 Laplace transform function
 Laplace transform table
 Laplace transform properties
 Laplace transform examples
Laplace transform converts a time domain function to sdomain function by integration from zero to infinity
of the time domain function, multiplied by e^{st}.
The Laplace transform is used to quickly find solutions for differential equations and integrals.
Derivation in the time domain is transformed to multiplication by s in the sdomain.
Integration in the time domain is transformed to division by s in the sdomain.
Laplace transform function
The Laplace transform is defined with the L{} operator:
Inverse Laplace transform
The inverse Laplace transform can be calculated directly.
Usually the inverse transform is given from the transforms table.
Laplace transform table
Function name  Time domain function  Laplace transform 

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

Constant  1  
Linear  t  
Power  t^{ n} 

Power  t^{ a} 
Γ(a+1) ⋅ s ^{(a+1)} 
Exponent  e^{ at} 

Sine  sin at 

Cosine  cos at 

Hyperbolic sine 
sinh at 

Hyperbolic cosine 
cosh at 

Growing sine 
t sin at 

Growing cosine 
t cos at 

Decaying sine 
e^{ at }sin ωt 

Decaying cosine 
e^{ at }cos ωt 

Delta function 
δ(t) 
1 
Delayed delta 
δ(ta) 
e^{as} 
Laplace transform properties
Property name  Time domain function  Laplace transform  Comment 

f (t) 
F(s) 

Linearity  a f (t)+bg(t)  aF(s) + bG(s)  a,b are constant 
Scale change  f (at)  a>0  
Shift  e^{at} f (t)  F(s + a)  
Delay  f (ta)  e^{as}F(s)  
Derivation  sF(s)  f (0)  
Nth derivation  s^{n}f (s)  s^{n1}f (0)  s^{n2}f '(0)...f^{ (n1)}(0)  
Power  t^{ n} f (t)  
Integration  
Reciprocal  
Convolution  f (t) * g (t)  F(s) ⋅ G(s)  * is the convolution operator 
Periodic function  f (t) = f (t+T) 
Laplace transform examples
Example #1
Find the transform of f(t):
f (t) = 3t + 2t^{2}
Solution:
ℒ{t} = 1/s^{2}
ℒ{t^{2}} = 2/s^{3}
F(s) = ℒ{f (t)} = ℒ{3t + 2t^{2}} = 3ℒ{t} + 2ℒ{t^{2}} = 3/s^{2} + 4/s^{3}
Example #2
Find the inverse transform of F(s):
F(s) = 3 / (s^{2} + s  6)
Solution:
In order to find the inverse transform, we need to change the s domain function to a simpler form:
F(s) = 3 / (s^{2} + s  6) = 3 / [(s2)(s+3)] = a / (s2) + b / (s+3)
[a(s+3) + b(s2)] / [(s2)(s+3)] = 3 / [(s2)(s+3)]
a(s+3) + b(s2) = 3
To find a and b, we get 2 equations  one of the s coefficients and second of the rest:
(a+b)s + 3a2b = 3
a+b = 0 , 3a2b = 3
a = 3/5 , b = 3/5
F(s) = 3 / 5(s2)  3 / 5(s+3)
Now F(s) can be transformed easily by using the transforms table for exponent function:
f (t) = (3/5)e^{2t}  (3/5)e^{3t}
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