Formula Sheet - Laplace Tranform
1. Definition of Laplace transform of f(t): L{f(t)}=
∞
Z
0
e−stf(t) dt.
This definition will not be provided during the quizzes/final exam.
2. L{C}=C
sfor any constant C
3. L{tn}=n!
sn+1 for n= 1,2,3· · ·
4. Leat=1
s−afor any constant a
5. L{sin (kt)}=k
s2+k2for any constant k
6. L{cos (kt)}=s
s2+k2for any constant k
7. L{sinh (kt)}=k
s2−k2for any constant k
8. L{cosh (kt)}=s
s2−k2for any constant k
9. L{f0(t)}=sF (s)−f(0)
10. L{f00(t)}=s2F(s)−sf (0) −f0(0)
11. L{f000(t)}=s3F(s)−s2f(0) −sf 0(0) −f00(0)
12. Lf(n)(t)=snF(s)−sn−1f(0) −sn−2f0(0) − · · · − f(n−1) (0)
13. First Translation Theorem: Leatf(t)=F(s)|s→s−awhere F(s) = L{f(t)}
14. Unit Step Function: U(t−a) = 0 if 0 ≤t<a
1 if t≥a
15. f(t) = g(t) if 0 ≤t < a
h(t) if t≥a⇒f(t) = g(t)−g(t)U(t−a) + h(t)U(t−a)
This formula will not be provided during quiz/examination.
16. f(t) =
g(t) if 0 ≤t<a
h(t) if a≤t<b
j(t) if t≥b
⇒f(t) = g(t)−g(t)U(t−a) + h(t)U(t−a)−h(t)U(t−b) + j(t)U(t−b)
This formula will not be provided during quiz/examination.
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