TRANSFORMADA DE LAPLACE
DEFINICI ´
ON: L[f(x)] = L[f] = Z+∞
0
e−zxf(x)dx
Transformadas de funciones elementales
L[K] = K
zL[x] = 1
z2
L[xn] = n!
zn+1 L[eax] = 1
z−a
L[sen wx] = w
z2+w2L[cos wx] = z
z2+w2
L[Sh wx] = w
z2−w2L[Ch wx] = z
z2−w2
Propiedades de la Transformada de Laplace
1. L[fn)(x)] = znL[f]−zn−1f(0) −zn−2f0(0) − · · · − zf n−2)(0) −fn−1) (0)
2. LZx
0
f(t)dt=L[f]
z
3. L[eaxf(x)] = F(z−a) siendo F(z) = L[f]
4. L[ua(x)f(x−a)] = e−azF(z) siendo F(z) = L[f], y ua(x) = (0 0 ≤x≤a
1x>a
5. L[xnf(x)] = (−1)nFn)(z) siendo F(z) = L[f]
6. L"f(x)
x#=Z+∞
z
F(t)dt siendo F(z) = L[f]
7. L[f∗g] = L[f]L[g] siendo f∗g(x) = Zx
0
f(t)g(x−t)dt
8. Si f(x+p) = f(x) entonces L[f] = 1
1−e−pz Zp
0
e−zxf(x)dx
