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Reference Guide

# Laplace Transforms - Reference Guides

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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
Solve
ODE &
invert
Laplace Transform
Inverse Transform
Transform of a Constant
Inverse of a Constant
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