If you look carefully at the above proofs they demo straight that the compound interest rate function and the Laplace transform are virtually  the same function. The only difference is the Laplace transform has an integration wrapped around it. The difference is the kernel of the compound interest rate function and the Laplace transform use a different letter for the exponent, that is s instead of r.

Therefore   : If s = r and P = 1

Therefore   : e^st = e^rt

Therfore     : st = rt

Therefore   : s = r

They are therefore same formula and thus the same thing.

If two apparently different observable phenomena are described by the same function, then they are the same thing everywhere including the financial. This also indicates that compound interest is by definition a complex number and thus can be translated to the complex plane and spectrographically analyzed. 

Markets are by definition elastic and therefore will have the properties of an elastic medium. So any impulse (transaction) will create point source interference that will propagate a wave front through the medium.

Young’s slit experiment. 

If many impulses that is to say trades are applied to the medium (market) they will form interference patterns as each wave front decays over time.

Where the interference is generated by the impulse responses from transactions.

Specifically markets are interference patterns which obscure the underlying wave functions that are at work, in other words the function of trend.

Using the laplace transform it is possible to measure the magnitude and direction of the trend which can be converted into a transfer function that can be used as a process control input to manage a market position of an automated portfolio system.

Next in this series Pythagoras & All Ords