temas de la unidad 2 de matematicas para ingenieriaDescripción completa
Descripción completa
Full description
Proyecto LaplaceDescripción completa
Descripción: ecuaciones diferenciales laplace
Ecuaciones DiferencialesDescripción completa
Mathematics
Laplace Transform
LAplace Transforms
Ecuaciones DiferencialesDescripción completa
ejercicios resueltos de la transformada de laplace.Descripción completa
lapFull description
Descripción completa
mathematicDeskripsi lengkap
Table of Laplace of Laplace Transforms
f ( t ) = L
-1
{F ( s )}
1
1.
1
3.
t n , n = 1, 2, 3,
7. 9.
11.
s n!
K
5.
s n +1 p
t
2 s a
sin ( at )
3 2
s + a 2 2
( s
sin ( at ) - at cos ( at )
+ a2 )
2
cos ( at ) - at sin ( at )
15.
sin ( at + b )
17.
sinh ( at )
19.
e
at
21.
e
at
23.
t ne at , n = 1, 2, 3,
25.
uc ( t ) = u ( t - c )
2
2
( s - a ) + b 2
( s - a ) - b2
1
f ( t ) t
33.
ò f ( t - t ) g (t ) d t
35.
f ¢ ( t )
37.
f
( t )
2
b
f ( t )
( n)
2
2
ct
0
2
s 2 - a 2 b
n!
( s - a ) e
e
n +1
t , p > -1
6.
t
8.
cos ( at )
10.
t cos ( at )
- cs
F ( s - c )
ò
12.
, n = 1, 2, 3,
1× 3 × 5
L
K
F ( u ) du
( 2n - 1)
2n s s
s 2 + a 2 s 2 - a 2
( s
+ a2 )
2
16.
cos ( at + b )
18.
cosh ( at )
20.
e
at
22.
e
at
24.
f ( ct ) d
2
( s + a ) s ( s + 3a ) ( s + a ) 2
2
2
2
2
2
s cos ( b ) - a sin ( b ) s 2 + a 2 s s 2 - a 2 s - a
cos ( bt )
2
( s - a ) + b 2 s - a
cosh ( bt )
2
( s - a ) - b2 1 æsö F ç ÷ c ècø
(t - c)
e
- cs
Dirac Delta Function
28.
uc ( t ) g ( t )
30.
t n f ( t ) , n = 1, 2, 3,
32.
2
2as 2
sin ( at ) + at cos ( at )
cos ( at ) + at sin ( at )
26.
p
n + 12
14.
¥
s
n - 12
s p +1
2
- cs
s F ( s )
Heaviside Function
t
2
s + a a
K
e
31.
2
2
sinh ( bt )
29.
4.
s - a G ( p + 1)
2
s sin ( b ) + a cos ( b )
sin ( bt )
u c (t ) f ( t - c )
2
( s + a ) s ( s - a ) ( s + a )
1
e
2a 3
2
13.
2
F ( s ) = L { f ( t )}
at
2.
2as
t sin ( at )
27.
f ( t ) = L -1 {F ( s )}
F ( s ) = L { f ( t )}
e K
ò
t 0
f ( v ) dv
L
{ g ( t + c )} n
( -1) F ( n) ( s ) F ( s ) s
F ( s ) G ( s )
34.
f ( t + T ) = f ( t )
sF ( s ) - f ( 0 )
36.
f ¢¢ ( t )
s n F ( s ) - s n -1 f ( 0 ) - s n - 2 f ¢ ( 0 )
- cs
L
ò
T 0
e
- st
f ( t ) dt
1 - e - sT s 2 F ( s ) - sf ( 0 ) - f ¢ ( 0 )
- sf ( n- 2) ( 0 ) - f ( n-1) ( 0 )
Table Notes 1. This list is not a complete listing of Laplace transforms and only contains some o f the more commonly used Laplace transforms and formulas.
2. Recall the definition of hyperbolic functions. cosh ( t ) =
e
t
+ e- t
sinh ( t ) =
2
e
t
- e-t
2
3. Be careful when using “normal” trig function vs. hyperbolic hyperbolic functions. The only 2 difference in the formulas is the “+ a ” for the “normal” trig functions becomes a 2 “- a ” for the hyperbolic functions! 4. Formula #4 uses the Gamma function which is defined as
G (t ) = ò
¥
e
- x t -1
0
x
dx
If n If n is a positive integer then,
G ( n + 1) = n ! The Gamma function is an extension of the normal factorial factorial function. Here are a couple of quick facts for the Gamma function
