Laplace Transformations for LETS and USURY bank accounts

     JCT: In "Advanced Engineering Mathematics" by Erwin Kreyszig, 
both the "LETSystem account equation" and the "Usury system account 
equation" are on the same page. (Page 148 Chapter 4, Laplace 
Transformations, Example #1 and Example #2). 
     I will denote an integral from 0 to infinity(oo) of the function 
f(t) by 0/oo {f(t)}. e is the base 2.71 for natural exponentiating.
     In Basic language, * is multiplication and ^ is exponentiation. 
     From page 148: 

"The Laplace Transformation of a function will be denoted by L [f(t)]. 
Hence, F(s) = L [f(t)] = 0/oo {f(t)*(e^(-st))*dt}. 

The described operation on f(t) is called the Laplace Transformation. 
We shall denote the original function by a lower case letter and its 
transform by the same letter in Capital. 

Example 1. 

Let f(t) = 1 when t > 0. Then:

L [f(t)] = L [1] = 0/oo {1*(e^(-st))*dt} = -e^(-st)/s from 0 to oo;

Hence, when s > 0, L [1] = 1/s 

Example 2. 

Let f(t) = e^(it) when t > 0 where i is a constant. Then 

L [f(t)] = L [e^(it)] = 0/oo {(e^(it))*(e^(-st))*dt} = e^(-(s-i)t)/(i-s) from 0 to oo; 

Hence, when s-i > 0, L [e^(it)] = 1/(s-i). 

     JCT: The first equation represents a stable LETS account (1/s). 
The second equation represents a positive feedback which generates 
exponential growth, the equation of a usury currency account 1/(s-i).
     It is interesting to note that the first two examples of the very 
simplest Laplace Transformations in his book are the linear LETS 
accounts and exponential usury accounts. 
     It is also interesting to note that the only difference between 
the two is that in a LETS banking system, i=0 so that there is no 
positive feedback on debt. This obeys all the religious prohibitions 
against exacting interest. 
     

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