BỘ GIÁO DỤC VÀ ĐÀO TẠO
VIỆN HÀN LÂM
KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM
viÖn to¸n häc
------
TỐNG THỊ HỒNG NGỌC
MỘT SỐ LỚP
PHƯƠNG TRÌNH TÍCH PHÂN KỲ DỊ
LUẬN VĂN THẠC SĨ TOÁN HỌC
Hà Nội – 2015
BỘ GIÁO DỤC VÀ ĐÀO TẠO
VIỆN HÀN LÂM
KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM
viÖn to¸n häc
TỐNG THỊ HỒNG NGỌC
MỘT SỐ LỚP
PHƯƠNG TRÌNH TÍCH PHÂN KỲ DỊ
CHUYÊN NGÀNH: TOÁN GIẢI TÍCH
MÃ SỐ: 60 46 01 02
LUẬN VĂN THẠC SĨ TOÁN HỌC
NGƯỜI HƯỚNG DẪN KHOA HỌC:
TS. NCVC. NGUYỄN VĂN NGỌC
Hà Nội – 2015
▼ö❝ ❧ö❝
▼ð ✤➛✉
✶
✶
✸
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶
✶✳✷
✶✳✸
✶✳✹
✶✳✺
❚➼❝❤ ♣❤➙♥ s✉② rë♥❣ ✈➔ ❣✐→ trà ❝❤➼♥❤ ❈❛✉❝❤② ❝õ❛ t➼❝❤ ♣❤➙♥ ✸
❇✐➳♥ ✤ê✐ ▼❡❧❧✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
❇✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✷ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ tr➯♥ ❦❤♦↔♥❣ ✈æ ❤↕♥
✷✺
✸ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ tr➯♥ ❦❤♦↔♥❣ ❤ú✉ ❤↕♥
✹✷
✷✳✶
✷✳✷
✷✳✸
✷✳✹
✷✳✺
✸✳✶
✸✳✷
✸✳✸
✸✳✹
✸✳✺
❉➝♥ ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ❣✐↔✐ ❜➡♥❣ ❜✐➳♥ ✤ê✐ ▼❡❧❧✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
P❤÷ì♥❣ tr➻♥❤ ❣✐↔✐ ❜➡♥❣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
P❤÷ì♥❣ tr➻♥❤ ❱♦❧t❡rr❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ♥❤➙♥ ❈❛✉❝❤② tr➯♥ t♦➔♥ trö❝
P❤÷ì♥❣ tr➻♥❤ ❆❜❡❧ ❧♦↕✐ ♠ët ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❆❜❡❧ ❧♦↕✐ ❤❛✐ ✳ ✳ ✳ ✳
P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ♥❤➙♥ ❈❛✉❝❤②
P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❈❛r❧❡♠❛♥♥ ✳ ✳ ✳ ✳ ✳
P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ❍✐❧❜❡rt ✳ ✳ ✳ ✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✷✺
✷✻
✷✾
✸✺
✹✵
✹✷
✹✺
✹✻
✹✾
✺✶
❑➳t ❧✉➟♥
✺✺
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦
✺✻
✐
tr ồ ỡ ồ t ỵ tt
ỳ ữỡ tr tr õ ữ t ữủ ự ữợ
t ỳ ữỡ tr õ ữủ ồ ữỡ tr t
Pữỡ tr t ởt q trồ ừ t
ồ ổ ử ỳ tr ỹ ữủ q
t ự t ữ sỹ tỗ t
ữỡ ú t t
ỵ tt tờ qt ữỡ tr t t t
ữủ ỹ ờ tớ ừ t ừ tr
ổ tr ừ trr r rt
t ữỡ tr t t t tr (a, b) õ s
b
u(x) +
K(x, y)u(y)dy = f (x), a < x < b,
a
tr õ u(x) t K(x, y) t ữủ
ồ ừ ữỡ tr f (x) ữủ
ồ ừ ữỡ tr t số ừ ữỡ tr
Pữỡ tr ữủ ồ ữỡ tr t ởt
tũ tở t số = 0, = 0, tữỡ ự
r ữỡ tr t a, b số ỳ ổ
ỷ ổ ữủ ố t
K(x, y) õ t tử tờ õ ý t tữớ õ
ố ợ ữỡ tr t t t
q t tợ t s
(a, b) ổ ỷ ổ ỏ K(x, y)
tử õ t tữớ õ K(x, x) = .
s t ồ
(a, b)
K(x, x) = .
ố ỗ ồ
ỳ ỏ K(x, y) õ t tữớ
tt ỳ tr ỳ ữỡ tr t tr ữủ
ồ ữỡ tr t ý t ồ t ừ
ử ừ t ồ t tự
ừ ởt số ợ ữỡ tr t ý t tr
ỗ ở s
ữỡ tự r t s rở t
ý ờ t rr
rt ỡ s ự ữỡ tr t ữỡ s
ữỡ Pữỡ tr t tr ổ r ởt
số ợ ữỡ tr t õ t ữủ ờ
t rr rt
ữỡ t ữỡ tr t ý tr ỳ
ợ rt rt
ữủ t ỹ tr t tr õ
t
ữủ t t ồ
ồ ổ t ữợ sỹ ữợ ừ t
ồ ổ t tọ ỏ t ỡ s s tợ
t t ữợ ở tổ tr sốt q tr tỹ
ổ ỡ ổ tr ỏ Pữỡ tr
q t ú ù tổ tr sốt q tr
ổ t ỡ ồ t
ũ ổ trỹ t ợ ồ
ồ t t ủ ở ú ù tổ tr sốt
q tr ồ t ự t
ở
ố ỗ ồ
❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶ ❚➼❝❤ ♣❤➙♥ s✉② rë♥❣ ✈➔ ❣✐→ trà ❝❤➼♥❤ ❈❛✉❝❤② ❝õ❛
t➼❝❤ ♣❤➙♥
•
✣→♥❤ ❣✐→ t➼❝❤ ♣❤➙♥ s✉② rë♥❣
❈â ❤❛✐ ❧♦↕✐ t➼❝❤ ♣❤➙♥ s✉② rë♥❣✿ ✭✶✮ t➼❝❤ ♣❤➙♥ ♠➔ ❤➔♠ ❞÷î✐ ❞➜✉ t➼❝❤
♣❤➙♥ ❝â ✤✐➸♠ ❣✐→♥ ✤♦↕♥ ✈æ ❝ò♥❣ tr♦♥❣ ❝➟♥ ❝õ❛ t➼❝❤ ♣❤➙♥ ✈➔ ✭✷✮ t➼❝❤
♣❤➙♥ ♠➔ ❝➟♥ ❝õ❛ t➼❝❤ ♣❤➙♥ ❧➔ ✈æ ❤↕♥✳ ❙ü tç♥ t↕✐ ❝õ❛ ❝↔ ❤❛✐ ❧♦↕✐ t➼❝❤
♣❤➙♥ ♣❤ö t❤✉ë❝ sü tç♥ t↕✐ ❝õ❛ ♠ët ❣✐î✐ ❤↕♥ ❤♦➦❝ ♥❤✐➲✉ ❣✐î✐ ❤↕♥✳
❚➼❝❤ ♣❤➙♥ s✉② rë♥❣ ✤÷ñ❝ ❣å✐ ❧➔ ❤ë✐ tö ♥➳✉ ♠å✐ ❣✐î✐ ❤↕♥ ❧✐➯♥ q✉❛♥
tç♥ t↕✐✱ ✈➔ ♣❤➙♥ ❦ý ♥➳✉ ♠ët tr♦♥❣ ❝→❝ ❣✐î✐ ❤↕♥ ❦❤æ♥❣ tç♥ t↕✐✳
✶✳ ◆➳✉ ❤➔♠ ❞÷î✐ ❞➜✉ t➼❝❤ ♣❤➙♥ f (t) ❧✐➯♥ tö❝ tr➯♥ ❦❤♦↔♥❣ [a, c) ✈➔
❝â ✤✐➸♠ ❣✐→♥ ✤♦↕♥ ✈æ ❝ò♥❣ t↕✐ ✤➛✉ ♠ót ❜➯♥ ♣❤↔✐ ❝õ❛ ❦❤♦↔♥❣✱
t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛ t➼❝❤ ♣❤➙♥ ❝õ❛ f (t) tr➯♥ ❦❤♦↔♥❣ [a, c) ❧➔
c
c−ε
f (t)dt = lim
ε↓0
a
f (t)dt,
a
♥➳✉ ❣✐î✐ ❤↕♥ tç♥ t↕✐✳
❚÷ì♥❣ tü✱ ♥➳✉ ❤➔♠ ❞÷î✐ ❞➜✉ t➼❝❤ ♣❤➙♥ ❝â ✤✐➸♠ ❣✐→♥ ✤♦↕♥ ✈æ
❝ò♥❣ t↕✐ ✤➛✉ ♠ót tr→✐ ❝õ❛ ❦❤♦↔♥❣✱ t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛ t➼❝❤ ♣❤➙♥
❝õ❛ f (t) tr➯♥ ❦❤♦↔♥❣ (c, b] ❧➔
b
b
f (t)dt = lim
ε↓0
c
✸
f (t)dt,
c+ε
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
♥➳✉ ❣✐î✐ ❤↕♥ tç♥ t↕✐✳
◆➳✉ ❤➔♠ ❞÷î✐ ❞➜✉ t➼❝❤ ♣❤➙♥ ❝â ✤✐➸♠ ❣✐→♥ ✤♦↕♥ ✈æ ❝ò♥❣ t↕✐
✤✐➸♠ tr♦♥❣ c ❝õ❛ ❦❤♦↔♥❣ [a, b]✱ t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛
b
c
f (t)dt =
b
f (t)dt +
a
f (t)dt,
a
c
♥➳✉ ❝↔ ❤❛✐ ❣✐î✐ ❤↕♥ ♠ët ♣❤➼❛ tç♥ t↕✐ ✤ë❝ ❧➟♣ ✈î✐ ♥❤❛✉✳
✷✳ ◆➳✉ ❤➔♠ ❞÷î✐ ❞➜✉ t➼❝❤ ♣❤➙♥ f (t) ❧✐➯♥ tö❝ tr➯♥ ❦❤♦↔♥❣ [c, +∞)✱
t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛ t➼❝❤ ♣❤➙♥ ❝õ❛ f (t) tr➯♥ ❦❤♦↔♥❣ ♥➔② ❧➔
+∞
b
f (t)dt = lim
b→+∞
c
f (t)dt,
c
♥➳✉ ❣✐î✐ ❤↕♥ tç♥ t↕✐✳
◆➳✉ ❤➔♠ ❞÷î✐ ❞➜✉ t➼❝❤ ♣❤➙♥ ❧✐➯♥ tö❝ tr➯♥ ❦❤♦↔♥❣ (−∞, c]✱ t❤➻
t❛ ✤à♥❤ ♥❣❤➽❛ t➼❝❤ ♣❤➙♥ ❝õ❛ f (t) tr➯♥ ❦❤♦↔♥❣ ♥➔② ❧➔
c
c
f (t)dt = lim
a→−∞
−∞
f (t)dt,
a
♥➳✉ ❣✐î✐ ❤↕♥ tç♥ t↕✐✳
◆➳✉ ❤➔♠ ❞÷î✐ ❞➜✉ t➼❝❤ ♣❤➙♥ ❧✐➯♥ tö❝ tr➯♥ ❦❤♦↔♥❣ (−∞, +∞)✱
t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛ t➼❝❤ ♣❤➙♥ ❝õ❛ f (t) tr➯♥ ❦❤♦↔♥❣ ♥➔② ❧➔
+∞
c
f (t)dt =
−∞
+∞
f (t)dt +
−∞
f (t)dt,
c
♥➳✉ ❝↔ ❤❛✐ ❣✐î✐ ❤↕♥ ♠ët ♣❤➼❛ tç♥ t↕✐ ✤ë❝ ❧➟♣ ✈î✐ ♥❤❛✉✳
▼ët t➼❝❤ ♣❤➙♥ ❝❤♦ tr÷î❝ ❝â t❤➸ ❧➔ ❝↔ ❤❛✐ ❧♦↕✐ t➼❝❤ ♣❤➙♥ s✉②
rë♥❣✳
• ●✐→ trà ❝❤➼♥❤ ❈❛✉❝❤② ❝õ❛ t➼❝❤ ♣❤➙♥
❚❛ t❤➜② r➡♥❣ ♥➳✉ ❤➔♠ ❞÷î✐ ❞➜✉ t➼❝❤ ♣❤➙♥ ❝â ✤✐➸♠ ❣✐→♥ ✤♦↕♥ ✈æ
❝ò♥❣ tr♦♥❣ ♣❤➛♥ tr♦♥❣ ❝õ❛ ❝➟♥ t➼❝❤ ♣❤➙♥ ❤♦➦❝ ♥➳✉ ❝➟♥ t➼❝❤ ♣❤➙♥
♠ð rë♥❣ tî✐ ✈æ ❝ò♥❣ t❤❡♦ ❝↔ ❤❛✐ ❤÷î♥❣✱ t❤➻ sü tç♥ t↕✐ ❝õ❛ t➼❝❤ ♣❤➙♥
♣❤ö t❤✉ë❝ sü tç♥ t↕✐ ❝õ❛ ❤❛✐ ❣✐î✐ ❤↕♥ ♠ët ❝→❝❤ ✤ë❝ ❧➟♣✳ ❚❤➟♠ ❝❤➼
✹
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
♥➳✉ ❦❤æ♥❣ ❝â ❣✐î✐ ❤↕♥ ♥➔♦ tr♦♥❣ ❤❛✐ ❣✐î✐ ❤↕♥ tç♥ t↕✐✱ ♠ët ❣✐î✐ ❤↕♥
✤è✐ ①ù♥❣ ✤ì♥ ❝â t❤➸ ✈➝♥ tç♥ t↕✐✳
❈❤♦ t➼❝❤ ♣❤➙♥ ❝â ❤➔♠ ❞÷î✐ ❞➜✉ t➼❝❤ ♣❤➙♥ f (t) ❝â ✤✐➸♠ ❣✐→♥ ✤♦↕♥
✈æ ❝ò♥❣ c tr♦♥❣ ♣❤➛♥ tr♦♥❣ ❝õ❛ ❝➟♥ t➼❝❤ ♣❤➙♥✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❣✐→ trà
❝❤➼♥❤ ❈❛✉❝❤② ❝õ❛ t➼❝❤ ♣❤➙♥ ❝õ❛ f (t) tr➯♥ ❦❤♦↔♥❣ [a, b] ❧➔
b
PV
c−ε
f (t)dt = lim
f (t)dt +
ε↓0
a
b
a
f (t)dt ,
c+ε
♥➳✉ ❣✐î✐ ❤↕♥ tç♥ t↕✐✳
❈❤♦ t➼❝❤ ♣❤➙♥ ❝â ❝➟♥ t➼❝❤ ♣❤➙♥ ♠ð rë♥❣ r❛ ✈æ ❝ò♥❣ t❤❡♦ ❝↔ ❤❛✐
❤÷î♥❣✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❣✐→ trà ❝❤➼♥❤ ❈❛✉❝❤② ❝õ❛ t➼❝❤ ♣❤➙♥ ✈î✐ f (t)
tr➯♥ ❦❤♦↔♥❣ (−∞, +∞) ❧➔
+∞
PV
+a
f (t)dt = lim
a→∞
−∞
f (t)dt ,
−a
♥➳✉ ❣✐î✐ ❤↕♥ tç♥ t↕✐✳
❚❛ s➩ ✈✐➳t P V ♣❤➼❛ tr÷î❝ ❝→❝ t➼❝❤ ♣❤➙♥ ♥❤÷ tr➯♥✳ ❈➛♥ ❝❤ó þ r➡♥❣
❝↔ ❤❛✐ ❣✐î✐ ❤↕♥ ♥➔② ✤÷ñ❝ ①→❝ ✤à♥❤ ❜➡♥❣ ❣✐î✐ ❤↕♥ ✤è✐ ①ù♥❣✳ ●✐î✐ ❤↕♥
❦❤æ♥❣ ✤è✐ ①ù♥❣ ❝â t❤➸ ❝ô♥❣ tç♥ t↕✐✱ ♥❤÷♥❣ ❣✐→ trà ❝õ❛ ♥â ❝â t❤➸
❦❤→❝ ✤✐✳
❳➨t ✈➼ ❞ö ♠✐♥❤ ❤å❛ s❛✉✳ ▼ët ♠➦t✱ t❛ ❝â
2
PV
0
1−ε
1
dt = lim
ε↓0
t−1
0
1
dt +
t−1
2
1
dt
1+ε t − 1
= 0.
▼➦t ❦❤→❝✱ t❛ ❝â
1−2ε
0
1
dt +
t−1
2
1
dt =
t
−
1
1+ε
1+2ε
1+ε
1
dt = ln 2.
t−1
❱✐➺❝ ❝❤å♥ 2ε ❧➔ ❜➜t ❦ý✳ ◆➳✉ t❛ ✤➣ ❝❤å♥ kε✱ t❤➻ ❣✐→ trà ❝õ❛ t➼❝❤ ♣❤➙♥
❧➔ ln k✳ ✣✐➸♠ q✉❛♥ trå♥❣ ð ✤➙② ❧➔ ❣✐→ trà ❝❤➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥ ❧➔ ♠ët
❣✐→ trà ❝ö t❤➸ ♠➔ ✤÷ñ❝ ❝❤å♥ tø ✈æ sè ❝→❝ ❣✐→ trà ❝â t❤➸✳
✺
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
▼ët ✈➼ ❞ö ❦❤→❝ ♠✐♥❤ ❤å❛ sü q✉❛♥ trå♥❣ ❝õ❛ ❣✐î✐ ❤↕♥ ✤è✐ ①ù♥❣ ♥❤÷
s❛✉✳ ▼ët ♠➦t✱ t❛ ❝â
+∞
PV
−∞
2t
dt = lim
a→∞
1 + t2
+a
−a
2t
dt
1 + t2
= 0,
✈➻ ❤➔♠ ❞÷î✐ t➼❝❤ ♣❤➙♥ ❧➔ ❧➫✳ ▼➦t ❦❤→❝✱ t❛ ❝â
+2a
−a
2t
dt =
1 + t2
+2a
+a
2t
1 + 4a2
dt = ln
1 + t2
1 + a2
→ ln 4,
❦❤✐ a → +∞✳ ❑❤æ♥❣ ❝â ❣➻ ✤➦❝ ❜✐➺t ✈➲ ❝→❝❤ ❝❤å♥ 2a ð ✤➙②✳ ◆➳✉ t❛
✤➣ ❝❤å♥ ka✱ t❤➻ ❣✐→ trà ❝õ❛ t➼❝❤ ♣❤➙♥ ❧➔ ln(k2).
• ❚➼❝❤ ♣❤➙♥ ●❛♠♠❛
❚➼❝❤ ♣❤➙♥ ●❛♠♠❛ ✭❤➔♠ ●❛♠♠❛✮ ✈î✐ ❜✐➳♥ ♣❤ù❝ z = x + iy,
(i2 = −1) ✤÷ñ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝
∞
e−t tz−1 dt, Rez > 0.
Γ(z) =
0
▼ët sè ❝æ♥❣ t❤ù❝ ❝ì ❜↔♥ ❝õ❛ t➼❝❤ ♣❤➙♥ ●❛♠♠❛
Γ(z + 1) = zΓ(z), Γ(n + 1) = n!, n ∈ N,
π
Γ(z)Γ(z + 1) =
, 0 < Rez < 1,
sin(πz)
√
1
1.3.5...(2n + 1) √
1
Γ
= π, Γ n +
=
π.
2
2
2n
•
❚➼❝❤ ♣❤➙♥ ❇❡t❛
❈â ♠ët sè ✤à♥❤ ♥❣❤➽❛ t÷ì♥❣ ✤÷ì♥❣ ❝õ❛ ❤➔♠ ❇❡t❛ B(p, q)✭❤➔♠ ❇❡t❛✮
✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ t❤❡♦ ❝æ♥❣ t❤ù❝
1
up−1 (1 − u)q−1 du,
B(p, q) =
0
tr♦♥❣ ✤â p ✈➔ q ❞÷ì♥❣ ✤➸ t➼❝❤ ♣❤➙♥ tç♥ t↕✐✳ ❇➡♥❣ ♣❤➨♣ ✤ê✐ ❜✐➳♥
t❤æ♥❣ t❤÷í♥❣ ❝❤➾ r❛ B(p, q) = B(q, p).
✻
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
◆➳✉ t❛ ✤➦t u = sin2(θ)✱ t❤➻ t➼❝❤ ♣❤➙♥ trð t❤➔♥❤
π/2
sin2p−1 (θ) cos2q−1 (θ)dθ.
B(p, q) = 2
0
◆➳✉ t❛ ✤➦t u = x/(1 + x)✱ t❤➻ t➼❝❤ ♣❤➙♥ trð t❤➔♥❤
∞
B(p, q) =
0
xp−1
dx.
(1 + x)p+q
❚❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤
B(p, q) =
Γ(p)Γ(q)
,
Γ(p + q)
✈î✐ ♠å✐ ❝→❝❤ ❝❤å♥ p > 0 ✈➔ q > 0✳ ❱➼ ❞ö✱ ♥➳✉ p + q = 1✱ t❤➻ t❛ ❝â
❤➺ t❤ù❝
π
B(p, 1 − p) = Γ(p)Γ(1 − p) =
sin(πp)
.
●✐→ trà Γ(1/2) = √π ✤÷ñ❝ rót r❛ ❜➡♥❣ ❝→❝❤ ✤➦t p = 1/2.
✶✳✷ ❇✐➳♥ ✤ê✐ ▼❡❧❧✐♥
•
✣à♥❤ ♥❣❤➽❛ ❜✐➳♥ ✤ê✐ ▼❡❧❧✐♥
●✐↔ sû f (t) ❧✐➯♥ tö❝ tr➯♥ ❦❤♦↔♥❣ (0, ∞) ✈➔ ♥â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
❦❤↔ t➼❝❤ t✉②➺t ✤è✐
∞
tσ−1 |f (t)|dt < +∞,
0
✈î✐ ❣✐→ trà ♣❤ù❝ s = σ + iτ ✳
P❤➨♣ ❜✐➳♥ ✤ê✐ ▼❡❧❧✐♥ ❝õ❛ f (t) ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐
∞
F (s) = M{f (t)} =
A
t
s−1
A→∞
0
✼
ts−1 f (t)dt.
f (t)dt = lim
0
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
◆➳✉ ❝↔ ❤❛✐ ✤✐➲✉ ❦✐➺♥ ❦❤↔ t➼❝❤
1
σ1 −1
t
✈➔
|f (t)|dt < +∞
0
∞
tσ2 −1 |f (t)|dt < +∞
1
✤ó♥❣ t❤➻ F (s) ❧➔ ❣✐↔✐ t➼❝❤ tr♦♥❣ ❞↔✐ ✈æ ❤↕♥ Σ = {s : σ1 < σ < σ2}✳
• P❤➨♣ ❜✐➳♥ ✤ê✐ ▼❡❧❧✐♥ ♥❣÷ñ❝ ❝õ❛ F (s) ✤÷ñ❝ ❝❤♦ ❜ð✐
f (t) = M−1 {F (s)} =
1
2πi
c+i∞
t−s F (s)ds,
c−i∞
✭✶✳✶✮
tr♦♥❣ ✤â σ1 < c < σ2✳ ❚↕✐ ♠ët ❣✐→ trà ❝õ❛ t ♠➔ f (t) ❣✐→♥ ✤♦↕♥✱ t➼❝❤
♣❤➙♥ ❤ë✐ tö tî✐ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝õ❛ ❣✐î✐ ❤↕♥ tr→✐ ✈➔ ❣✐î✐ ❤↕♥ ♣❤↔✐✱
tù❝ ❧➔ (f (t+) + f (t−))/2.
• ❚➼❝❤ ❝❤➟♣ ▼❡❧❧✐♥ ❝õ❛ ❤➔♠ f (t) ✈➔ g(t) ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐
∞
(f ∗ g)(t) =
0
1
f
u
t
g(u)du.
u
◆➳✉ F (s) = M{f (t)} ✈➔ G = M{g(t)}✱ t❤➻ M{(f ∗g)} = F (s)G(s)✳
✶✳✸ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r
•
✣à♥❤ ♥❣❤➽❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❤✉➟♥ ✈➔ ♥❣÷ñ❝
◆➳✉ f (x) ✈➔ f (x) ❧➔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ tø♥❣ ✤♦↕♥ ✈➔ ♥➳✉ f (x) ❦❤↔
t➼❝❤ t✉②➺t ✤è✐ tr➯♥ ❦❤♦↔♥❣ (−∞, +∞)✱ t❤➻ f (x) ❝â ❜✐➸✉ ❞✐➵♥
1
f (x) =
π
+∞
+∞
f (t) cos[s(t − x)]dtds,
0
−∞
✈î✐ ♠å✐ ❣✐→ trà x ♠➔ f (x) ❧✐➯♥ tö❝✳ ◆❣♦➔✐ r❛✱ ♥➳✉ x ❧➔ ❣✐→ trà ♠➔ f (x)
❝â ✤✐➸♠ ❜÷î❝ ♥❤↔② ❣✐→♥ ✤♦↕♥ t↕✐ x✱ t❤➻ t➼❝❤ ♣❤➙♥ ❤ë✐ tö tî✐ ❣✐→ trà
tr✉♥❣ ❜➻♥❤ ❝õ❛ ❣✐î✐ ❤↕♥ tr→✐ ✈➔ ❣✐î✐ ❤↕♥ ♣❤↔✐ ❝õ❛ f (x) t↕✐ x✱ tù❝ ❧➔
(f (x−) + f (x+))/2.
❚➼❝❤ ♣❤➙♥ ❦➨♣ ♥➔② ❝â ❞↕♥❣ t❤❛② t❤➳ ♠➔ ❝â t❤➸ ✤÷ñ❝ ✈✐➳t ❞÷î✐ ❞↕♥❣
✽
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
♠ô ♣❤ù❝✳ ❱➻ cos θ = e
+iθ
+ e−iθ
2
t❛ ❝â
+∞
1
f (x) =
2π
+∞
e
−isx
−∞
e+ist f (t)dtds.
−∞
❚❛ ❣✐↔ sû r➡♥❣ s ✈➔ t tr♦♥❣ ❝→❝ ❜✐➸✉ ❞✐➵♥ t➼❝❤ ♣❤➙♥ tr➯♥ ❧➔ ❝→❝ ❜✐➳♥
t❤ü❝✳
❈â ♥❤✐➲✉ ❝→❝❤ ✤➸ t→❝❤ ❞↕♥❣ ♠ô ♣❤ù❝ ❝õ❛ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r t❤➔♥❤
♠ët ❝➦♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r✳ ❈→❝❤ tê♥❣ q✉→t ❧➔ ✈✐➳t
F (s) =
|b|
(2π)1−a
+∞
f (t)eibst dt
−∞
✈➔
1
|b|(2π)1+a
f (x) =
+∞
F (s)e−ibxs ds,
−∞
tr♦♥❣ ✤â a ✈➔ b ❧➔ t❤❛♠ sè t❤ü❝✳ ❚❛ ❝❤å♥ a = 0 ✈➔ b = 1✳ ❱î✐ ❝→❝❤
❝❤å♥ ♥➔②✱ ❝➦♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r trð t❤➔♥❤
1
F (s) = F{f (t)} = √
2π
✈➔
f (x) = F
−1
1
{F (s)} = √
2π
+∞
eist f (t)dt
✭✶✳✷✮
e−ixs F (s)ds.
✭✶✳✸✮
−∞
+∞
−∞
◆➳✉ ❝❤å♥ |b| = (2π)1−a t❤➻ t❛ ❝â ❝➦♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❤✉➟♥✲♥❣÷ñ❝
s❛✉ ✤➙②
+∞
eist f (t)dt,
F (s) = F{f (t)} =
−∞
f (x) = F
−1
1
{F (s)} =
2π
+∞
e−ixs F (s)ds.
−∞
✭✶✳✹✮
✭✶✳✺✮
❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❤✉➟♥ ✈➔ ♥❣÷ñ❝ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ t÷ì♥❣ ù♥❣ ❜ð✐ ❝→❝
❝æ♥❣ t❤ù❝ ✭✶✳✹✮ ✈➔ ✭✶✳✺✮ ❝ô♥❣ t❤÷í♥❣ ✤÷ñ❝ sû ❞ö♥❣✳
✾
s t ồ
ố ỗ ồ
t t ừ ờ rr
f (x) t tt ố t F (s)
+
1
|F (s)|
2
|f (x)|dx < +.
ữỡ tỹ F (s) t tt ố t f (x)
+
1
|f (x)|
2
|F (s)|ds < +.
P ờ rr F (s) tử f (x) tử
tứ ú r F (s) tỗ t t F (s) 0 s
ổ tữớ t t ờ rr ừ ừ
f (t) t ờ rr ừ f (t) ổ tự
F{f (n) (t)} = (is)n F (s)
ú ữợ tt f (n)(t) trỡ tứ ú t tt
ố tt f (t), f (t), . . . , f (n1)(t) tử ồ ỡ
trt t t t tt ố tr
(, +)
ừ ờ rr
f (t) g(t) t tr ồ ỳ [a, b]
|f (t u)g(u)| t ợ ộ t (, +) t t
rr ừ f (t) g(t) ữủ
1
(f g)(t) =
2
+
f (t u)g(u)du,
ởt tữỡ ữỡ q ờ
1
(f g)(t) =
2
+
f (u)g(t u)du.
t ữủ tọ ử f (t) g(t) t
s t ồ
ố ỗ ồ
ữỡ ử t tự r t ữủ
+
1/2
+
|f (tu)g(u)|du
2
|f (u)| du
1/2
+
2
|g(u)| du
.
ờ rr ỹ ý ỳ ữỡ tr t
ý q t t t t
ờ rr ừ t t F (s) = F{f (t)}
G(s) = F{g(t)}
tự Prs
t ỵ t rr t r
F{(f g)(t)} = F (s)G(s).
õ tữớ ữủ t ữợ t t ữ s
+
+
ist
e
f (u)g(t u)du.
F (s)G(s)ds =
ử t t = 0 g(u) = (f (u)) t G(s) = F (s)
+
+
2
|f (t)|2 dt.
|F (s)| ds =
tự ỳ f (t) ờ rr ừ õ F (s) ữủ
ồ tự Prs ứ õ s r f (t) t ữỡ
F (s) t ữỡ
ờ rrs rrs
f (t) t tr [0, +) t
ờ rr ừ rở ố ự ừ õ õ t t ữủ
rở ừ f (t) (, +) ữủ
t fE (t) = f (|t|) rở ừ f (t) (, +) ữủ
t fo(t) = signum(t)f (|t|) tr õ signum(t)
t ữỡ t
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
◆➳✉ f (t) ❝❤➤♥✱ t❤➻ ❜✐➸✉ ❞✐➵♥ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ❞↕♥❣ ✤ì♥ ❣✐↔♥
f (x) =
2
π
∞
∞
cos(xs) cos(st)f (t)dtds.
0
0
❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✭✶✳✷✮ trð t❤➔♥❤ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❤❡♦ ❤➔♠ ❝♦s✐♥
FC (s) = FC {f (t)} =
∞
2
π
cos(st)f (t)dt (s > 0)
0
✭✶✳✶✵✮
✈➔ F{f (t)} = FC {f (t)}.
❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷ñ❝ ✭✶✳✸✮ trð t❤➔♥❤ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷ñ❝ t❤❡♦
❤➔♠ ❝♦s✐♥
f (x) =
FC−1 {FC (s)}
∞
2
π
=
cos(xs)FC (s)ds (x > 0).
0
✭✶✳✶✶✮
◆➳✉ f (t) ❧➫✱ t❤➻ ❜✐➸✉ ❞✐➵♥ ❋♦✉r✐❡r t❤ø❛ ♥❤➟♥ ❞↕♥❣ ✤ì♥ ❣✐↔♥
2
f (x) =
π
∞
∞
sin(xs) sin(st)f (t)dtds.
0
0
❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✭✶✳✷✮ ❞➝♥ tî✐ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r t❤❡♦ ❤➔♠ s✐♥
FS (s) = FS {f (t)} =
∞
2
π
sin(st)f (t)dt (s > 0)
0
✭✶✳✶✷✮
✈➔ F{f (t)} = iFS {f (t)}.
❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷ñ❝ ✭✶✳✸✮ ❞➝♥ tî✐ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷ñ❝ t❤❡♦
❤➔♠ s✐♥
f (x) =
FS−1 {FS (s)}
=
2
π
∞
sin(xs)FC (s)ds (x > 0).
0
✭✶✳✶✸✮
✶✳✹ ❇✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡
•
✣à♥❤ ♥❣❤➽❛✳
P❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❦❤æ♥❣ ❝❤➾ ❧➔ ♠ët
❝æ♥❣ ❝ö ❝ü❝ ❦➻ ❤ú✉ ➼❝❤ ✤➸ ❣✐↔✐ ♥❤ú♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳
✶✷
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
t✉②➳♥ t➼♥❤ ♠➔ ❝á♥ ❝â ❣✐→ trà t÷ì♥❣ ✤è✐ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ ♥❤ú♥❣ ♣❤÷ì♥❣
t➼❝❤ ♣❤➙♥ ❱♦❧t❡rr❛ t✉②➳♥ t➼♥❤ ❝õ❛ ♠ët ❧♦↕✐ ♥❤➜t ✤à♥❤✳
❈❤♦ f (t) ①→❝ ✤à♥❤ tr➯♥ [0, ∞)✳ ❇✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ f (t) ✤÷ñ❝ ❝❤♦
❜ð✐ t➼❝❤ ♣❤➙♥ s✉② rë♥❣
∞
A
e−st f (t)dt.
e−st f (t)dt = lim
F (s) := L{f(t)} =
A→∞
0
0
❚➼❝❤ ♣❤➙♥ s➩ tç♥ t↕✐ ♥➳✉ f (t) ❧✐➯♥ tö❝ tø♥❣ ♠↔♥❤ tr➯♥ [0, A] ✈î✐ ♠å✐
❆ ✈➔ ❝â ❝➜♣ t➠♥❣ ❦❤æ♥❣ q✉→ ❞↕♥❣ ♠ô✳ ✭◆❤➢❝ ❧↕✐ ❤➔♠ f (t) ❧✐➯♥ tö❝
tø♥❣ ♠↔♥❤ tr➯♥ [0, A] ♥➳✉ ♥â ❧✐➯♥ tö❝ t↕✐ ♥❣♦↕✐ trø ♠ët sè ❤ú✉ ❤↕♥
❝→❝ ✤✐➸♠ ❣✐→♥ ✤♦↕♥ ❬✵✱ ❆❪✮✳ ❍➔♠ f (t) ❝â ❝➜♣ t➠♥❣ ❞↕♥❣ ♠ô ♥➳✉ tç♥
t↕✐ ❝→❝ ❤➡♥❣ sè ❛✱ ❝ ✈➔ ♠ s❛♦ ❝❤♦ |f (t)| ≤ c.eat ✱ ✈î✐ ♠å✐ t ≥ m✳
❈→❝ ✈➼ ❞ö✳ ✣➸ ♠✐♥❤ ❤å❛ ❝❤♦ ✤à♥❤ ♥❣❤➽❛✱ ①➨t ♠ët sè ✈➼ ❞ö s❛✉ ✤➙②✳
❱➼ ❞ö ✶✳✹✳✶✳ ❳➨t ❤➔♠ sè ✤ì♥ ✈à ❍❡❛✈✐s✐❞❡
0 ♥➳✉ t < 0,
σ0 (t) =
1 ♥➳✉ t ≥ 0.
❇✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ σ0 ❧➔
∞
F (p) =
e
0
✈î✐ Rep > 0 ✳
❱➼ ❞ö ✶✳✹✳✷✳
−pt
1
d (t) = − e−pt
p
t=∞
t=0
1
= ,
p
❇✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ ❤➔♠ f (t) = eαt ♥❤÷ s❛✉
∞
e−pt eαt dt =
F (p) =
0
1
e(α−p)t
α−p
∞
=
t=0
1
,
α−p
✈î✐ Re (p − α) > 0 ✳
❱➼ ❞ö ✶✳✹✳✸✳ ❇✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ ❤➔♠ f (t) = tn ❧➔
∞
1
e t dt = −
p
∞
tn d e−pt
pt n
F (p) =
0
✶✸
0
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
∞
1 n −pt ∞
= − t e
−n
tn−1 e−pt
t=0
p
0
∞
n
tn−1 e−pt dt
=
p 0
n!
= . . . = n+1 , Re p > 0.
p
❱➼ ❞ö ✶✳✹✳✹✳
α∈Q
❚❛ ❝â
✳
❚➻♠ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ ❤➔♠ f (t) = tα, α > 1,
∞
F (p) =
∞
e
0
=
L{tn } =
n!
sn+1
,
1
pα+1
uα du
pα p
0
∞
Γ (α + 1)
,
e−u uα du =
pα+1
0
−pt α
t dt =
L{eat } =
1
,
s−a
e−u
L{sin(at)} =
s2
a
.
+ a2
❇✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ ❝→❝ ✤↕♦ ❤➔♠ f (n)(t) ❝õ❛ f (t) ❝â t❤➸ ✤÷ñ❝ ❜✐➸✉
t❤à tr♦♥❣ ♥❤ú♥❣ sè ❤↕♥❣ ❝õ❛ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ f (t)✳ ❈æ♥❣ t❤ù❝
❝❤➼♥❤ ①→❝ ❧➔
n−1
L{f
(n)
n
f (m) (0)sn−1−m .
(t)} =s L{f(t)}−
m=0
❚❤ü❝ t➳ ♥➔② ❧➔ ❧þ ❞♦ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝â t❤➸ ✤÷ñ❝ ❞ò♥❣ ✤➸ ❣✐↔✐
♥❤ú♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ t✉②➳♥ t➼♥❤ ✈î✐ ❤➺ sè ❤➡♥❣✳
❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡
❚➼♥❤ ❝❤➜t ✶✳✹✳✶✳ ❈❤♦ ❝→❝ ❤➔♠ ❣è❝ fk ❝â ❝→❝ ❝❤➾ sè t➠♥❣ ❧➔ λk ✱ ❜✐➳♥
✤ê✐ ▲❛♣❧❛❝❡ ❧➔ Fk , k = 1, 2, ..., n. ❑❤✐ ✤â ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ ❤➔♠
tê ❤ñ♣ t✉②➳♥ t➼♥❤ f ❝õ❛ ❝→❝ ❤➔♠ fk
n
f (t) =
ck fk (t) ,
k=1
✶✹
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
ck ❧➔
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
❤➡♥❣ sè✱ ❧➔ ❤➔♠ F ✤à♥❤ ❜ð✐
n
✭✶✳✶✹✮
ck Fk (p).
F (p) =
k=1
❱î✐ ♠✐➲♥ ①→❝ ✤à♥❤ Re p > max αk .
❈❤ù♥❣ ♠✐♥❤✳ ❙✉② r❛ tø ✤à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t t✉②➳♥ t➼♥❤ ❝õ❛ t➼❝❤
♣❤➙♥✳
❱➼ ❞ö ✶✳✹✳✺✳ ❚r♦♥❣ ♠ö❝ tr÷î❝✱ t❛ ❝â
L eαt =
1
, Re (p − α) > 0.
p−α
✣➥♥❣ t❤ù❝ tr➯♥ ❧➔ ✈✐➳t t➢t✱ ✈✐➳t ❝❤➦t ❝❤➩ ❧➔ L [t → eαt] = p → p −1 α .
◆❤÷♥❣ ♥➳✉ ❦❤æ♥❣ ♥❤➛♠ ❧➝♥✱ s❛✉ ♥➔② t❛ s➩ ✈✐➳t ❞↕♥❣ t➢t ❝❤♦ t❤✉➟♥
t✐➺♥✳
❚ø t➼♥❤ ❝❤➜t ✶ ✈➔ ❦➳t q✉↔ ♥â✐ tr➯♥✱ t❛ s➩ t➻♠ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛
❝→❝ ❤➔♠ t❤æ♥❣ ❞ö♥❣ s❛✉ ✤➙②
✭❛✮ L [cos βt] = L 12 eiβt + e−iβt = 21 p −1 iβ + p +1 iβ
❱➟②
p
L [cos βt] =
p2 + β 2
, Re p > |Imβ| .
✭❜✮ ❚÷ì♥❣ tü✱ t❛ ❝â
L [sin βt] =
β
, Re p > |Imβ| .
p2 + β 2
✭❝✮
L [cosh βt] = L
1 βt
e + e−βt
2
=
L [sinh βt] = L
1 βt
e − e−βt
2
=
p
,
− β2
Rep > |Re β| .
β
,
p2 − β 2
Re p > |Re β| .
p2
✭❞✮
✶✺
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
❚➼♥❤ ❝❤➜t ✶✳✹✳✷✳ ❈❤♦ ❤➔♠ ❣è❝ f
✈➔ c > 0 ❧➔ ❤➡♥❣ sè✳ ❑❤✐ ✤â
❝â ❝❤➾ sè t➠♥❣ ❧➔ λ0✱ L [f ] = F (p)✱
1
p
L [t → f (ct)] = p → F
, Re p > cα0 .
c
c
✭✶✳✶✺✮
❈❤ù♥❣ ♠✐♥❤✳
∞
∞
1
e−pt f (ct) dt =
c
L [f (ct)] =
0
1
p
e−pu/c f (u) du = F
.
c
c
0
❈❤♦ L [f (t)] = F (p) , Re p > a0✳ ✣➦t
0
♥➳✉ t < τ ,
fτ (t) =
f (t − τ ) ♥➳✉ t ≥ τ .
❚➼♥❤ ❝❤➜t ✶✳✹✳✸✳
❑❤✐ ✤â
L (fτ ) = p → e−pτ F (p) ,
Re p > α0 .
✭✶✳✶✻✮
❈❤ù♥❣ ♠✐♥❤✳
∞
∞
e−pt fτ (t) dt =
L [fτ ] (p) =
e−pt f (t − τ )
τ
0
∞
f (u) e−p(u+τ ) du = e−pτ F (p) .
=
0
❚➼♥❤ ❝❤➜t ✶✳✹✳✹✳ ❈❤♦ L (f ) = F ✱ f
sè✳ ❑❤✐ ✤â
❝â ❝❤➾ sè t➠♥❣ ❧➔ α0, λ ❧➔ ❤➡♥❣
L eλt f (t) = F (p − λ) , Re p > α0 + Re λ.
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â
∞
L eλt f (t) =
e(λ−p)t f (t) = F (p − λ) .
0
✶✻
✭✶✳✶✼✮
s t ồ
ố ỗ ồ
ứ t t ử trữợ t s t
ờ ừ ởt tổ ử s
ử
L et cost =
p
,
(p )2 + 2
Re p > |Im| + Re .
L et sin t =
,
(p )2 + 2
Re p > |Im| + Re .
L et tn =
n!
,
(p )n+1
Re p > Re .
L (f ) = F sỷ f (k) tỗ t
ố f (k1) (0+) tỗ t k = 1, n t t õ
t
L f
(n)
=p
n
f (0+ ) f + (0+ )
f (n 1) (0+ )
F (p)
ããã
.
p
p2
pn
ự ỷ ử ổ tự t tứ t
tr ữủ ú ợ n = 1 sỷ q r
ú ợ n = 1, N õ
L f (N +1)
(N )
= L (f )
= p
N
f (n+1) (0+ )
f (0+ ) f (0+ )
L [f ] (p)
ããã
p
p2
pn
L [f ] = pF (p) f 0+ ,
s r
L f
(N +1)
=p
N +1
f (0+ ) f (0+ )
f (N ) (0+ )
F (p)
ã ã ã N +1
.
p
p2
p
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
❚❤❡♦ ♥❣✉②➯♥ ❧þ q✉✐ ♥↕♣✱ t❛ ❝â ✤♣❝♠✳
❱➼ ❞ö ✶✳✹✳✼✳ ❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ s❛✉ ✤➙②
y + 2y − 3y = e−t ,
y (0) = y (0) = 0.
✣➦t Y = L [y]✱ ❧➜② ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ✈➔
sû ❞ö♥❣ t➼♥❤ ❝❤➜t ✶✳✹✳✺✱ t❛ ❝â
p2 Y (p) + 2pY (p) − 3Y (p) =
s✉② r❛
1
,
p+1
1
1
1
+
+
4 (p + 1) 8 (p − 1) 8 (p + 3)
1
1
1
= L − e−t + et + e−3t .
4
8
8
Y (p) = −
❱➟②
1
1
1
y (t) = − e−t + et + e−3t .
4
8
8
❚➼♥❤ ❝❤➜t ✶✳✹✳✻✳ ❈❤♦ L (f ) = F ✱ f ❝â ❝❤➾ sè t➠♥❣
❧➔ α0✳ ❚❛ ❝â
L [(−t)n f (t)] = F (n) (p) , n ∈, Re p > α0 .
✭✶✳✶✾✮
❈❤ù♥❣ ♠✐♥❤✳ ❉➵ t❤➜② r➡♥❣ ❤➔♠ t → (−t)nf (t) ❝â ❝ò♥❣ ❝❤➾ sè t➠♥❣
✈î✐ f ✳ ❚❛ ❝â
∞
e−pt (−t) f (t) dt,
F (p) =
0
❞♦ ✤â
L [(−t) f (f )] = F (p) ,
Re p > α0 .
❇➡♥❣ ♣❤➨♣ q✉✐ ♥↕♣✱ t❛ s✉② r❛ ✤÷ñ❝ ✭✶✳✶✾✮✳
❱➼ ❞ö ✶✳✹✳✽✳
L [t sin βt] = −L [(−t) sin βt]
d
2pβ
β
= −
=
,
dp p2 + β 2
(p2 + β 2 )2
✶✽
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
✈➔
L t2 sin βt
= −L [(−t) sin βt]
d
2pβ
6p2 β − 2β 3
= −
=
.
dp (p2 + β 2 )2
(p2 + β 2 )3
❈❤♦ L (f ) = F ✈➔ f ❧✐➯♥ tö❝✳
❑❤✐ ✤â✱ →♥❤ ①↕ t → 0t f (τ )dτ ❝ô♥❣ ❧➔ ❤➔♠ ❣è❝ ✭♥➳✉ f ❧✐➯♥ tö❝ t❤➻
→♥❤ ①↕ ♥➔② ❧➔ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ f ✮ ✈➔
❚➼♥❤ ❝❤➜t ✶✳✹✳✼✳
t
L
f (τ )dτ
=
0
F (p)
.
p
✭✶✳✷✵✮
❈❤ù♥❣ ♠✐♥❤✳ ✣➦t g (t) = 0t f (τ )dτ t❤➻ g ❧✐➯♥ tö❝✱ s✉② r❛ ✤♦ ✤÷ñ❝✳
●å✐ λ0 ❧➔ ❝❤➾ sè t➠♥❣ ❝õ❛ f ✱ t❤➻ ✈î✐ ♠å✐ 0 < ε < 1✱ t❛ ❝â
t
|g (t)| ≤
t
0
=
e(α0 +ε)τ dτ
|f (τ )|dτ ≤ M
0
t
M (α0 +ε)τ
e
α0 + ε
< M1 e(α0 +ε)t .
τ =0
❱➟② g ❧➔ ❤➔♠ ❣è❝✳ ✣➦t G = L (g)✱ t❤➻
F = L (f ) = L (g) = pG (p) ,
s✉② r❛ ✤♣❝♠✳
❚➼♥❤ ❝❤➜t ✶✳✹✳✽✳ ●✐↔ sû R (f ) = F
✤â
❚r♦♥❣ ✤â ✱
f (t)
L
=
t
∞
z
lim p
p = Rez→∞
✱ ✈➔ t → f (t)
❧➔ ❤➔♠ ❣è❝✳ ❑❤✐
t
∞
F (u)du.
p
✭✶✳✷✶✮
.
❈❤ù♥❣ ♠✐♥❤✳ ✣➦t g (t) = f (t)t , G = L (g)✳ ❚❤❡♦ t➼♥❤ ❝❤➜t ✶✳✹✳✻ t❤➻
G (p) = L [(−t)] g (t) = −L (f ) = −F.
❱➟② G ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ −F ✳
✶✾
▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t♦→♥ ❤å❝
❚è♥❣ ❚❤à ❍ç♥❣ ◆❣å❝
◆❣♦➔✐ r❛✱ g ❧➔ ❤➔♠ ❣è❝✱ ✭❣✐↔ sû ❝❤➾ sè t➠♥❣ ❝õ❛ ♥â ❧➔ β ✮ ♥➯♥
∞
∞
e−(Rez)t |g (t)| dt ≤ M
|G (z)| ≤
0
0
(−Rez+β+1)t
= M
e(−Rez+β+1)t dt
e
−Rez + β + 1
∞
=
t=0
M
,
Rez − β − 1
tr♦♥❣ ✤â✱ Rez − β − 1 > 0✳ ❙✉② r❛
lim G (z) = 0,
Rez→∞
✈➔
∞
−G (p) = −G (p) + lim G (z) =
(−F (u)) du.
Rez→∞
0
❚ù❝ ❧➔ ✭✶✳✷✶✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
❱➼ ❞ö ✶✳✹✳✾✳
sin t
L
=
t
∞
0
du
π
=
− arctan p.
u2 + 1
2
⑩♣ ❞ö♥❣ t➼♥❤ ❝❤➜t ✶✳✹✳✼✱ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ ❤➔♠ ❙✐ ✤à♥❤ ❜ð✐
t
sin τ
dτ,
τ
Sit =
0
❧➔
1
sin t
1 π
L [Si] = L
=
− arctan p .
p
t
p 2
❚➼❝❤ ❝❤➟♣ ▲❛♣❧❛❝❡✳ ◆➳✉ f (t) ✈➔ g(t) ❦❤↔ t➼❝❤ tr➯♥ [0; ∞) t❤➻ t➼❝❤
❝❤➟♣ ❝õ❛ f (t) ✈➔ g(t) ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ t➼❝❤ ♣❤➙♥
t
(f ∗ g)(t) =
f (t − u)g(u)du.
0
✷✵
s t ồ
ố ỗ ồ
ỵ sỷ L (f ) = F L (g) = G f
g ữủt
ố õ số t 0 0 tử tứ ú tr ồ
ỳ ừ R+ t f g tr R trt
t tr (, 0) t t f g ụ ố õ
số t 0 max {0, 0}
L [f g] = F ã G.
ự ợ ồ t > 0, > 0
t
t
f ( ) g (t ) d
|(f g) (t)| =
|f ( ) g (t )| d
0
0
t
t
e(0 0 ) d
e0 e0 (t ) d = M e0 t
M
0
0
0 0,
0 < 0.
t tự s ũ õ ữủ t trỹ t t
f g ố õ số t 0 max {0, 0}
t t õ
M1 e0 t
M2 e0 t
t
ept
L [(f g) (t)] =
0
=
f ( ) g (t ) d dt
0
ept g (t ) dt
f ( ) d
0
f ( ) ept d
= G (p)
0
= F (p) ã G (p) .
s t ồ
ố ỗ ồ
ổ tự ờ ữủ
ố f trỡ tứ ú tr ồ ỳ ừ ỷ
trử t 0, số t 0. õ
x+i
1
f (t) =
2i
ept F (p) dp,
x > 0 .
xi
tr ữủ t tr ổ
tự õ t ổ tự
r ổ tự tt F ờ ừ
ởt ố f (t) trữợ t r F tọ
õ õ t ờ ừ ởt ố
õ õ ỵ ữợ ự ữủ ọ q
ỵ F tọ s
t tr Re p > 0.
|p| tr ộ Re p > 0, t t
t arg p
,
.
2 2
ợ ồ x > 0 tỗ t số ữỡ s
x+i
|F (x + iy)| dy M .
xi
õ tr Re p > 0 ờ ừ
x+i
1
f (t) =
2i
ept F (p) dp,
x > 0 .
xi
ỵ ữợ t t ố ừ ởt q
t ổ ỹ