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Laplace formula sheet, Cheat Sheet of Mathematics

Formula sheet in define the laplace transform, first translation theorem, unit step functions and definition of convolutions.

Typology: Cheat Sheet

2021/2022

Uploaded on 02/07/2022

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gaurishaknar 🇺🇸

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Formula Sheet - Laplace Tranform
1. Definition of Laplace transform of f(t): L{f(t)}=
Z
0
estf(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
safor 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
s2k2for any constant k
8. L{cosh (kt)}=s
s2k2for 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)sn1f(0) sn2f0(0) · · · f(n1) (0)
13. First Translation Theorem: Leatf(t)=F(s)|ssawhere F(s) = L{f(t)}
14. Unit Step Function: U(ta) = 0 if 0 t<a
1 if ta
15. f(t) = g(t) if 0 t < a
h(t) if taf(t) = g(t)g(t)U(ta) + h(t)U(ta)
This formula will not be provided during quiz/examination.
16. f(t) =
g(t) if 0 t<a
h(t) if at<b
j(t) if tb
f(t) = g(t)g(t)U(ta) + h(t)U(ta)h(t)U(tb) + j(t)U(tb)
This formula will not be provided during quiz/examination.
1
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Formula Sheet - Laplace Tranform

  1. Definition of Laplace transform of f (t): (^) L {f (t)} =

∫^ ∞

0

e−stf (t) dt.

This definition will not be provided during the quizzes/final exam.

2. L {C} =

C

s

for any constant C

  1. (^) L {tn} =

n! sn+^

for n = 1, 2 , 3 · · ·

4. L

eat

s − a

for any constant a

  1. (^) L {sin (kt)} =

k s^2 + k^2

for any constant k

  1. (^) L {cos (kt)} =

s

s^2 + k^2

for any constant k

  1. (^) L {sinh (kt)} =

k s^2 − k^2

for any constant k

  1. (^) L {cosh (kt)} =

s

s^2 − k^2

for any constant k

  1. (^) L {f ′(t)} = sF (s) − f (0)
  2. (^) L {f ′′(t)} = s^2 F (s) − sf (0) − f ′(0)
  3. (^) L {f ′′′(t)} = s^3 F (s) − s^2 f (0) − sf ′(0) − f ′′(0)

12. L

f (n)(t)

= snF (s) − sn−^1 f (0) − sn−^2 f ′(0) − · · · − f (n−1)(0)

  1. First Translation Theorem: (^) L

eatf (t)

= F (s)|s→s−a where F (s) = (^) L {f (t)}

  1. Unit Step Function: (^) U (t − a) =

0 if 0 ≤ t < a 1 if t ≥ a

  1. 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.

  1. 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.

  1. Second Translation Theorem (version 1): (^) L {f (t − a) (^) U (t − a)} = e−as^ L {f (t)}

This formula is easier to apply for finding inverse-Laplace transform.

  1. Second Translation Theorem (version 2): (^) L {f (t) (^) U (t − a)} = e−as^ L {f (t + a)}

This formula is easier to apply for finding Laplace transform.

  1. (^) L {U (t − a)} =

e−as s

  1. (^) L {tnf (t)} = (−1)n^

dn dsn^

F (s) where F (s) = (^) L {f (t)}

  1. Definition of convolution: f (t) ∗ g(t) =

∫^ t

0

f (τ )g(t − τ ) dτ

  1. f (t) ∗ g(t) = g(t) ∗ f (t)
  2. (^) L {f (t) ∗ g(t)} = (^) L {f (t)} · L {g(t)}

24. L

∫^ t

0

f (τ ) dτ

F (s)

s

  1. Let f (t + T ) = f (t) for all t ≥ 0 be periodic with period T > 0. Then

L {f^ (t)}^ =^

1 − e−sT

∫T

0

e−stf (t) dt