permacharts.com

LAPLACE TRANSFORMS • A-838-9 1© 2003-2012 Mindsource Technologies Inc.

l e a r n • r e f e r e n c e • r e v i e w

TM

permacharts

PURPOSE

TRIGONOMETRIC FUNCTIONS

EXPONENTIAL FUNCTIONS

LIMIT THEOREMS

OPERAT I O N S

Laplace Transforms

Laplace Transforms

ft Lft fte dt

Iiefsds Fs

Taas

Iat a

Aft f t F s F s

st

b

st

aib

aib

s

() ()

[]

=

()

() ( )

()

()

±

() ()

±

∞−

→∞

+

−

+

∫

∫

0

121

1

2

Dirac Function]

/

π

δ

lim

[

(()

() ()

()

()

−

() ()()

−

()

−

()

−

()

() ()

−

()

() ()

−

()

−

∫

af t aF s

M f t

aaF as

Mfft d FsFs

Dabe ab s a

Dft sFs f

ft sFs sf f

t

at

1

0212

2

0

0

τττ

'

'''

00

1

1

0

()

() ()

() ( )

∫

∫∞

fd sFs

tft F d

t

s

ττ

δδ

lim lim lim

ts s

ft sFs f sFs

→→∞ →∞

()

=

() ()

=

()

0

12

0

lim lim

ts

ft sFs

→∞ →

()

=

()

0

lim

s

Fs

→∞

()

=0

sin

cos

sin sin cos

cos cos sin

cos

cos cos

sin sin

at a

sa

at s

sa

at b sba b

sa

at b sbab

sa

at a

ss a

at bt

bas

sasb

aatbbt

abs

sa

22

22

22

22

2

22

22

2222

222

22

1

+

+

+

()

+

+

+

()

−

+

−+

()

−−

()

+

()

+

()

−−

()

+

( ))

+

()

−−

()

+

()

+

()

−−

()

+

()

+

()

+

+

()

++

()

+−

()

sb

aatbbt

abs

sasb

aab bt

bs a

ss b

at sa

ss a

at bt abs

sabsab

22

22

223

2222

2

22

22

222

22

2222

2

4

2

cos cos

cos

cos

sin sin

esa

ts

s

ts

s

s

at a

sa

at s

sa

tat as

sa

e

at

a

±

−+

++

()

−

−

−

()

−

1

6

2

222

22

22

22

2

ln ln

ln ln

sinh

cosh

sinh

γ

πγ

ea

ss a

ee ab

sasb

ae be abs

sasb

ae

at

at bt

at bt

±

−

−+

()

−−

−

()

−

()

−−

()

−

()

−

()

1

1

−−−

−−

−

−−

()

+

()

+

()

−

()

−−

()

−

()

+

()

+

()

+

()

+

+

()

−

=

at bt

bt ct

at

a

be abs

sasb

abe ace cbsa

sbsc

ebt sa

sa b

e

cosh 22

05772156

γ

Euler's Constant = .

The purpose of Laplace transforms is to:

Convert ordinary differential equations (ODE’s) to

algebraic equations

Convert partial differential equations (PDE’s) to

ODE’s

The resulting equations in the Laplace domain (s) are

solved and inverted back into the time domain (t).

Example 1: Example 2:

INITIAL VALUE THEOREMS FINAL VALUE THEOREM

TRANSFORM LIMIT

dy

dt ay

sY y aY

yt y e

u

at

+=

−+=

=−

0

00

0

()

() ( )

(

Transform

Solve for Y(s) & invert

u

tku

x

sU u k dU

dx

uu x

t

=

−=

=

0

022

2

2

2

2

()

()

∂

∂

∂

∂

erfc

Solve

ODE &

invert

Laplace Transform

Inverse Transform

Transform of a Constant

Inverse of a Constant

Addition/Subtraction

Multiplication

Division

Differentiation

Integration

Transform

Operation f(t) F(s)

f(t) F(s)

f(t) F(s) f(t) F(s)

Note: Repeated

application of Laplace

transforms on PDE’s is uncommon because of possible

problems concerning boundary conditions

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