ♠ët ✈➔✐ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ f (ξ)
✶✳ f (ξ) ❧➔ ❤➔♠ ❜à ❝❤➦♥✱ ✈➻
+∞
1
e−ix.ξ f (x)dx
f (ξ) = √
2π −∞
+∞
1
√
e−ix.ξ |f (x)| dx
2π −∞
+∞
1
=√


|f (x)| 2 sin(
−∞
+∞

xy
) dx
2

+∞

|f (x)| dx.

|f (x)| dx + yR

+
−∞

+R

−∞

−R

❱î✐ ε > 0 ✱ t❛ ❝â t❤➸ ❝❤å♥ R ✤õ ❧î♥ ✈➔ y ✤õ ❜➨ ✤➸ ❜✐➸✉ t❤ù❝ ❝ë♥❣ ❧↕✐ ❧➜②
tê♥❣ ❝✉è✐ ❝ò♥❣ ♥❤ä ❤ì♥ ε✳
✸✳ ◆➳✉ c1 ✈➔ c2 ❧➔ ❝→❝ sè t❤ü❝ t❤➻

(c1 f1 + c2 f2 ) = c1 f1 + c2 f2 .
✹✳ Df (x) = −iξ f (x) ✈î✐ Df (x) ∈ L1 (R)✳

A→∞ 2π −A
1
= lim √
(e−ixξ f (x))A
−A − iξ
A→∞ 2π

A

e−ixξ f (x)dx

a

= −iξ f (ξ).
✭❞♦ ❣✐î✐ ❤↕♥ ❜à tr✐➺t t✐➯✉✱ ❧ ❂ ✲♠ ❂ ✵ ✮✳

✶✳✷✳✷ ✣à♥❤ ♥❣❤➽❛ ✭❚➼❝❤ ❝❤➟♣✮✳ ❚➼❝❤ ❝❤➟♣ ❝õ❛ ❝→❝ ❤➔♠ f, g ❦þ ❤✐➺✉ ❧➔
f ∗ g ✈➔ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛
+∞

(f ∗ g)(x) =

f (y)g(x − y)dy.
−∞

✶✳✷✳✸ ✣à♥❤ ❧þ✳ ❚➼❝❤ ❝❤➟♣ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✤➙②
✶✮ ∀f, g ∈ L1 (R), f ∗ g ∈ L1 (R) ✈➔ f ∗ g

1


=
−∞
+∞

−∞
+∞

≤

|f (y)|
−∞

= f

1

|g(x − y)|dx dy
−∞

g 1.

✷✮ ❱î✐ x ∈ R t❛ ❝â
+∞

(f ∗ g)(x) =

f (y)g(x − y)dy
−∞
+∞



✭✶✳✶✽✮

❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❝→❝ ❤➔♠ g, h ∈ L1 (R)✳ ❑❤✐ ✤â g, h ∈ L∞ (R) ✭❞♦ t➼♥❤


✶✹

❝❤➜t ✶✮✳ ❚❛ ❝â
+∞

+∞

g(x)h(x)dx =
−∞

g(x)

1
√
2π

+∞

+∞

−∞

1
=√

+∞

+∞

1
=√
2π

+∞

e−ixξ g(x)h(ξ)dxdξ.

−∞

❙✉② r❛
+∞

+∞

g(x)h(x)dx =
−∞

✭✶✳✶✾✮

g(ξ)h(ξ)dξ.
−∞

❚❛ ❧↕✐ ❝â
+∞


1 −
2
t
4t
√
e
=
e
d( tx)
t
−∞
ξ2
ξ2
1 − √
π −
= √ e 4t π =
e 4t . (t > 0)
t
t
❙✉② r❛
+∞

ξ2
π −
e 4t (t > 0).
t

2

e−ixξ−tx dx =

✭✶✳✷✵✮

−∞

▲➜② f (x) ∈ L1 (R) ∩ L2 (R) ✈➔ ✤➦t g(x) := f (−x). ❳➨t

h := f ∗ g ∈ L1 (R) ∩ L2 (R).
❚❛ ❝â
+∞
1
e−ix.ξ (f ∗ g)(x)dx
h(ξ) = f ∗ g(ξ) = √
2π −∞
+∞
+∞
1
−ixξ
e
f (y)g(x − y)dy dx
=√
2π −∞
−∞
+∞
+∞
1
−ixξ
=√
e
f (y)
e−iξ(x−y) g(x − y)d(x − y) dy


2

2π f

.

❱➻ h ❧✐➯♥ tö❝ ♥➯♥

1
lim √
ε→0 2ε
❉♦ h =

√

2

2π f

+∞

ξ2
√
−
h(ξ)e 4ε dξ = 2πh(0).

−∞

≥ 0 ♥➯♥ ❦❤✐ ❝❤♦ ε → 0+ tr♦♥❣ ✭✶✳✷✵✮ t❛ ❝â


2

f (x) dx = h(0) =

+∞

f (y)g(−y)dy =
−∞

−∞

✣✐➲✉ ♥➔② t÷ì♥❣ ✤÷ì♥❣ ✈î✐ f

2

= f

|f (y)|2 dy.

−∞
2.

❉♦ ✤â f = f . ❚÷ì♥❣ tü t❛

❝ô♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ f ∨ = f .

✶✳✷✳✺ ✣à♥❤ ♥❣❤➽❛ ✭✣à♥❤ ♥❣❤➽❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ L2(R) ✮✳ ❚❛ ✤à♥❤
♥❣❤➽❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r f ❝õ❛ f ∈ L2 (R) ♥❤÷ s❛✉
❈❤♦ ♠ët ❞➣② {fk }∞

✐✐✮ Dα f =

L2 (R)✱

fˆgˆdξ ✱

−∞
(iξ)α f

✈î✐ ♠é✐ ❝❤➾ sè α ♥❣✉②➯♥ ❞÷ì♥❣ s❛♦ ❝❤♦ Dα f ∈


✶✼

✐✐✐✮ f ∗ g =

√

2π f g ✱

✐✈✮ f = (f )∨ .
❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❈❤♦ f, g ∈ L2 (R) ✈➔ α ∈ C ✳ ❑❤✐ ✤â✱ t❤❡♦ ✣à♥❤ ❧þ ✶✳✷✳✹✱
t❛ ❝â f + αg

2

= f + αg 2 . ❑❤❛✐ tr✐➸♥ t❛ ✤÷ñ❝
+∞

+∞

¯
αf g + α
¯ f g¯ dξ.

αf¯g + α
¯ f g¯ dx =
−∞

−∞

❱î✐ α = 1 t❤➻
+∞

+∞

f¯g + f g¯ dx =

¯
f g + f g¯ dξ.

✭✶✳✷✶✮

¯
−if g + if g¯ dξ.

✭✶✳✷✷✮

−∞

−∞

+∞
1
=√
e−ixξ Dα f (x)dx
2π −∞
(−1)α +∞ α −ixξ
= √
Dx e
f (x)dx
2π −∞
+∞
1
e−ix.ξ (iξ)α f (x)dx
=√
2π −∞
= (iξ)α f (ξ).

❇➡♥❣ ❝→❝❤ t✐➳♥ tî✐ ❣✐î✐ ❤↕♥✱ ❝æ♥❣ t❤ù❝ tr➯♥ s➩ ✤ó♥❣ ♥➳✉ Dα f ∈ L2 (R)✳
✐✐✐✮ ❱î✐ f (x), g(x) ∈ L1 (R) ∩ L2 (R) ✈➔ ξ ∈ R t❛ ❝â

1
f ∗ g(ξ) = √
2π
1
=√
2π
1
=√
2π



+∞

+∞

f (y)

e−i(x−y).ξ g(x − y)dx dy

−∞

f (y)dy g(ξ)

−∞

2π f (ξ)g(ξ).
2

✐✈✮ ❈è ✤à♥❤ y ∈ R, ε > 0 ✈➔ ✤➦t gε (ξ) := eiξy−εξ . ❚❛ ❝â

1
gε (ξ) = √
2π
1
=√
2π

+∞

2

1
f eiyξ−εξ dξ = √
2ε

+∞
−∞

−(x − y)2
−
4ε
f (x)e
dx.

✭✶✳✷✸✮


✶✾

❱➳ ♣❤↔✐ ❝õ❛ ✭✶✳✷✸✮ ❞➛♥ tî✐

1
√
2π
❱➟② (f )∨ = f ✳

√

2πf (y) ❦❤✐ ε → 0+ ✳ ❙✉② r❛

+∞


t
a

f (n) (s)
ds, a
(t − s)α+1−n

t

T, n − 1 < α < n,

t

T, α = n.

✷✳✶✳✷ ◆❤➟♥ ①➨t✳ ✶✮ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ n = 1✱ t❛ ❝â
C (α)
a Dt f (t)

1
=
Γ(1 − α)

t
a

f (s)
ds, a
(t − s)α

❍ì♥ ♥ú❛ f ❧➔ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝➜♣ n + 1 tr➯♥ ✤♦↕♥ [a, T ] ✈î✐ T > a
t❤➻
(α)

n
lim C
a Dt f (t) = f (t), ∀t ∈ [a, T ].

α→n

✭✷✳✷✮

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â

f (n) (a)(t − a)n−α
α→n
Γ(n − α + 1)
t
1
(t − τ )n−α f (n+1) (τ )dτ
+ lim
α→n Γ(n − α + 1) a

(α)

lim C
a Dt f (t) = lim

α→n



=
k=0

∞

k=0

dn (λtn )k
dtn Γ(kn + 1)

nk(nk − 1).....(nk − n + 1)λk tn(k−1)
.
Γ(kn + 1)

❈❤ó þ r➡♥❣

Γ(kn + 1) = knΓ(kn) = kn(kn − 1)Γ(kn − 1)
= ... = kn(kn − 1)...(kn − n + 1)Γ(kn − n + 1).


✷✷

❚❛ ❝â
∞
C (α)
0 Dt f (t)

=λ
k=1


dn
α k
dsn (λs )

t
0

Γ(αk + 1)(t − s)α+1−n

0

t dn E (λsα )
dsn α,1
ds
(t − s)α+1−n

ds.

❚÷ì♥❣ tü ♥❤÷ tr÷í♥❣ ❤ñ♣ α ❧➔ sè ♥❣✉②➯♥ t❛ ❝â
C (α)
0 Dt f (t)

λ
=
Γ(n − α)
=

λ
Γ(n − α)

0 Dt u(x, t)

= −aux (x, t), x > 0, t > 0, α ∈ (0, 1),

u(1, t) = f (t), t

✭✷✳✸✮
✭✷✳✹✮

0,

✭✷✳✺✮

u(x, 0) = lim u(x, t) = 0,
x→∞

tr♦♥❣ ✤â 0 Dtα u ❧➔ ✤↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦ ✈î✐ ❜➟❝ α (0 < α

1) ✤÷ñ❝

①→❝ ✤à♥❤ ❜ð✐✿
t
1
g (s)
=
ds, 0
Γ(1 − α) 0 (t − s)α
dg(t)
α
, α = 1.

1
2

.

R

❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ f (t) ✤÷ñ❝ ✈✐➳t ❧➔
∞

1
fˆ(ω) = √
2π
✈➔

·

p

f (t)e−iωt dt,

−∞

❧➔ ❦þ ❤✐➺✉ ❝❤✉➞♥ Hp ✱ tù❝ ❧➔

f

p

(1 + ω 2 )p |fˆ(ω)|2 dω

ω

απ
| ω |α (cos απ
2 − i sin 2 ),

ω < 0.

0,

✭✷✳✶✵✮

❚ø ✭✷✳✽✮ ✈➔ ✭✷✳✾✮ t❛ ❞➵ ❞➔♥❣ t➼♥❤ ✤÷ñ❝
1

α

uˆ(x, ω) = e a (iω) (1−x) fˆ(ω),
1
α
1
uˆ(x, ω) = − (iω)α e a (iω) (1−x) fˆ(ω).
a

✭✷✳✶✶✮
✭✷✳✶✷✮





ux tứ u(1, t) = f (t) ởt t t ổ õ
qt t t r t t ữỡ õ

õ t
ởt tỹ ờ t tr ỷ tt t
số tr ừ t ử t t õ
s ờ rr

uc (x, ) = u(x, )max ,



uc,x (x, ) = u(x, )max ,



ừ õ

tr õ max trữ tr [max , max ] tự

max () =

1 [max , max ]
0
/ [max , max ].

õ õ õ t t ữủ sỷ ử
ờ rr ữủ

1

u(x, .) − uδc (x, .)

u(x, .) − uc (x, .) + u(x, .) − uδc (x, .) ,

✭✷✳✶✾✮

✈➔ s❛✐ sè ❝õ❛ ✤↕♦ ❤➔♠ ❝õ❛ ♥â ❧➔

ux (x, .) − uδc,x (x, .)

ux (x, .) − uc.x (x, .)
+ uc,x (x, .) − uδc,x (x, .) .

✭✷✳✷✵✮

❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ s➩ ✤÷❛ r❛ ✤→♥❤ ❣✐→ tè❝ ✤ë ❤ë✐ tö ❝❤♦ u(x, .)−uc (x, .)
✈➔ ux (x, .) − uc.x (x, .) ♥❤í ✈✐➺❝ ❝❤å♥ t❤➼❝❤ ❤ñ♣ t➛♥ sè ❝❤➦t ❝öt ωmax ✈➔
♠ët ❣✐↔ t❤✐➳t ✈➲ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ u ✈➔ ✤↕♦ ❤➔♠ ux ❝õ❛
♥â✳

✷✳✸✳✶ ✣à♥❤ ❧þ✳ ✭❬✺❪✮ ●✐↔ sû u ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✸✮✕✭✷✳✺✮✱ uδc ❧➔
♥❣❤✐➺♠ ❝❤➾♥❤ ❤â❛ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✶✼✮ ✈î✐ ❞ú ❦✐➺♥ ❜à ♥❤✐➵✉ f δ ✈➔ f δ
t❤ä❛ ♠➣♥ f δ − f

δ✳
1

✭✶✮ ◆➳✉ u(0, .)

E 2


−p
α

.

✭✷✳✷✷✮

✷✳✸✳✷ ✣à♥❤ ❧þ✳ ✭❬✺❪✮ ●✐↔ sû u ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✸✮✕✭✷✳✺✮✱ uδc,x
❧➔ ✤↕♦ ❤➔♠ ❝õ❛ ♥❣❤✐➺♠ ❝❤➾♥❤ ❤â❛ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝ ✭✷✳✶✽✮ ✈î✐ ❞ú
❦✐➺♥ ❜à ♥❤✐➵✉ f δ ✈➔ f δ − f
✭✶✮ ◆➳✉ u(0, ·)

δ✳
1

E 2
E ✤ó♥❣ ✈➔ ωmax ✤÷ñ❝ ❝❤å♥ ❧➔ ωmax = (a sec απ
2 ln δ )

t❤➻ ✈î✐ ♠å✐ x ∈ (0, 1) t❛ ❝â ✤→♥❤ ❣✐→

uδc,x (x, ·) − ux (x, ·)

1 + sec

απ E
ln
2
δ