How It Works...

Evaluating "how it works" requires that you apply a simple sequence of formulae readily understood by any 8-th grade pre-algebra student. Those formulae are presented below; To evaluate, set up the initial matrix by evaluating the first table of formulae in order. Then, for each resulting data point, evaluate using the second table of formulae (again, in order).

Voila! "How it works" becomes immediately obvious!
For those without the elemental but requisite math background, a simple layman's explanation can be seen by clicking the link below::

Keep it simple...

 Initial Operations


 N

 F(s)

  f ( t ) , t > 0
 1.1  

 definition of a Laplace transform

y(t)

 1.2  Y(s)  inversion formula

 1.3    first derivative

 1.4    second derivative

 1.5  

 nth derivative


 1.6  

 integration


 1.7  F(s)G(s) convolution integral 

 1.8    
 1.9   shifting in the s-plane

 
 1.10  

 f(t) has period T, such that

f( t + T ) = f (t)

 1.11  

  g(t) has period T, such that

g(t + T ) = - g(t)

 Follow-on Operations


 N

 F(s)

 f ( t ) , t > 0

 2.1
 1

 

unit impulse at t = 0

 2.2
 s

 

double impulse at t = 0

 2.3
   

 2.4
  unit step 

u(t)

 2.5
   

 2.6
   t

 2.7a
  , n=1, 2, 3,.  

 2.7b
  , n=1, 2, 3,.  

 2.8
  , k is any real number > 0


the Gamma function is given in Appendix A

 2.9
   

 2.10
 
 


 2.11   , n=1, 2, 3,.  
 2.12    
 2.13    
 2.14    
 2.15  
 
 2.16a    
 2.16b    
 2.17    
 2.18

 

 
 2.19    
2.20     
 2.21    
 2.22    

 2.23    
 2.24    
 2.25    
 2.26    
 2.27    
 2.28    
 2.29    
 2.30    
 2.31    
 2.32    
 2.33    

 2.34