Ba’zi sodda funksiyalarning tasvirlari

1
0 t
Xevisayda funksiyasining tasviri (1) ga ko’ra quyidagicha aniqlanadi:
F(t)=
0
e
-pt
f(t)dt =
0
e
-pt
0
(t)dt=
0
e
-pt
1
dt=-e
-pt
p
1
0
|
=-0+
p
1
=
p
1
.
Shunday qilib
0
(t)
1/p yoki 1
1/p (2)
2. f(t)=sint funksiya tasviri ham (1) ga ko’ra aniqlanadi:
p
e
e
e
e
e
e
e
e
e
e
e
e
e
2
2
0
pt
-
0
pt
-
2
0
pt
-
0
pt
-
pt
-
pt
-
0
pt
-
0
pt
-
0
pt
-
-pt
-pt
0
pt
-
0
pt
-
1
1
F(t)
bundan
F(t)
1
=
F(t)
ekaligidan
dt
sint
=
F(t)
sin
1
sin
sint
1
sin
cos
U
cos
-
cos
cos
-
=
cos
sin
U
=
dt
sint
=
f(t)dt
=
F(t)
p
p
tdt
tdt
p
p
t
V
tdt
dV
dt
p
dU
p
t
tdt
p
t
t
V
tdt
dV
dt
p
dU
Shunday qilib sint
2
1
1
p
+
(3)
3. f(t)=cost funksiya tasvirini topamiz:
p
e
e
e
e
-
e
e
e
e
e
e
e
e
e
2
2
0
pt
-
0
pt
-
2
0
pt
-
0
pt
-
pt
-
pt
-
0
pt
-
0
pt
-
0
pt
-
-pt
-pt
0
pt
-
0
pt
-
1
p
F(t)
bundan
F(t)
p
=
F(t)
ekaligidan
dt
cost
=
F(t)
cos
p
cos
p
cost
p
cos
sin
p
U
p
sin
sin
p
sin
=
sin
cos
p
U
=
dt
cost
=
f(t)dt
=
F(t)
p
p
tdt
tdt
t
V
tdt
dV
dt
dU
t
tdt
t
t
V
tdt
dV
dt
dU
Demak cost funksiyani tasviri
2
1
p
p
yoki cost
2
1
p
p
(4)
3. Laplas tasvirining ba’zi xossalari.
1.
Chiziqlilik xossasi.
Agar F
k
(t)
f
k
(t) (k=1,2,3,...,n) bo’lib, s
k
lar
o’zgarmaslar bo’lsa , u
holda
n
k
1
c
k
F
k
(t)
n
k
1
c
k
f
k
(t) munosabat o’rinli
bo’ladi.
2. O’xshashlik teoremasi. Agar a>0 va F(t)
f(t) bo’lsa , u holda
a
1
F(
a
p
)
f(at) bo’ladi.
3. Originalning kechikish teoremasi. Agar
>0 bo’lsa, u holda F(t)
f(t)
dan f(t-
)
e
-
t
F(t) kelib chiqadi.
4. Tasvirning sinish teoremasi. Agar F(t)
f(t) bo’lsa, u holda istalgan
uchun e
-
t
f(t)
F(t+
) kelib chiqadi.
5. Originalni differensiallash. Agar f(t) funksiya [0,
] da uzluksiz,
differensiallanuvchi va f
(t) hosila tasvir mavjudligining 1,2,3 shartlarini
qanoatlantirib, F(t)
f(t) bo’lsa, quyidagilar o’rinli bo’ladi.
a) pF(p)-f(0)
f
(t) Xususiy holda f(0)=0 bo’lsa pF(p)
f
(t) bo’ladi.
b)
Agar f
n
(t) mavjud bo’lsa va tasvir mavjudligining shartlarini qanoatlantirsa
f
(n)
(t)
t
n
F(t)-[ t
n-1
f(0)+ t
n-2
f
(0)+ t
n-3
f
(0) +...+ tf
(n-2)
(0)+f
(n-1)
(0) ],
agar f(0)=f
(0)=...=f
(n-1)
(0)=0 bo’lsa, f
(n)
(t)
t
n
F(t) bo’ladi.
6. Originalni integrallash. Agar F(p)
f (t) bo’lsa,
t
0
f(t)dt
ning tasviri
t
0
f(t)dt
p
p
F
)
(
bo’ladi.
7. Tasvirni integrallash. Agar F(p)
f (t) bo’lsa,
0
F(t)dt
t
t
f
)
(
bo’ladi.
8. Tasvirni differensiallash. Agar F(p)
f (t) bo’lsa , u holda
a)
F
(p)
-tf(t) ; b) F
(n)
(p)
(-1)
n
t
n
f(t) bo’ladi.
Misol.
f(t)=sinat funksiya tasvirini topamiz: F(p)=
0
sinat
e
-
pt
dt=1/a
1/[(p/a)
2
+1] =a/(p
2
+a
2
) Demak sinat
a/(p
2
+a
2
) (5)
Shunga o’hshash f(t)=cosat funksiyani tasviri quyidagicha bo’ladi: cosat
p/(p
2
+a
2
) (6)
Misol.
f(t)=3sin4t-2cos5t funksiya tasviri topilsin.
Yechish: (5) va (6) dan L
f(t)
=3
4/(p
2
+16)-2
p/(p
2
+25)=12/(p
2
+16)-
2p/(p
2
+25).
Misol.
F(t)=5/(t
2
+4)+20t/(t
2
+9) tasvir funksiya berilganda boshlangich funksiyani
toping.
Yechish: F(t)=5/2
2/(p
2
+4)+20
p/(p
2
+9)
5/2
sin2t+20cos3t=f(t)
Demak f(t)=5/2
sin2t+20
cos3t.
Misol.
f(t)=e
-at
funksiyani tasviri topilsin. F(t) =
0
e
-pt
e
-at
dt ==
0
e
-
(p+a)t
dt=1/(p+a) (7)
Misol.
f(t)=e
at
funksiyani tasviri topilsin. F(t) =
0
e
-pt
e
at
dt ==
0
e
-(p-a)t
dt=1/(p-a)
(8)
Misol.
f(t)=shat=1/2
(e
at
-e
-at
) funksiyani tasviri topilsin.
(7) va (8) dan 1/2
(e
at
-e
-at
)
1/2
[1/(1-p)-1/(p+a)]=a/(p
2
-a
2
)
Demak F(t)=a/(p
2
-a
2
) yoki F(t)
f(t) ya’ni shat
a/(p
2
-a
2
)
(9)
Shunga o’xshash chat=(e
at
+e
-at
)/2
funksiya tasviri chat
p/(p
2
-a
2
)
(10)
Misol.
F(t)= 7/(p
2
+10p+41) tasvir funksiyadan boshlangich funksiya topilsin.
Yechish: F(t)= 7/(p
2
+10p+41)= 7
4/[4
((p+5)
2
+4
2
)] Demak
7
4/[4
((p+5)
2
+4
2
)
7/4
e
-5t
sin4t. yoki F(t)
7/4
e
-5t
sin4t=f(t)
Misol.
F(t)=(p+3)/(p
2
+2p+10) tasvir funksiyadan boshlang’ich
funksiya topilsin.
F(t)=(p+3)/(p
2
+2p+10)=[(p+1)+2]/[(p+1)
2
+9]=(p+1)/[(p+1)
2
+3
2
]+2/[(p+1)
2
+3
2
]=
=(p+1)/[(p+1)
2
+3
2
]+2/3
3/[(p+1)
2
+3
2
]
Demak F(t)
e
-t
cos3t+2/3
e
-t
sin3t f(t)= e
-t
cos3t+2/3
e
-t
sin3t.
4. ORIGINAL-TASVIR JADVALI.
f(t)
F(t)=
0
e
-pt
f(t)dt
1. 1
2. sinat
3. cosat
4. cosa(t-t
0
)
5. e
-at
6. shat
7. chat
8. e
-
t
sinat
9. e
-
t
cosat
10. t
n
11. tsinat
12. tcosat
13. te
-
t
14. (sinat-atcosat)/2a
3
15. t
n
f(t)
16.
t
0
f(t)dt
17.
t
t
f
)
(
18. f(t-t
0
)
19. f
(t)
1. 1/p
2. a/(p
2
+a
2
)
3. p/(p
2
+a
2
)
4. pe
-pto
/(p
2
+a
2
)
5. 1/p+a
6. a/(p
2
-a
2
)
7. p/(p
2
-a
2
)
8. a/[(p+
)
2
+a
2
]
9. (p+
)/[(p+
)
2
+a
2
]
10. n!/p
n+1
11. 2pa/(p
2
+a
2
)
2
12. -(a
2
-p
2
)/(p
2
+a
2
)
2
13. 1/(p+
)
2
14. 1/(p
2
+a
2
)
2
15. (-1)
n
d
n
F(t)/dt
n
16. F(p)/p
17.
0
F(t)dt
18. e
-pto
F(t)
19. tF(t)-f(0)