Laplace Transform Table - MIT OpenCourseWare

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Laplace Transform Table Properties and Rules Function

Transform

Already know. � ∞

f (t)e−st dt

f (t)

F (s) =

a f (t) + b g(t)

a F (s) + b G (s)

(Linearity)

F (s − z)

(s-shift)

zt

e f (t)

0−

(Definition)

Will learn in a future session. f � (t)

sF (s) − f (0− )

f �� (t)

s 2 F ( s ) − s f (0− ) − f � (0− )

f (n) ( t )

sn F (s) − sn−1 f (0− )−

t f (t)

− F � (s)

tn f (t)

(−1)n F (n) (s)

u(t − a) f (t − a)

e−as F (s)

u(t − a) f (t)

( f ∗ g)(t) = � t+ 0−

f (τ ) dτ

e � t+ 0−

− as

L( f (t + a))

f (t − τ ) g(τ ) dτ F (s) s

� ∞ s

(t-translation) (t-translation)

F (s) G (s)

Interesting, but not included in this course. f (t) t

· · · − f ( n −1) (0− )

F (σ) dσ

(The function table is on the next page.)

(integration rule)

Laplace Transform Table

OCW 18.03SC

Function Table Function

Transform

Region of convergence

1/s

Re(s) > 0

Already know. 1 e

at

1/(s − a)

Re(s) > a

2

Re(s) > 0

t

1/s

tn

n!/sn+1 2

Re(s) > 0 2

cos(ωt)

s/(s + ω )

sin(ωt)

ω/(s2 + ω 2 )

zt

Re(s) > 0 Re(s) > 0 2

2

e cos(ωt)

(s − z)/((s − z) + ω )

Re(s) > Re(z)

ezt sin(ωt)

ω/((s − z)2 + ω 2 )

Re(s) > Re(z)

δ(t)

1

all s

δ(t − a)

e−as

all s

s/(s2 − k2 )

Re(s) > k

k/(s2 − k2 )

Re(s) > k

cosh(kt) =

ekt

+ e−kt

2 kt e − e−kt sinh(kt) = 2 Will learn in this session. 1 (sin(ωt) − ωt cos(ωt)) 2ω 3 t sin(ωt) 2ω

( s2

1 (sin(ωt) + ωt cos(ωt)) 2ω

s2 ( s2 + ω 2 )2

Re(s) > 0

u(t − a)

e−as /s

Re(s) > 0

tn e at

n!/(s − a)n+1

Re(s) > a

1 + ω 2 )2 s ( s2 + ω 2 )2

Re(s) > 0 Re(s) > 0

Will learn in a future session.

Interesting, but not included in this course. 1 1 √ √ s π t t a

Γ ( a + 1) s a +1

2

Re(s) > 0 Re(s) > 0

MIT OpenCourseWare http://ocw.mit.edu

18.03SC Differential Equations��

Fall 2011 ��

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Laplace Transform Table - MIT OpenCourseWare

Laplace Transform Table Properties and Rules Function Transform Already know. � ∞ f (t)e−st dt f (t) F (s) = a f (t) + b g(t) a F (s) + b G (s)...

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