"
def
F(s) = [ f (x)] =
+"e- sx f (x)dx
0
Zestawienie ważniejszych własności przekształcenia Laplace a
[f(x)] = F(s)
Twierdzenie nr f(x) F(s) Wzór nr
III f (ax), a > 0 1 s 10
F( )
a a
IV F(s + a) 15
e-ax f (x)
V f (x - a) 0 d" a < x 20
e-as F(s)
VI f (x + a) a > 0 a
28
e-as[F(s) - f (x)e-sxdx]
+"
0
VII f (x + T ) = f (x) T 29
1
f (x)e- sxdx
+"
1- e-Ts 0
XII f '(x) 36
sF(s) - f (0+ )
n-1
XIII (n) 37
f (x)
s(n)F(s) -
"sn-1-½ f (½ ) (0+ )
½ =0
y
39
Y
y' 39
sY - y(0+ )
y'' 39
s2Y - sy(0+ ) - y'(0+ )
y''' 39
s3Y - s2 y(0+ ) - sy'(0+ ) - y''(0+ )
x
XIV 1 40
[ f (x)]
f (t)dt
+"
s
0
n
XV 41
d
xn f (x)
(-1)n [ f (x)]
dsn
xf (x) d 42
- [ f (x)]
ds
2
x2 f (x) d
[ f (x)]
ds2
"
XVI f (x) 43
[ f (x)]ds
x
+"
s
XXII f (x) " g(x) 50
[ f (x)]Å" [g(x)]
TABLICE TRANSFORMAT LAPLACE A
Uwaga: w drugiej kolumnie pominiÄ™to czynnik ·(x)
nr f(x) F(s)
1. 1
1
s
2. 1
eax
s - a
3. x
1
-
1
a
e 1+ as
a
4. x
1
s2
5. sin ax a
s2 + a2
6. b
e-ax sin bx
(s + a)2 + b2
7. sinh ax a
s2 - a2
8. b
e-ax sinh bx
(s + a)2 - b2
9. 1 1
(eax -1)
a s(s - a)
10. x
1
-
a
1- e s(1+ as)
11. 1
xeax
(s - a)2
12. x
1
-
1
a
xe
(1+ as)2
a2
13. 1
eax - ebx
(s - a)(s - b)
a - b
14. cos ax
s
s2 + a2
15. s + a
e-ax cosbx
(s + a)2 + b2
16. cosh ax s
s2 + a2
17. s + a
e-ax cosh bx
(s + a)2 + b2
18. s
(1+ ax)eax
(s - a)2
19. x
s
-
1
a
(a - x)e
(1+ as)2
a3
20. s
aeax - bebx
(s - a)(s - b)
a - b
21. 1 1
x2
2
s3
22. 1 1
(eax -1- ax)
a2 s2 (s - a)
23. x
1
-
a
ae + x - a
s2 (1+ as)
24. 1 1
[1+ (ax -1)eax ]
a2 s(s - a)2
25. x
1
-
a + x
a
1- e
s(1+ as)2
a
26. 1
1 beax - aebx
+
s(s - a)(s - b)
ab ab(a - b)
27. 1
eax -[1+ (a - b)x)ebx
(s - a)(s - b)2
(a - b)2
28. x
1
-
1
a
x2e
(s - a)3
2
x
29. 1
-
1
a
x2e
(1+ as)3
2a3
30. 1 1
(1- cos ax)
a2 s(s2 + a2 )
31. 1 1
(cosh ax -1)
a2 s(s2 - a2 )
32. 1 s
(x + ax2 )eax
2
(s - a)3
33. 1
s2
(1+ 2ax + a2x2 )eax
2
(s - a)3
34. x
s
x x2 - a
( - )e
(1+ as)3
a3 2a4
35. x
s2
1 2x x5 - a
( - + )e
(1+ as)3
a3 a4 2a5
36.
s2 + 2a2
cos2 ax
s(s2 + 4a2 )
37.
s2 - 2a2
cosh2 ax
s(s2 - 4a2 )
38.
2a2
sin2 ax
s(s2 + 4a2 )
39.
2a2
sinh2 ax
s(s2 - 4a2 )
40. 1 a a a a
a3
(cosh xsin x - sinh x cos x)
2 2 2 2 2
s4 + a4
41. a a
a2s
sin xsinh x
2 2
s4 + a4
42. 1 a a a a
as2
(cos xsinh x + sin x cosh x)
2 2 2 2 2
s4 + a4
43. a a
s3
cos x cosh x
2 2
s4 + a4
44. 1
a3
[sinh ax - sin ax]
2
s4 - a4
45. 1
a2s
[cosh ax - cos ax]
2
s4 + a4
46. 1
as2
[sinh ax + sin ax]
2
s4 - a4
47. 1
s3
[cosh ax + cos ax]
2
s4 - a4
Przepisał
Jan Jagła