✶

▼Ö❈ ▲Ö❈
❚r❛♥❣

▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶
▲❮■ ◆➶■ ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷
❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✶ ❍➔♠ ●❛♠♠❛ ✈➔ ❤➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✷ ✣↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
❈❤÷ì♥❣ ✷✳ ❳→❝ ✤à♥❤ ♥❣✉ç♥ ♥❤✐➺t tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥
♥❤✐➺t ❜➟❝ ♣❤➙♥ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➦t ❝öt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷✳✶ ●✐î✐ t❤✐➺✉ ❜➔✐ t♦→♥ ✈➔ ✤→♥❤ ❣✐→ ê♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷✳✷ ❈❤➾♥❤ ❤â❛ ❝❤➦t ❝öt ✈➔ ✤→♥❤ ❣✐→ s❛✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵



r ỳ ữỡ tr r
t út ữủ sỹ q t ự ừ ồ
ổ sỷ ử ữỡ tr r ữủ ự
ử t ổ ổ t t tr s t t ỵ õ ồ
õ s t t tr tr
ữỡ tr t ữủ ự rở r ợ
ở ữ ỵ ỹ sỹ tỗ t t t
ữỡ số ữ ữỡ tỷ ỳ ữỡ
s ỳ ụ ữủ ử t tr
tr ởt số t ố ử t ởt tr tổ t
ỳ ỳ số t ỗ ổ
ữủ t ú t ú tứ ỳ ờ s

t ữỡ t ửt
ừ

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 ồ ỡ
t ổ tr ở ổ t ồ t t
ú ù t tr sốt tớ ồ t t
ữỡ ố ũ t ỡ ỗ
t tr ợ ồ t ở t
ú ù ở t tr sốt q tr ồ t ự
ũ õ ố ữ ổ tr ọ
ỳ t sõt ú tổ rt ữủ ỳ ỵ
õ õ ừ t ổ ữủ
t ỡ

t




❈❍×❒◆● ✶

▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❇✃ ❚❘Ñ
❈❤÷ì♥❣ ♥➔② ♥❤➡♠ ♠ö❝ ✤➼❝❤ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥
♥ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ✷✱ ❝❤õ ②➳✉ ✤÷ñ❝ ❝❤ó♥❣ tæ✐ t❤❛♠ ❦❤↔♦ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✺❪✳

✶✳✶ ❍➔♠ ●❛♠♠❛ ✈➔ ❤➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r
✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❍➔♠ ❣❛♠♠❛ Γ ❤❛② t➼❝❤ ♣❤➙♥ ❊✉❧❡r ❧♦↕✐ ✷ ❧➔ t➼❝❤

0

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

✭✶✳✷✮

0

❱➻ ✤↕✐ ❧÷ñ♥❣ (cos(y ln t) + i sin(y ln t)) ❜à ❝❤➦♥ ♥➯♥ ❞➵ ♥❤➟♥ t❤➜② r➡♥❣ t➼❝❤
♣❤➙♥ ✭✶✳✷✮ ❤ë✐ tö ✈î✐ ♠å✐ x > 0 ✈➔ ♠å✐ y ∈ R✳

✶✳✶✳✸ ✣à♥❤ ❧þ✳ ❍➔♠ ❣❛♠♠❛ Γ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉
✹


✺

✶✮ Γ(z + 1) = zΓ(z), ∀z ∈ C, ❘❡z > 0✱
✷✮ Γ(1) = 1✱
✸✮ Γ(n + 1) = n!, ∀n ∈ N∗✱
✹✮ Γ 12 = √π✱
√
✺✮ Γ n + 12 = 2(2n)!
π✱
2n n!
✻✮ ❱î✐ ♠å✐ sè t❤ü❝ x > 0 t❛ ❝â
∞

Γ(x) = 2

= zΓ(z).
2) ❚❛ ❝â
∞

Γ(1) =

e−t t0 dt = −e−t

t=∞

= 1.
t=0

0

3) ❙û ❞ö♥❣ ❝→❝ t➼♥❤ ❝❤➜t 1) ✈➔ 2)✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❝❤➜t 3)
❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣✳

4) ❚r÷î❝ ❤➳t t❛ t➼♥❤ t➼❝❤ ♣❤➙♥ I =
I=u

∞ −x2
dx✳
0 e
∞
−u2 t2

e

✣➦t x = ut, u > 0✱ t❛ ❝â


e−u t dt du

0
∞

=

∞

∞
0

2

e−u

(1+t2 )

0

dt
π
= .
2
1+t
4

udu dt


❇➡♥❣ ❝→❝❤ t❤❛② z =

1
✈➔♦ ✭✶✳✹✮ t❛ ✤÷ñ❝
2
∞

1
2

Γ

=2

2

e−u du = 2I =

√

π.

0

5) ❚ø t➼♥❤ ❝❤➜t 1) ✈➔ t➼♥❤ ❝❤➜t 4) t❛ ❝â
Γ n+

1
2



n−

6) ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❤➔♠ ❣❛♠♠❛ t❛ ❝â
∞

Γ(x) =

e−t tx−1 dt.

0

❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ✤ê✐ ❜✐➳♥ t = u2 ✈î✐ u ❜✐➳♥ t❤✐➯♥ tø ✵ ✤➳♥ ∞ t❛ ❝â
∞

Γ(x) =
0

∞

=
0

e−t tx−1 dt

∞

=2
0



fn (z) =
0

❇➡♥❣ ❝→❝❤ ✤ê✐ ❜✐➳♥ τ =

n

t
1−
n

tz−1 dt.

✭✶✳✻✮

s❛✉ ✤â sû ❞ö♥❣ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ❝❤ó♥❣ t❛

t
n

✤➦t ✤÷ñ❝
1

(1 − τ )n τ z−1 dτ

z

fn (z) = n


❈❤ó þ r➡♥❣

lim

n→∞

n

t
1−
n

= e−t .

❉♦ ✤â ♠ö❝ ✤➼❝❤ t✐➳♣ t❤❡♦ ❝õ❛ ❝❤ó♥❣ t❛ ❧➔ ❝❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝
n

t
1−
lim fn (z) = lim
n→∞
n→∞ 0
n
∞
t
=
lim 1 −
n
0 n→∞


0

t
− 1−
n

e−t tz−1 dt − fn (z)
n

∞
z−1

t

dt +
n

e−t tz−1 dt.

✭✶✳✾✮




> 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

n0 t õ

n

0
n

e
N

õ
n

e
N

t

tz1 dt
n

t
1
n

t

+

n

t
1
n

tx1 dt
n
N






1
n



0

ỵ ữủ ự


✾

✶✳✶✳✺ ◆❤➟♥ ①➨t✳ ◆❤í t➼♥❤ ❝❤➜t 1) tr♦♥❣ ✣à♥❤ ❧þ ✶✳✶✳✸✱ ♥❣÷í✐ t❛ ❝â t❤➸
✤à♥❤ ♥❣❤➽❛ ❤➔♠ ❣❛♠♠❛ Γ(z) ❝❤♦ ♠å✐ z ∈ C ♠➔ z = 0, −1, −2, · · · ❈ö
t❤➸✱ ♥➳✉ z ∈ C ♠➔ −m < ❘❡z

−m + 1 ✈î✐ m ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣

♥➔♦ ✤â t❤➻ t❛ ①→❝ ✤à♥❤ Γ(z) t❤❡♦ ❝æ♥❣ t❤ù❝

Γ(z) =

Γ(z + m)
.
z(z + 1) · · · (z + m − 1)

✭✶✳✶✺✮

✶✳✶✳✻ ◆❤➟♥ 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!
=

Eα (z) =
k=0

zk
, α > 0, z ∈ C.
Γ(αk + 1)

❍➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r ❤❛✐ t❤❛♠ sè
∞

Eα,β (z) =
k=0

❧➔ ❤➔♠ ❝â ❞↕♥❣
✭✶✳✶✻✮

❧➔ ❤➔♠ ❝â ❞↕♥❣

zk
, α > 0, β > 0, z ∈ C.
Γ(αk + β)

✭✶✳✶✼✮




t ợ ồ z C z = 0 t õ



5) E2,2 (z ) =
k=0




k=0


zk
= ez ,
k!

k=0


z 2k
=
(2k + 1)

k=0

z 2k+1
sinh(z)
=
,
(2k + 1)
z



=
(k + 1)! z

2 z2

=
e

( k2 + 1)

2

2

et dt = ez r(z).

z

ú t (, ) ( > 0, 0 <

) ữớ ỗ t

s
arg = | |


arg




exp(ζ 1/α )ζ (1−β)/α
1
dζ, z ∈ G− (ε, µ)
2απi γ(ε,µ)
ζ −z
1
Eα,β (z) = z (1−β)/α exp z 1/α
α
1
exp(ζ 1/α )ζ (1−β)/α
dζ, z ∈ G+ (ε, µ).
+
2απi γ(ε,µ)
ζ −z
Eα,β (z) =

✭✶✳✶✾✮

✭✶✳✷✵✮

❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ |z| < ε t❤➻
z
< 1, ζ ∈ γ(ε, µ).
ζ

✭✶✳✷✶✮

❉♦ ✤â ✈î✐ 0 < α < 2 ✈➔ |z| < ε t❛ ❝â
∞


exp(ζ 1/α )ζ (1−β)/α
γ(ε,µ)

zk

ζ −z

z
ζ

dζ.

dζ
✭✶✳✷✷✮

◆❤í ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✽✮✱ t➼❝❤ ♣❤➙♥ ð ✭✶✳✷✷✮ ❤ë✐ tö t✉②➺t ✤è✐ ✈➔ ①→❝ ✤à♥❤ ♠ët
❤➔♠ ❜✐➳♥ z ❣✐↔✐ t➼❝❤ tr♦♥❣ ❝→❝ ♠✐➲♥ G− (ε, µ) ✈➔ G+ (ε, µ)✳ ▼➦t ❦❤→❝✱ ✈î✐
πα
, min{π, πα} ✱ ❤➻♥❤ trá♥ |z| < ε ♥➡♠ tr♦♥❣ ♠✐➲♥ G− (ε, µ)✳
♠é✐ µ ∈
2
❉♦ ✤â✱ t➼❝❤ ♣❤➙♥ ✭✶✳✷✷✮ ❜➡♥❣ Eα,β (z) ❦❤æ♥❣ ❝❤➾ tr♦♥❣ ♠✐➲♥ ❤➻♥❤ trá♥

|z| < ε ♠➔ tr♦♥❣ t♦➔♥ ❜ë ♠✐➲♥ G− (ε, µ) ♥➯♥ t❛ ❝â ❝æ♥❣ t❤ù❝ ✭✶✳✶✾✮✳
❇➙② ❣✐í ❝❤ó♥❣ t❛ ❧➜② z ∈ G+ (ε, µ)✳ ❑❤✐ ✤â ✈î✐ ❜➜t ❦ý ε1 > |z| t❛ ❝â

z ∈ G− (ε1 , µ)✳ ❙û ❞ö♥❣ ❝æ♥❣ t❤ù❝ ✭✶✳✶✾✮ t❛ ❝â
Eα,β (z) =

1


♠➣♥

πα
< µ < min{π, πα}
2

t❤➻ ✈î✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ p tò② þ t❛ ❝â ✤→♥❤ ❣✐→
1
Eα,β (z) = z (1−β)/α exp z 1/α −
α
|z| → ∞, | arg(z)|

p

k=1

z −k
+ O |z|−1−p
Γ(β − αk)

✭✶✳✷✺✮

µ.

❈❤ù♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ ❜➢t ✤➛✉ ❝❤ù♥❣ ♠✐♥❤ ❝æ♥❣ t❤ù❝ ✭✶✳✷✺✮ ❜➡♥❣ ❝→❝❤
❝❤å♥ ϕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥

πα
< µ < ϕ < min{π, πα}.

exp z 1/α
α
p
1
−
exp(ζ 1/α )ζ (1−β)/α+k−1 dζ
2απi γ(1,ϕ)

z −k

k=1

+

1
2απiz p

exp(ζ 1/α )ζ (1−β)/α+p ζ.
γ(1,ϕ)

✭✶✳✷✽✮


✶✸

❈❤ó þ r➡♥❣

1
2απi



γ(1,ϕ)

µ, |z| > 1).

❈❤ó♥❣ t❛ ❤➣② ✤→♥❤ ❣✐→ t➼❝❤ ♣❤➙♥

Ip (z) =

1
2απiz p

✈î✐ |z| ✤õ ❧î♥ ✈➔ | arg(z)|

exp(ζ 1/α )ζ (1−β)/α+p ζ,

✭✶✳✸✶✮

γ(1,ϕ)

µ✳ ❱î✐ |z| ✤õ ❧î♥ ✈➔ | arg(z)|

µ t❛ ❝â

min |ζ − z| = |z| sin(ϕ − µ).
ζ∈γ(1,ϕ)

❉♦ ✤â ✈î✐ |z| ✤õ ❧î♥ ✈➔ | arg(z)|

|Ip (z)|

πα
< µ < min{π, πα}
2


✶✹

t❤➻ ✈î✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ p tò② þ t❛ ❝â ✤→♥❤ ❣✐→
p

Eα,β (z) = −
k=1

z −k
+ O |z|−1−p
Γ(β − αk)

|z| → ∞, µ

| arg(z)|

✭✶✳✸✸✮

π.

❈❤ù♥❣ ♠✐♥❤✳ ❚r÷î❝ ❤➳t✱ ❝❤ó♥❣ t❛ ❧➜② ϕ t❤ä❛ ♠➣♥
πα
< ϕ < µ < min{π, πα}.
✭✶✳✸✹✮
2

✈➔ µ

| arg(z)|
|Ip (z)|

| arg(z)|

π ✳ ❉♦ ✤â ✈î✐ |z| ✤õ ❧î♥

π t❛ ✤↕t ✤÷ñ❝

|z|−1−p
2πα sin(ϕ − µ)

exp(ζ 1/α ) ζ (1−β)+p ζ.

✭✶✳✸✻✮

γ(1,ϕ)

❚ê ❤ñ♣ ✭✶✳✸✺✮ ✈➔ ✭✶✳✸✻✮ t❛ ✤↕t ✤÷ñ❝ ❝æ♥❣ t❤ù❝ ✭✶✳✸✸✮✳

✶✳✷ ✣↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦
✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ f ❧➔ ♠ët ❤➔♠ sè ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝➜♣ n ∈ N∗
tr➯♥ [a, T ] (T > a)✳

✣↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦ ✈î✐ ❜➟❝ α > 0 ❝õ❛ ♠ët

❤➔♠ f tr➯♥ ✤♦↕♥ [a, T ] ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉
C (α)

C (α)
a Dt f (t)

1
=
Γ(1 − α)

t
a

f (s)
ds, a
(t − s)α

t

T, 0 < α < 1.

✭✶✳✸✼✮

✷✮ ❱î✐ n ∈ N∗ ✱ α ∈ R t❤ä❛ ♠➣♥ n − 1 < α < n✱ m ∈ N ✈➔ f ❧➔ ❤➔♠ ❦❤↔
✈✐ ❧✐➯♥ tö❝ ❝➜♣ n + m t❤➻ t❛ ❝â
C (α) C m
a Dt f (t)
a Dt

(α+m)

=C
a Dt

α→n Γ(n − α + 1) a

lim C
a Dt f (t) = lim

α→n

t

=f

(n)
n

f (n+1) (τ )dτ

(a) +
a

= f (t), ∀t ∈ [a, T ].


❈❍×❒◆● ✷

❳⑩❈ ✣➚◆❍ ◆●❯➬◆ ◆❍■➏❚ ❚❘❖◆● P❍×❒◆● ❚❘➐◆❍
❚❘❯❨➋◆ ◆❍■➏❚ ❇❾❈ P❍❹◆ ❇➀◆● P❍×❒◆● P❍⑩P
❈❍➄❚ ❈Ö❚
❈❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ❝→❝ ❦➳t qõ❛ tr♦♥❣ ❜➔✐ ❜→♦ ❬✹❪ ❝ô♥❣
♥❤÷ ✤➲ ①✉➜t ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët ✈➔✐ ❦➳t qõ❛ ♠î✐✳


❇➔✐ t♦→♥ ✭✷✳✶✮ ❧➔ ❜➔✐ t♦→♥ t❤✉➟♥ ❦❤✐ f (x) ✤➣ ✤÷ñ❝ ❝❤♦ t❤➼❝❤ ❤ñ♣✳
❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tæ✐ q✉❛♥ t➙♠ tî✐ ✧❜➔✐ t♦→♥ ♥❣✉ç♥ ♥❣÷ñ❝✧
❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮✳ ❚ù❝ ❧➔ ❜➔✐ t♦→♥ ①→❝ ✤à♥❤ ♥❣✉ç♥ ♥❤✐➺t ❝❤÷❛ ✤÷ñ❝
❜✐➳t f (x) ❞ü❛ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ✈➔ ♠ët t❤æ♥❣ t✐♥ ❜ê s✉♥❣ ✈➲ ♥❣❤✐➺♠
❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ t↕✐ t❤í✐ ✤✐➸♠ t = T

u(x, T ) = g(x), 0 < x < 1.
✶✻

✭✷✳✷✮


✶✼

❚r♦♥❣ ❝→❝ ù♥❣ ❞ö♥❣ ❝ö t❤➸✱ ❞ú ❦✐➺♥ ✤➛✉ ✈➔♦ g(x) ✤÷ñ❝ ❝✉♥❣ ❝➜♣ ❜ð✐
✤♦ ✤↕❝ ♥➯♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ s❛✐ sè✳ ❉♦ ✤â✱ t❤❛② ✈➻ ❜✐➳t ❝❤➼♥❤ ①→❝ g(x)
t❛ ❝❤➾ ❜✐➳t ❞ú ❦✐➺♥ ✤♦ ✤↕❝ g δ ∈ L2 (0, 1) t❤ä❛ ♠➣♥

gδ − g
✈î✐

·

✭✷✳✸✮

δ,

❧➔ ❝❤✉➞♥ L2 ✈➔ ♠ù❝ s❛✐ sè δ > 0 ✤➣ ✤÷ñ❝ ❜✐➳t✳

✷✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✹❪✮✣↕♦ ❤➔♠ ❈❛♣✉t♦ ❜➟❝ α ✤÷ñ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣

t
σ

z(s)
ds, 0 < α < 1.
(t − s)α

✭✷✳✺✮

✷✳✶✳✷ ▼➺♥❤ ✤➲✳ ✭❬✺❪✮ ●✐ú❛ ✤↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦ ✈➔ ✤↕♦ ❤➔♠ ❜➟❝
♣❤➙♥ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❝â ♠è✐ q✉❛♥ ❤➺ s❛✉
α
σ Dt z(t)

=

z(σ)
1
+σ ∂tα z(t).
α
Γ(1 − α) (t − σ)

✷✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✹❪✮❍➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r ❤❛✐ t❤❛♠ sè

✭✷✳✻✮
✤÷ñ❝ ①→❝ ✤à♥❤

❜ð✐
∞


n

✈î✐ ♠å✐ t

0

✈➔

1 = Eα,1 (0) > Eα,1 (−t) > 0, t > 0.

✭✷✳✶✵✮


✶✽

❈❤ó þ r➡♥❣ ♥❣❤✐➺♠ u(x, t) ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✮ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ ❞÷î✐
❞↕♥❣
t

u(x, t) =

v(x, t; τ )dτ,

✭✷✳✶✶✮

0

tr♦♥❣ ✤â v(x, t; τ ) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ s❛✉

 α

t

v(x, t; τ )dτ

u(x, t) =
0
∞

t
1−α
1Eα,1 (−k 2 π 2 (t − τ )α )dτ fk Xk .
0 Dτ

=

✭✷✳✶✸✮

0

k=1

▼➦t ❦❤→❝✱ t❛ ❜✐➳t r➡♥❣ ✭①❡♠ ❬✸❪✮
t
1−α
1Eα,1 (−k 2 π 2 (t − τ )α )dτ fk Xk
0 Dτ
0
t

(t − τ )α−1 Eα.α (−k 2 π 2 (t − τ )α )dτ fk Xk .

k2π2

❉➵ t❤➜② r➡♥❣ A ❧➔ t♦→♥ tû ❝♦♠♣❛❝t ✈î✐ ❝→❝ ❣✐→ trà ❦ý ❞à {σk }∞
k=1 ✤÷ñ❝
①→❝ ✤à♥❤ ❜ð✐

1 − Eα,1 (−k 2 π 2 τ α )
σk =
k2π2

✈➔

1 − Eα,1 (−k 2 π 2 τ α )
(g, Xk ) =
(f, Xk ).
k2π2
1
❉♦ ✤â t❛ ❝â (f, Xk ) = (g, Xk ) ✈➔
σk
∞
−1

f (x) = A

g(x) =
k=1

k2π2
(g, Xk )Xk .
1 − Eα,1 (−k 2 π 2 τ α )


✭✷✳✶✼✮




tr õ E số ữỡ

ã

ỵ tr ổ

H p (0,1)

H p (0, 1) ữủ
1
2



f

H p (0,1)

(1 + k 2 )p | fk |2

=




ự ỷ ử t tự r t
õ


f

2

=
k=1


=
k=1


=
k=1






k=1

k=1

k=1



gk

2p
p+2

2
p+2

p

1 E,1 (k 2 2 T )

2p

4

| gk | p+2 | gk | p+2

k=1

1 E,1 (k 2 2 T )
k22

2

| gk |2

p+2
2

p+2



(1 + k 2 )p | fk |2
k=1

2p
p+2

gk

2p
p+2

2p
p+2

)

p+2
p

p
p+2



2p
p+2


.

t f1(x) f2(x) ừ t ỗ
ữủ ợ ỳ g1 (x) g2 (x) tữỡ ự t s
ú

f1 (ã) f2 (ã)
2
1 E,1 ( 2 T )

p
p+2

f1 (ã) f2 (ã)

2
p+2
H p (0,1)

g1 (ã) g2 (ã)

p
p+2

.

ó r r g1 (ã) g2 (ã) 0 t f1 (ã) f2 (ã) 0
t qừ ờ õ ổ sỹ ờ ừ
số tr t ợ ỳ

✷✳✷✳✶ ✣à♥❤ ❧þ✳ ✭❬✹❪✮ ●✐↔ sû fδ,K (x) ❧➔ ♥❣❤✐➺♠ ❝❤➾♥❤ ❤â❛ ❝õ❛ ❜➔✐ t♦→♥
✭✷✳✶✮✲✭✷✳✷✮

✈î✐ ❞ú ❦✐➺♥ ❜à ♥❤✐➵✉ gδ (x) ✈➔ f (x) ❧➔ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ✈î✐ ❞ú
❦✐➺♥ ❝❤➼♥❤ ①→❝ g(x)✳ ◆❣♦➔✐ r❛ t❛ ❣✐↔ t❤✐➳t t❤➯♠ r➡♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✷✳✸✮
✈➔ ✭✷✳✶✼✮ ✤÷ñ❝ t❤ä❛ ♠➣♥✳ ◆➳✉ t❤❛♠ sè ❝❤➾♥❤ ❤â❛ ✤÷ñ❝ ❝❤å♥ ❧➔ K = [γ]
✭[γ] ❧➔ ❦þ ❤✐➺✉ ♣❤➛♥ ♥❣✉②➯♥ ❝õ❛ γ ✮ ✈î✐
E
δ

γ=

1
p+2

✭✷✳✷✶✮

t❤➻ ✤→♥❤ ❣✐→ s❛✉ ✤➙② ✤ó♥❣
f (·) − fδ,K (·)

1+

π2
1 − Eα,1 (−π 2 T α )

2

p

E p+2 δ p+2 .

=
k=K+1
K

+
k=1
∞

K

k2π2
(g δ , Xk )Xk
2
2
α
1 − Eα,1 (−k π T )

k=1
K

k=1
K

k=1

k2π2
(g, Xk )Xk
1 − Eα,1 (−k 2 π 2 T α )
k2π2
(g δ , Xk )Xk

(g − g δ , Xk )Xk
k=1

k2π2
1 − Eα,1 (−π 2 T α )

δ.




K

K + 1 t t ữủ



22

1 E,1 ( 2 T )
2
2
2
p+2 p+2 .
1+
E
1 E,1 ( 2 T )

p E +


D p+2 ,



H p (0,1)

p ồ ữ

t ỳ tr t t ử t
t ú tổ tr t ồ t số

m


(g , Xk )Xk

Pm g =



k=1

K = K(, g ) ừ t

(I PK )g



(I PK1 )g , > 1.


(g, Xk )Xk
k=K
∞

=
k=K
∞

=
k=K

1 − Eα,1 (−k 2 π 2 T α )
(f, Xk )Xk
k2π2
1 − Eα,1 (−k 2 π 2 T α )
2 −p
2 p
2 (1 + k ) 2 (f, X )X
(1
+
k
)
k
k
k2π2

1 − Eα,1 (−k 2 π 2 T α )
2 −p
sup
(1

.

❇ê ✤➲ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳

✷✳✷✳✺ ✣à♥❤ ❧þ✳ ✭❬✹❪✮ ●✐↔ sû fδ,K (x) ❧➔ ♥❣❤✐➺♠ ❝❤➾♥❤ ❤â❛ ❝õ❛ ❜➔✐ t♦→♥

✭✷✳✶✮✲✭✷✳✷✮

✈î✐ ❞ú ❦✐➺♥ ❜à ♥❤✐➵✉ gδ (x) ✈➔ f (x) ❧➔ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ✈î✐ ❞ú
❦✐➺♥ ❝❤➼♥❤ ①→❝ g(x)✳ ◆❣♦➔✐ r❛ t❛ ❣✐↔ t❤✐➳t t❤➯♠ r➡♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✷✳✸✮
✈➔ ✭✷✳✶✼✮ ✤÷ñ❝ t❤ä❛ ♠➣♥✳ ◆➳✉ t❤❛♠ sè ❝❤➾♥❤ ❤â❛ K ✤÷ñ❝ ❝❤å♥ ❧➔ ♥❣❤✐➺♠
❝õ❛ ✭✷✳✷✺✮ t❤➻ t❛ ❝â ✤→♥❤ ❣✐→ s❛✉
f (·) − fδ,K (·)

2

p

CE p+2 δ p+2 ,

✭✷✳✸✵✮


✷✺

tr♦♥❣ ✤â
p
−2
π2
π 2 (τ + 1)


H p (0,1)

=

f k Xk
k=K+1

H p (0,1)

∞

(1 + k 2 )p fk2

=

1
2

✭✷✳✸✷✮

E.

k=K+1

❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝ ✈➔ ✭✷✳✷✺✮ t❛ ❝â

Af − AfK = (I − PK )g
(I − PK )g δ + (I − PK )(g − g δ )
(I − PK )g δ + (I − PK )(g − g δ )