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.